14 Confidence Intervals

14.1 Interval Estimation and Quantifying Uncertainty

The primary concern with point estimation is that, regardless of the procedure we use, we never expect that \(\widehat{\theta}\) will equal the true parameter, \(\theta\). Instead, by using statistically valid estimation techniques, we hope to have estimators that are close to the truth. Moreover, using the sampling distribution, or assessing the MSE of an estimator can quantify how close is close. Still, even the most performant estimators are unlikely to exactly equal the truth. What’s more, reporting the estimated value will not, on its own, convey any information about the reliability of the estimator. To solve these issues we can turn to interval estimation. Specifically, with interval estimation we form estimators that do not give a single value for a parameter of interest, but rather, a range of plausible values.

Definition 14.1 (Interval Estimator) An interval estimator is a mathematical procedure (function) that takes in observable data (from a sample) and outputs an estimated range for a parameter value. Interval estimators are specific to the parameter that they are attempting to estimate.

There are several types of interval estimators, including confidence intervals, credible intervals, likelihood intervals, and prediction intervals. Confidence intervals are rooted in frequentist probability theory, the framework we have been exploring throughout, and will serve as the primary focus of this chapter.¹ With any interval estimator the goal is to not only provide an estimate of the likely value of a parameter, but also to quantify the uncertainty around this estimate. Providing the interval communicates that, while it is suspected that the parameter is near the point estimate, it is not possible to be entirely sure of exactly where. When reported intervals are more narrow, there is a higher level of precision attributed to the estimate. When reported intervals are wider, there is less certainty ascribed to the estimate. As a result, interval estimates convey a significant amount of information about the parameter, and the estimation procedure that was used.

14.2 Confidence Intervals in General

Ideally, we would like to be able to form an interval that has a direct probability statement associated with it. For instance, we may wish to form an interval, \((a, b)\), such that \(P(\theta \in (a, b))\) is known explicitly. Unfortunately, in the frequentist paradigm, \(\theta\) is not a random quantity. The statement \(P(\theta \in (a, b))\) is akin to writing \(P(2 \in (4, 6))\): this probability is either \(0\) or it is \(1\).² We need to hold in mind that the uncertainty in estimation does not stem from the parameter value, \(\theta\), and instead stems from the estimator, \(\widehat{\theta}\) (and its sampling distribution).

The statement \(P(\widehat{\theta} \in (a, b))\) can be interpreted as a sensible probability, where here the probability is taken with respect to the sampling distribution. To form our confidence intervals, we start with this idea. Specifically, we attempt to find values \(a\) and \(b\) such that \(P(a \leq \widehat{\theta} \leq b) = p\) for some suitably high threshold, \(p\). This interval, \((a, b)\), can be thought of as an interval that will contain the estimated value with high probability. If \(p=0.95\) then, based on the sampling distribution, there is a \(95\%\) chance that the computed estimator will fall between \(a\) and \(b\).

Note that this interval, \((a, b)\) is not yet an interval for the true parameter; it is an interval for our estimator. However, it will almost always be the case that the endpoints of the interval, \(a\) and \(b\), depend on the parameters of the sampling distribution. Recall that the sampling distribution’s parameters are connected to the population distribution’s parameters, and so we can view \(a\) and \(b\) as depending directly on \(\theta\). To emphasize this point, we write \[P\left(a(\theta) \leq \widehat{\theta} \leq b(\theta)\right) = p.\] It is important to remember that, in this expression, the only random quantity is \(\widehat{\theta}\). However, we have an expression that we know will be true with high probability, that contains both \(\theta\) and \(\widehat{\theta}\). We can exploit this to form confidence intervals.

The lower bound of our interval occurs where \(a(\theta) = \widehat{\theta}\). Note that \(a\) is some function of \(\theta\), and so we can think of the inverse function of \(a\), and apply it to both sides.³ That is, \[a^{-1}(a(\theta)) = a^{-1}(\widehat{\theta}) \implies \theta = a^{-1}(\widehat{\theta}).\] Similarly, applying the same rationale with \(b^{-1}\) to the upper bound on the interval provides a secondary bound, \(\theta = b^{-1}(\widehat{\theta})\).

Example 14.1 (Car Washes for Charity) Charles and Sadie have decided that, to raise money for the local animal shelter, they will run a car wash. They recognize that the main cost in running the car wash is the water used for it, and that the key variable in terms of throughput of their cash wash is the time it takes for a car to be cleaned. They are trying to determine how to price and time their washes. Charles suggests that they should aim for using between \(100\) and \(300\) liters to wash a car, and Sadie notes that the amount of water being used is likely \(5T + 10\) liters, where \(T\) is the time in minutes.

Given this information, what is the equivalent interval for the time per car through their car wash?

Solution

Note that we are told \[100 \leq 5T + 10 \leq 300.\] We can invert this interval to find an interval for \(T\) instead. Thus, we get \(100 \leq 5T + 10\) which means \(T \geq 18\). Similarly, \(5T + 10 \leq 300\) means that \(T \leq 58\). Thus, if the interval for water use is \((100, 300)\), then the interval for time will be \((18, 58)\) minutes.

Taken together this means that whenever \(\widehat{\theta}\) falls in the interval with endpoints \(a(\theta)\) and \(b(\theta)\), it is equivalent to say that \(\theta\) falls in the interval with endpoints \(a^{-1}(\widehat{\theta})\) and \(b^{-1}(\widehat{\theta})\).⁴ Thus, if we know that \(\widehat{\theta}\) falls between \(a(\theta)\) and \(b(\theta)\) with probability \(p\), this is mathematically equivalent to saying that \(\theta\) falls between \(a^{-1}(\widehat{\theta})\) and \(b^{-1}(\widehat{\theta})\) with probability \(p\) as well. This inverted interval gives the confidence interval for \(\theta\).

Definition 14.2 (Confidence Interval) A confidence interval for the parameter \(\theta\) is an interval given by \((L(\widehat{\theta}), U(\widehat{\theta})\) such that \[P(L(\widehat{\theta}) \leq \theta \leq U(\widehat{\theta})) = p,\] for some set level of \(p\). Often the confidence interval will be expressed in terms of \(p=1-\alpha\), where \(\alpha\) is referred to as the level of significance. It is important to note that in this expression, \(L(\widehat{\theta})\) and \(U(\widehat{\theta})\) are random quantities, where \(\theta\) is fixed. Correspondingly, the confidence probability can be written as \[P(L(\widehat{\theta}) \leq \theta \leq U(\widehat{\theta})) = P(L(\widehat{\theta}) \leq \theta, U(\widehat{\theta}) \geq \theta) = p.\]

Conceptually, a confidence interval provides an interval that, prior to conducting our estimation, we know there is a set threshold for the probability that the true parameter falls within its bounds. In our framing of the confidence intervals the random variables are \(U(\widehat{\theta})\) and \(L(\widehat{\theta})\) corresponding to \(a^{-1}(\widehat{\theta})\) and \(b^{-1}(\widehat{\theta})\). These can be viewed more generally as \(U(X_1, X_2, \dots, X_n)\) and \(L(X_1, X_2, \dots, X_n)\), being functions of the random sample. In this sense, both \(U\) and \(L\) are point estimators themselves. However, instead of estimating the parameter value directly, they are estimating upper and lower bounds for the parameter value. This is important to keep in mind since, fundamentally, it is the estimator that is random,⁵ not the parameter.

It is theoretically possible that there are many estimators, \(U\) and \(L\), that would satisfy the given probability statements. We generally need some manner of choosing which interval to use. The most important feature of any confidence interval is the length, which is to say \(U(\widehat{\theta}) - L(\widehat{\theta})\). All else equal, we prefer shorter intervals, since these give a more precise estimate of the true parameter value. We typically constrain our confidence interval definition to be such that \(U(\widehat{\theta}) - L(\widehat{\theta})\) is as small as possible, and select estimators that achieve this optimality.

Example 14.2 (Charles Tufts a Rug) Charles has become very intrigued by tufting, making rugs using yarn, but is still learning, and has not fully calibrated the amount of yarn required for each project. For the size of rugs that Charles has been making, the internet states that the amount of yarn used per rug (measured in kg) can be thought of as coming from a \(N(5, 1)\) distribution.

Charles wants to use this information to be confident in the average amount of yarn used, per rug, for the next \(9\) rugs made. With the help of Sadie, they determine three possible intervals:

\((4, 5)\).
\((5, 6)\).
\((4.77596, 5.22404)\).

Work out the approximate level of confidence that they should have in each interval. Note that \(\Phi(0.67212) \approx 0.7492463\).
Which of these intervals is the best to take as the basis of a confidence interval for Charles to use? Explain.

Solution

Note that the next \(9\) rugs will have a sampling distribution given by \(N(5, \frac{1}{9})\). We can compute the probability of each interval on this basis. Specifically, \[P(4 \leq \overline{X} \leq 5) = P(\frac{4-5}{1/3} \leq Z \leq 0) = \Phi(0) - \Phi(-3).\] Note that, using the empirical rule we know that this is going to be approximately \(\frac{0.997}{2} = 0.4985\). The calculation for the second interval proceeds exactly the same way, giving \(\Phi(3) - \Phi(0) \approx 0.4985\) by the empirical rule. For the third interval we take \[\begin{split}P(4.77596 \leq \overline{X} \leq 5.22404) = P(\frac{4.77596-5}{1/3} \leq Z \leq \frac{5.22404-5}{1/3}) \\ = \Phi(0.67212) - \Phi(-0.67212).\end{split}\] From the question hint we have that \(\Phi(0.67212) \approx 0.7492463\), and \(\Phi(-0.67212) \approx 1 - 0.7492463.\) Thus, taken together the coverage of the third interval is \[0.7492463 - (1 - 0.7492463) = 0.4984926 \approx 0.4985.\] All three intervals have the same coverage.
The third interval will be preferred. Each of the intervals gives the same coverage probabilities and the same level of confidence can be placed in all three of them. Thus, the main point of differentiation stems from the width of the intervals. The first two have a width of \(1\), whereas the third is \(0.448\) wide. This is less than half the width, with the same overall level of confidence. This allows Charles to have a more precise estimate of the amount of yarn used, without sacrificing the level of confidence in that estimate.

14.3 Interpretation of Confidence Intervals

The construction of the confidence interval follows a fairly natural idea. We want an interval with a high probability of containing the truth, and so we start with an interval for our estimator and invert this to have an interval for the parameter. This procedure, while intuitive in construction, obscures the nuance in the probability statements that we can make. We must be very careful in the interpretation of a confidence interval.

Recall that the random component in the interval is the endpoints, \(U\) and \(L\). These endpoints are estimators, and so the probability distribution associated with them is the sampling distribution of the estimators. Sampling distributions capture the behaviour of an estimator when we repeatedly draw samples and compute estimates on those samples. Thus, \(P(L \leq \theta \leq U) = p\) means that, if we were to repeatedly draw samples from the population, use those samples to compute \(L(\widehat{\theta})\) and \(U(\widehat{\theta})\), and then check whether \(L(\widehat{\theta}) \leq \theta \leq U(\widehat{\theta})\) holds, in the long-run \(p\) proportion of those runs would satisfy this condition. This is how we must interpret the confidence intervals.

Formally, a \(100p\%\) confidence interval interpreted by noting that, were this procedure to be repeated across many, many samples, the proportion of these samples that contain the true parameter will tend towards \(p\) as the number of samples tends towards \(\infty\). Note that this is not a statement regarding any specific confidence interval, but rather, it is a statement regarding the procedure of constructing the confidence intervals.

As an alternative framing, we can consider applying the procedure in practice. Suppose that we have yet to take our sample, but we know that we will form the confidence interval as \(\left(L(\widehat{\theta}), U(\widehat{\theta})\right)\). Prior to observing any data, we know that this procedure will have a \(p\) probability of containing the true parameter, \(\theta\). However, once we collect data and compute the values, we get \(L(\widehat{\theta}) = \ell\) and \(U(\widehat{\theta}) = u\) for some fixed values \(\ell\) and \(u\). Now, the statement \(P(\ell \leq \theta \leq u)\) is either equal to \(1\), if \(\ell \leq \theta \leq u\) holds, or it is equal to \(0\) otherwise. We do not know whether it is \(0\) or it is \(1\), but we do know that the probability is not \(p\).

Example 14.3 (The Probability of Rolling a Six) Charles is working hard to truly understand the valid, and invalid, ways of interpreting confidence intervals. Sadie, who seems to have understood the concepts more thoroughly, is trying to help. Charles is particularly hung up on the idea that the process itself can be interpreted probabilistically, even when the specific interval that is formed cannot. Sadie, recognizing that this is the hang-up, works through the following thought experiment with Charles. Sadie asks the following questions.

Suppose that I have a fair die in my hand. Before I have rolled it, what is the probability that the die shows a \(6\)?
Once the die is rolled, and the die is showing a \(5\), what is the probability that the die shows a \(6\)?
If instead, after the die were rolled, it showed a \(6\), what is the probability that the die shows a \(6\)?
What can we say about the probability that the die shows a \(6\), in general, after it is rolled?
How does this relate to the confidence interval interpretation?

Solution

Before the die is rolled, because it is a fair die, the probability it shows a \(6\) is \(\dfrac{1}{6}\). This is precisely what we mean by the fact that the die is fair.
If the die is rolled, and it shows a \(5\), the probability that it equals \(6\) is \(0\). That is, once it does not equal \(6\), we know for certain it is not \(6\).
If the die is rolled, and it shows a \(6\), the probability that it equals \(6\) is \(1\). That is, once it does equal \(6\), we know for certain that it is \(6\).
After the die is rolled, the probability will be either \(0\) or it will be \(1\), depending on whether it was a \(6\) or not. We do not know until we have observed it whether it is \(0\) or it is \(1\), but necessarily one of those probabilities is the truth.
In a confidence interval, the probability is not attributed to the specific interval, but rather to the process. Relating to the die example, the specific realization that is shown after the die is rolled is analogous to the specific interval. We know that depending on whether we see a \(6\) or it is not the probability is \(1\) or \(0\). Similarly, once computed, a specific interval has either \(0\) or \(1\) probability of containing the truth. Still, the process through which the interval was formed produces a predictable probability of forming intervals with the true value, just as the process of rolling the die produces predictable probabilities of showing \(6\).

The confidence threshold in a confidence interval relates to the procedure, not to any specific interval produced by that procedure. That is, we can put our confidence in constructing intervals in this way knowing that, if we always follow this procedure, over time our intervals will be correct in a specific proportion of cases. However, we cannot use this to make claims about any specific confidence interval. We do not know in any given sample whether the interval is one that contains the truth, or one that does not. In fact, we cannot know this, without learning what the true parameter value is.

This makes confidence intervals clunky to interpret. Notably, the interpretation of a confidence interval never makes reference to the specific interval that we have computed. Instead, the interpretation of the confidence interval gives a description of the procedure that created the interval, and ascribes a level of confidence to this underlying procedure.⁶

Further misconceptions of confidence intervals commonly occur. For instance, individuals will commonly interpret a confidence interval to mean that \(p\) proportion of the sample data lie within the interval. This is incorrect for two reasons. First, the confidence interval is a statement regarding an estimator or parameter value, rather than the sample value itself. From the same sample you can calculate different confidence intervals for different parameters.⁷ Second, this interpretation further relies on the thought that a confidence interval is interpreted specific to a particular sample. This is not the case.

Alternatively, individuals may misinterpret a confidence interval as stating that, were we to repeatedly sample from the population and compute the estimator, in \(p\) proportion of these trials the estimated value will be contained in the interval we have computed. This misinterpretation is appealing as it seems to use the repeated draws from the population. However, note two concerns. First, this interpretation also relies on giving interpretation to the specific interval that we have computed, which the correct interpretation cannot. Second, this interpretation makes a probability statement regarding the estimator, rather than the parameter (or the endpoints of the interval). Remember that \(U\) and \(L\) are the random components, and they are calibrated such that \(P(L \leq \theta \leq U) = p\). This is not a probability statement relating to where the estimate is likely to show up, rather one regarding the true value.

Example 14.4 (Bird Watching Confidence) Sadie and Charles have continued their bird watching adventures. As their statistical knowledge has expanded, they have continued to integrate this new-found knowledge into their trips. Lately, they have begun to consider confidence intervals, and so with a little help from their friend Garth, they work out the confidence interval for the average number of rare birds at a particular location that they enjoy traveling to. They find that the \(90\%\) confidence interval is given by \((1.4, 8.5)\). Seeing as confidence intervals are still new to them, they start trying to work out the interpretation of this statement together. Help them out by answering their questions, with explanations.

Charles wonders, “Does this mean that, if we go, there is a \(90\%\) chance we will see between \(1.4\) and \(8.5\) rare birds?”
Sadie responds, questioning “Perhaps it means that, if we go many, many, many times, in \(90\%\) of cases we will see in this range?”
Charles questions, “Could it mean that, from the sample data we have collected, \(90\%\) of observations fall between \(1.4\) and \(8.5\)?”
Sadie follows up, “Or that, if we were to perform the sampling many times, and calculate the sample average for each of those, then \(90\%\) of sample averages should fall between \(1.4\) and \(8.5\)?”
What does it actually mean?

Solution

This does not mean that there is a \(90\%\) chance we will see between \(1.4\) and \(8.5\) birds. The concern here is that we cannot apply the interpretation to any specific interval. Instead, we need to be thinking about the procedure generating the interval. There is a \(90\%\) chance that the procedure generates an interval that is valid, but we cannot know about any specific interval we are looking at.
This statement is just a rephrasing of Charles’ first statement. To have a \(90\%\) chance of seeing between \(1.4\) and \(8.5\) rare birds is to say that, if you observe birds many, many times, then in \(90\%\) of those cases you observe in that range. It is an incorrect interpretation for the same reason as above.
No. Confidence intervals are never making comments about what has been observed in the sample, they are instead designed to be inferential about the population at large. We are not capturing information about past samples. The other important note is that, this is a confidence interval for the mean, not for the observations themselves.
Again, this is a tempting misinterpretation. Here, Sadie has the correct idea of repeatedly performing experiments, however, the idea is not that in repeated experiments the current interval will contain the sample mean. Rather, in repeated experiments, the computed interval will contain the true mean. That is, once again, we need to focus on the population mean not the sample values, and we need to focus on the procedure, not the specific interval.
This confidence interval is not interpreted on its own. However, we can say that, were we to continue to make observations and use these observations to compute confidence intervals, following the exact same procedure, in \(90\%\) of those cases we would find an interval that contains the truth. We can also say, practically, that there is a fair amount of variability in the number of birds that are likely to be seen, and so Charles and Sadie should likely prepare themselves for the possibility of wide swings between different trips. An alternative way of interpreting the interval is to note that, before any data were analyzed, the probability that they would calculate an interval containing the truth was \(0.9\).

In practice, the utility of reporting a confidence interval is that it conveys a sense of uncertainty. Wider intervals suggest that we should put less assurance in the value of the point estimate than more narrow intervals. Moreover, given a confidence interval, there is little reason to think that one value in this interval is more likely to correspond to the truth than any other value. We can use the intervals to determine whether decisions that are made on the basis of estimation would be made under all scenarios implied by the interval. If conclusions differ assuming that \(\theta = U\) compared to \(\theta = L\), care should be put into making more conservative decisions, since based on the sample there is little reason to differentiate between these points.

14.4 Confidence Intervals for the Mean

While confidence intervals can be derived for any parameter of interest, as previously discussed, we are very commonly interested in the means of populations. Moreover, owing to the Central Limit Theorem, the sampling distribution of a sample mean is often easy to approximate. We will primarily focus on confidence intervals for the means of distribution. We differentiate the procedure based on whether the population is itself normally distributed or not.

14.4.1 Confidence Intervals in Normal Populations

If our population is normally distributed then the sample mean is known to be exactly normally distributed as well. The sampling distribution depends on both the population mean, \(\mu\), and on the population variance, \(\sigma^2\). Suppose that we wanted to determine \(P(a \leq \overline{X} \leq b)\), for some values \(a\) and \(b\), then we know that \[P(a \leq \overline{X} \leq b) = P\left(\frac{a - \mu}{\sigma/\sqrt{n}} \leq Z \leq \frac{b - \mu}{\sigma/\sqrt{n}}\right) = \Phi\left(\frac{b - \mu}{\sigma/\sqrt{n}}\right) - \Phi\left(\frac{a - \mu}{\sigma/\sqrt{n}}\right).\]

If \(\sigma^2\) is known then we can use the percentiles of the standard normal distribution to solve for what \(a\) and \(b\) would have to be. Specifically, if we take the two endpoints to be symmetric, then we want to solve for \[\Phi\left(\frac{b - \mu}{\sigma/\sqrt{n}}\right) = 1 - \frac{1-p}{2},\quad\text{and}\quad\Phi\left(\frac{a - \mu}{\sigma/\sqrt{n}}\right) = \frac{1-p}{2}.\] If we did this then we would find that \[P(a \leq \overline{X} \leq b) = 1 - \frac{1-p}{2} - \frac{1-p}{2} = p,\] as desired.

To solve for these quantities, we note that these are exactly the quantiles of the standard normal distribution. That is, \[\begin{align*}\Phi\left(\frac{b - \mu}{\sigma/\sqrt{n}}\right) &= 1 - \frac{1-p}{2} \implies \frac{b - \mu}{\sigma/\sqrt{n}} = Z_{1 - \frac{1-p}{2}}, \\ \Phi\left(\frac{a - \mu}{\sigma/\sqrt{n}}\right) &= \frac{1-p}{2} \implies \frac{a - \mu}{\sigma/\sqrt{n}} = Z_{\frac{1-p}{2}}.\end{align*}\] Note that, by symmetry, \(Z_{(1-p)/2} = -Z_{1 - (1-p)/2}\), so we take \[Z_{(1-p)/2}\frac{\sigma}{\sqrt{n}} + \mu \leq \overline{X} \leq Z_{1-(1-p)/2}\dfrac{\sigma}{\sqrt{n}} + \mu.\] This interval can then be inverted to give an interval for \(\mu\). Namely, \[\begin{align*} Z_{(1-p)/2}\frac{\sigma}{\sqrt{n}} + \mu &\leq \overline{X} \\ \implies \mu &\leq \overline{X} - Z_{(1-p)/2}\frac{\sigma}{\sqrt{n}} \\ &= \overline{X} + Z_{1 - (1-p)/2}\frac{\sigma}{\sqrt{n}} \\ \overline{X} \leq Z_{1-(1-p)/2}\dfrac{\sigma}{\sqrt{n}} + \mu \\ \implies \overline{X} - \leq Z_{1-(1-p)/2}\dfrac{\sigma}{\sqrt{n}} &\leq \mu. \end{align*}\]

These two bounds give rise to the confidence intervals for the mean of a normal population, when \(\sigma\) is known.

Confidence Intervals for the Mean (Known \(\sigma\) in a Normal Population)

If a population is known to be normally distributed, with a known population standard deviation, \(\sigma\), then a \(100p\%\) confidence interval for the mean is given by \[(\overline{X} - \leq Z_{1-(1-p)/2}\dfrac{\sigma}{\sqrt{n}}, \overline{X} + Z_{1 - (1-p)/2}\frac{\sigma}{\sqrt{n}}).\] This is often expressed as \[\overline{X} \pm Z_{(1-p)/2}\frac{\sigma}{\sqrt{n}}.\]

Common Normal Critical Values

When introducing the normal distribution, we discussed the percentiles of the normal as normal critical values. We discussed how these can be found using the qnorm function in R. In most introductory statistics references, they will contain tables of normal critical values, giving the mappings between the \(Z\) values and their probabilities. The following are commonly used \(Z\) values, provided for reference.

\(\alpha\)	\(Z_{\alpha}\)	\(\alpha\)	\(Z_{\alpha}\)
0.005	-2.58	0.995	2.58
0.010	-2.33	0.990	2.33
0.025	-1.96	0.975	1.96
0.050	-1.64	0.950	1.64
0.100	-1.28	0.900	1.28

Example 14.5 (Charles’ Nap Times) Charles is a frequent napper. Sadie, on the other hand, cannot take naps at all. Sadie often has a significant amount of boredom while Charles naps away, and so Sadie starts to record the lengths of the naps that Charles takes, in minutes. Over the last \(5\) days, Charles’ naps have been: \[33.2, \ 29.6, \ 29.7, \ 31.7, \ 29.1.\]

What is an estimate for the average length of Charles’ naps?
Sadie assumes that Charles’ naps are normally distributed, with a variance of \(4\). Find a \(90\%\) confidence interval for the average nap length.
Using the same assumptions, find a \(95\%\) confidence interval for the average nap length.
How would an \(80\%\) confidence interval compare? A \(99.9\%\) interval? Explain.

Solution

We can use the sample average to estimate the population average. This is given by \[\frac{1}{5}\left(33.2 + 29.6 + 29.7 + 31.7 + 29.1\right) = 30.66\]
Assuming normality and with known variance, we can take \[\overline{X} \pm Z_{(1-p)/2}\frac{\sigma}{\sqrt{n}}.\] Here, \(\sigma = 2\), \(\sqrt{n} = \sqrt{5}\), \(\overline{X} = 30.66\). For a \(90\%\) confidence interval, we have \((1 - 0.9) / 2 = 0.05\), and thus \(Z_{(1-p)/2} = -1.64\). We get \[30.66 \pm 1.64\frac{2}{\sqrt{5}} = (29.19, 32.14).\]
For a \(95\%\) confidence interval, we change the critical value to be given by \(Z_{(1-0.95)/2} = -1.96\). Thus, \[30.66 \pm 1.96\frac{2}{\sqrt{5}} = (28.91, 32.41).\]
As the confidence level increases, the width of the confidence level must also increase. Thus, we should find that an \(80\%\) confidence interval is more tightly centered around \(30.66\) and a \(99.9\%\) confidence interval is more widely spread out. Indeed, using R we can find these to be \((29.51, 31.81)\) for the \(80\%\) interval, and \((27.72, 33.6)\) for the \(99.9\%\) interval.

If the variance is not known, then this procedure cannot be followed directly. An obvious alteration to make is to substitute \(s^2\) for \(\sigma^2\) in the expression for the confidence interval. However, if we consider the quantity \[\frac{\overline{X} - \mu}{s/\sqrt{n}},\] this will no longer follow a normal distribution. Instead, assuming that \(\overline{X}\) is normally distributed, this ratio follows a \(t\) distribution with \(n-1\) degrees of freedom. Otherwise, the same procedure can be followed, this time needing the critical values from the \(t_{n-1}\) distribution rather than the standard normal distribution.

Confidence Intervals for the Mean (Unknown \(\sigma\) in a Normal Population)

If a population is known to be normally distributed, with an unknown population standard deviation, then a \(100p\%\) confidence interval for the mean is given by \[(\overline{X} - \leq t_{n-1, 1-(1-p)/2}\dfrac{s}{\sqrt{n}}, \overline{X} + t_{n-1, 1 - (1-p)/2}\frac{s}{\sqrt{n}}).\] This is often expressed as \[\overline{X} \pm t_{n-1, (1-p)/2}\frac{s}{\sqrt{n}}.\]

Example 14.6 (Charles Continues Napping) Charles has maintained the pattern of daily naps. Sadie, understanding more about confidence intervals, decides to revisit the intervals for the length of Charles’ naps. This time, instead of assuming that the variance is known to be \(4\), Sadie wants to leverage the same data as was used to find the mean. Specifically, Sadie has recorded that over a \(5\) day stretch, the length of Charles’ naps have been: \[33.2, \ 29.6, \ 29.7, \ 31.7, \ 29.1.\] While Sadie does not want to presume that the variance is known, for now, the assumption of normality still feels reasonable.

What is the sample variance?
Using the sample variance, how could Sadie construct a \(90\%\) confidence interval for the mean?
What critical value would need to be used to calculate a \(95\%\) confidence interval in this case? A \(99\%\) one?

Solution

For the sample variance, we calculate \[\begin{split}\frac{1}{5-1}\left((33.2 - 30.66)^2 + (29.6-30.66)^2 + (29.7 - 30.66)^2 \right.\\ \left. + (31.7 - 30.66)^2 + (29.1 - 30.66)^2\right) = 3.003.\end{split}\]
Since the population is normally distributed, with an unknown variance, Sadie can use the form \[\overline{X} \pm t_{n-1,(1-p)/2}\frac{s}{\sqrt{n}},\] to form confidence intervals. Here, \(\overline{X} = 30.66\), \(n = 5\), \(s = \sqrt{3.003}\), and \(p=0.9\). Thus, the critical value to use is the \(t_{4, 0.05}\) value, and the form of the interval will be \[30.66 \pm t_{4, 0.05}\sqrt{\frac{3.003}{5}} = 30.66 \pm t_{4, 0.05}\sqrt{0.6006}.\] Using R, we can call qt(0.05, 4) to determine that the critical value is approximately -2.13, to fill in for the specific value of \(t_{4, 0.05}\).
Here we would still reference the \(t\) distribution with \(4\) degrees of freedom, but now instead of the \(0.05\) percentile, we would need to look at either the \(0.025\) or \(0.005\) percentiles. This would give, \(t_{4, 0.025} \approx -2.78\) and \(t_{4, 0.005} \approx -4.6\).

In both scenarios, the confidence interval forms a symmetric interval around the estimated mean, \(\overline{X}\), and is based on critical values from the relevant sampling distribution. If \(n\) is sufficiently larger than \(t_{n-1, \alpha} \approx Z_{\alpha}\), and so in these cases it is sometimes acceptable to use the \(Z\)-based intervals, as though \(\sigma\) were known. These intervals are justified on the basis of the population being normally distributed. What if this is not an assumption we can make?

14.4.2 Confidence Intervals in Non-Normal Populations

When populations are non-normal, there are two possible techniques for forming confidence intervals for the mean. In the first, we can attempt to use the assumed population distribution to derive an exact confidence interval for the mean, based on the true sampling distribution. This would need to be worked out for every individual population distribution, and would be valid only for that distribution. Alternatively, if our sample sizes are large enough, then we can use the Central Limit Theorem to indicate that \(\overline{X}\) will be approximately normally distributed. In this case, we can follow the same procedures outlined above. Note that doing so will not give exact confidence intervals, instead settling for approximate confidence intervals. These are often reliable enough in practice.

Approximate Confidence Intervals for the Mean

For a sufficiently large sample size, such that Central Limit Theorem applies, a \(100p\%\) confidence interval for the mean is given by \[(\overline{X} - \leq t_{n-1, 1-(1-p)/2}\dfrac{s}{\sqrt{n}}, \overline{X} + t_{n-1, 1 - (1-p)/2}\frac{s}{\sqrt{n}}).\] This is often expressed as \[\overline{X} \pm t_{n-1, (1-p)/2}\frac{s}{\sqrt{n}}.\]

Example 14.7 (Charles’ Naps are Not Normal) Sadie realizes that the assumption that Charles’ nap lengths are normally distributed may not be correct, and begins to wonder how this may impact the calculations that have been done before. Recall that previously Sadie had recorded \(5\) nap lengths for Charles, giving \[33.2, \ 29.6, \ 29.7, \ 31.7, \ 29.1.\]

Can Sadie use these data, assuming that they are not normal, to compute a confidence interval for the mean length of Charles’ naps? Explain.
Suppose that Sadie records data over two months, giving \(n=60\). Over this time, the mean nap length is \(31.10\) and the standard deviation is \(8.61\). How may Sadie use this information to construct a \(100p\%\) confidence interval for the length of the nap.
Given that \(n\) is fairly large, what will be the approximate value of the critical value needed for a \(95\%\) confidence interval? Use this to write down an approximate \(95\%\) confidence interval.

Solution

No. The issue is that, without normality, we rely on the Central Limit Theorem to form the confidence intervals. However, in order for the CLT to apply, we must have a large enough sample. A sample of size \(5\) is not going to be large enough to invoke the CLT, and as a result, we either need to make other assumptions about the population, or else collect more data.
In this case, we can use the CLT to form the confidence interval. Namely, we get \[\overline{X} \pm t_{n-1, (1-p)/2}\frac{s}{\sqrt{n}} = 31.10 \pm t_{59, (1-p)/2}\frac{8.61}{\sqrt{60}}.\] Here the critical values come from the \(t_{59}\) distribution, and could be computed for any level of interest.
When \(n\) is large, we get that \(t_{n, \alpha} \approx Z_{\alpha}\). Because of this, we can take the critical value to be approximately \(Z_{0.025} = -1.96\).⁸ Using this value we get \[31.10 \pm 1.96\frac{8.61}{\sqrt{60}} = (30.92, 35.28).\]

We previously discussed how, in the event of having data from a binomial experiment, we can view the estimator of the success probability, \(\widehat{p}\) as a sample mean. This means that, supposing that we wish to find a confidence interval for a proportion, we can follow fundamentally the same procedure. A key distinction here is that, in the case of binomial data, the estimated variance is not \(s^2\) but rather \(n\widehat{p}(1 - \widehat{p})\). Practically, this means that we can use the standard normal critical values for our confidence intervals, rather than the \(t\) values.⁹

Approximate Confidence Intervals for a Proportion

For a sufficiently large sample size, such that Central Limit Theorem applies, a \(100p\%\) confidence interval for a proportion is given by \[(\widehat{p} - Z_{1-(1-p)/2}\sqrt{\dfrac{\widehat{p}(1-\widehat{p})}{n}}, \overline{X} + Z_{1 - (1-p)/2}\sqrt{\dfrac{\widehat{p}(1-\widehat{p})}{n}}).\] This is often expressed as \[\widehat{p} \pm Z_{(1-p)/2}\sqrt{\dfrac{\widehat{p}(1-\widehat{p})}{n}}.\]

14.5 Confidence Intervals for the Variance

It is important to recognize that, while the formulae for confidence intervals for the means are used frequently, the process of forming a confidence interval is equivalent regardless of the parameter that is under consideration. Generally, a confidence interval can be found by first finding an interval around the estimator, and then inverting this interval to form one around the parameter value. The key step in this process is determining the sampling distribution for the relevant statistic. As such, when learning confidence intervals, it is best to focus on the overarching procedure and understand where each formula is coming from, rather than specifically pattern matching to find which formula to use. When considering the mean, we use the \(Z\) or \(t\) critical values depending on whether the sampling distribution follows a normal or a \(t\) distribution. Recognizing this enables confidence intervals for any parameters for which the sampling distribution is known.

As an example of how this procedure can be applied, consider taking a normal population, and forming a confidence interval for the variance. In order to do so, we would first want to find an interval around \(S^2\) that would hold with probability \(p\), and then invert this interval. This relies on knowledge of the sampling distribution of \(S^2\).

The Sampling Distribution of \(S^2\)

If the population is normally distributed, with mean \(\mu\) and variance \(\sigma^2\), then the sampling distribution of \[S^2 = \frac{1}{n-1}\sum_{i=1}^n (x_i - \overline{x})^2,\] can be derived by noting that \[\frac{(n-1)}{\sigma^2}S^2 \sim \chi^2_{n-1}.\] This is, notably, not precisely the sampling distribution of \(S^2\) directly, however, it can be manipulated to give probability statements regarding \(S^2\).

Using this result, we can then form a confidence interval for \(S^2\). We start by finding constants, \(a\) and \(b\), such that \[P(a \leq S^2 \leq b) = p \implies P\left(\frac{(n-1)a}{\sigma^2} \leq \frac{(n-1)S^2}{\sigma^2} \leq \frac{(n-1)b}{\sigma^2}\right) = p.\] Noting the sampling distribution, we require \[\frac{(n-1)a}{\sigma^2} = \chi^2_{n-1,(1-p)/2} \quad\text{and}\quad \frac{(n-1)b}{\sigma^2} = \chi^2_{n-1,1-(1-p)/2}.\] Unlike the normal distribution, the chi-squared distribution is not symmetric, and the critical values will not be equal in magnitude. We can take this interval for \(S^2\), and solve it for \(\sigma^2\) instead. Doing so gives \[\chi^2_{n-1,(1-p)/2} \leq \frac{(n-1)S^2}{\sigma^2} \implies \sigma^2 \leq \frac{(n-1)S^2}{\chi^2_{n-1,(1-p)/2}} = U(S^2),\] as the upper bound on the confidence interval, and \[\frac{(n-1)S^2}{\sigma^2} \leq \chi^2_{n-1,1-(1-p)/2} \implies L(S^2) = \frac{(n-1)S^2}{\chi^2_{n-1,1-(1-p)/2}} \leq \sigma^2,\] as the lower bound.

Confidence Intervals for the Sample Variance

For a normally distributed population, a confidence interval for the variance is formed by taking \[\left(\frac{(n-1)S^2}{\chi^2_{n-1,1-(1-p)/2}}, \frac{(n-1)S^2}{\chi^2_{n-1,(1-p)/2}}\right).\] Here, the critical values from the \(\chi^2\) distribution are given by the \(1-(1-p)/2\) and \((1-p)/2\) quantiles, respectively. This is sometimes expressed using a different definition for the critical values.

Note that this derivation follows directly from knowledge of the sampling distribution of \(S^2\). Had we been given a different result, we could have worked through the same procedure, resulting in different interval formula with the same underlying process. This specific result importantly relies on the assumption that the population is normally distributed. Were we in a situation where this is not the case, the formula does not hold, regardless of the size of the sample. To find confidence intervals for sample variances in non-normal populations, we require results regarding their sampling distributions.

It is also worth considering how we may use this result to form confidence intervals for standard deviations. Generally speaking, confidence intervals are not invariant to transformations. By this we mean that, even though \(\sigma = \sqrt{\sigma^2}\), taking the confidence interval \((\sqrt{L(S^2)}, \sqrt{U(S^2)})\) will not result in an exact confidence interval for \(\sigma\). Despite the fact that this will not generally be exactly correct, this is the procedure we tend to use. That is, if we have a confidence interval for a parameter, \(\theta\), given by \[(L(\widehat{\theta}), U(\widehat{\theta})),\] and we wish to have a confidence interval for some transformation, \(g(\theta)\), we will typically use \[\left(g(L(\widehat{\theta})), g(U(\widehat{\theta}))\right).\] As long as \(g\) is a nicely behaved function, this will typically result in an acceptably close approximation.¹⁰ Supposing that we have a confidence interval for \(\sigma^2\), then we can form a confidence interval for \(\sigma\) simply by taking the square root of the endpoints.

Example 14.8 (Charles’ Tufting Supplier Variability) Charles has been continuing to learn to tuft. One issue that has been coming up is in the variability of the durability of the yarn across different batches from the same supplier. Charles has found that it is important to approximately match the tensile strength of yarns on the same rug, otherwise it can lead to suboptimal results. Curious about how much variability there truly is in the tensile strength from the main supplier, Charles takes the following measurements: \[20.29, 14.78, 13.9, 15.5, 20.05, 14.72.\]

What is the estimated variance of the yarn strength from this supplier?
How can Charles use this information to form a \(95\%\) confidence interval for the strength of the yarn?
Supposing that a confidence interval is found for the variance as \((3.24, 49.0)\), what is a confidence interval for the standard deviation? Is this exact? Explain.

Solution

For the sample variance, we first need to get the sample mean. This is given by \[\frac{1}{6}\left(20.29 + 14.78 + 13.9 + 15.5 + 20.05 + 14.72\right) = 16.54.\] Then, we take \[\begin{split}\frac{1}{6-1}\left((20.29 - 16.54)^2 + (14.78 - 16.54)^2 + (13.9 - 16.54)^2 \right. \\ \left. + (15.5 - 16.54)^2 + (20.05 - 16.54)^2\right) = 8.16876.\end{split}\]
To form the confidence interval in this setting we note that, assuming the observations are normally distributed, then the scaled sample mean will follow a \(\chi^2_{n-1}\) distribution, where \(n=6\). Thus, we take \[\left(\frac{(n-1)S^2}{\chi^2_{n-1,1-(1-p)/2}}, \frac{(n-1)S^2}{\chi^2_{n-1,(1-p)/2}}\right)\] where \(s^2 = 8.16876\), \(n=6\), and \(p=0.95\). Plugging in, this gives \[\left(\frac{(5)(8.16876)}{\chi^2_{5,0.975}}, \frac{(5)(8.16876)}{\chi^2_{5,0.025}}\right).\] If we use R to compute the critical values, this gives \((3.1828399, 49.1376676)\).
Given the interval \((3.24, 49.0)\), we can approximate a confidence interval for the standard deviation by square rooting the endpoints. This gives \((1.8, 7)\). This is not exact, since confidence intervals are not typically invariant to transformations. Instead, this will be approximate, but will typically be close enough in practice.

14.6 Margin of Error and Sample Sizing

When communicating the results of a scientific study, it is typically considered best practice to report confidence intervals alongside point estimates. This conveys the uncertainty in the estimate, and it gives the individual consuming the results the ability to make decisions based on the sensitivity of these results to the reported values. Knowing that you are likely to report a confidence interval, you can use the form of the confidence interval to make planning decisions regarding the study.

When forming confidence intervals for the mean, all the intervals we considered took the form \(\overline{X} \pm \Omega\), where \(\Omega\) depended on the underlying variability, the sampling distribution, the degree of confidence, as well as the sample size. We call this term, the margin of error.

Definition 14.3 (Margin of Error) In a symmetric confidence interval, the margin of error is defined to be the distance from the point estimate to the upper or lower confidence bounds. That is, a confidence interval for \(\theta\) can be expressed as \[\widehat{\theta} \pm \text{M.E.}(\widehat{\theta}),\] where \(\text{M.E}(\widehat{\theta})\) corresponds to the margin of error. The width of a confidence interval is always \(2\times\text{M.E.}\).

As noted previously, all else equal we would prefer a confidence interval that is less wide. This is equivalent to saying that we prefer a smaller margin of error. We can consider how the various choices that we make may influence the overall width of the confidence interval. Consider a normal population, with known variance. In this setting \[\text{M.E.}(\overline{X}) = Z_{1-(1-p)/2}\frac{\sigma}{\sqrt{n}}.\] The three components of this formula are \(p\), \(\sigma\), and \(n\). Note that, as \(p\) increases, \(1-(1-p)/2\) also increases, and so \(Z_{1-(1-p)/2}\) will also increase. This means that if we want a higher \(p\), the confidence interval must get wider.¹¹ Similarly, if \(\sigma\) is larger, then the confidence interval will also be wider. This is not under control of the researcher, but it is to say that in more varied populations, confidence intervals will tend to be wider.¹² Finally, we see that as \(n\) increases, the width of the confidence interval will decrease. This is a choice that the researcher gets to make.

Noting that as the sample size increases the width of the confidence interval decreases, we can use this relationship to decide what sample size our study should rely on. Specifically, if a researcher knows the variance (or has an estimated value for the sample variance), then they can determine what \(n\) is required in order to achieve a set margin of error. For instance, perhaps you want to have a \(99\%\) confidence interval that will give a margin of error of no more than \(0.01\). By setting the margin of error equal to \(0.01\), and taking \(p = 0.99\), you can determine that \[n \geq \left(Z_{0.995}\frac{\sigma}{0.01}\right)^2.\] As long as the value for \(\sigma\) is correct, this will guarantee the desired result.

Example 14.9 (Tensile Strength Averages) Charles has been continuing to deal with high variability in the tensile strength of the main yarn supplier. There is another possible supplier to use, however, Charles wants to be very sure that the average tensile strength of the new supplier’s yarn is acceptable as well. Charles will be happy if a \(90\%\) confidence interval is within \(0.1\) of the true average tensile strength. From experience and word-of-mouth, Charles suspects that the new supplier has a tensile strength variability of approximately \(4\).

How many balls of yarn should Charles buy in order to form the desired confidence interval?
What if Charles were content with being within \(1\) of the true average?
Suppose that instead of estimating the average, Charles wants to estimate the proportion of balls of yarn from the new supplier with a strength that exceeds that of the old supplier. To do so, Charles buys pairs of balls of yarn, and compares the strength of each of them, reporting whether the new supplier is higher or lower. Approximately how many pairs of balls of yarn would be required so that the proportion can be estimated to within \(0.01\) of the true proportion, with \(95\%\) confidence?

Solution

Here, Charles wants a margin of error equal to \(0.1\). We know that, in this case, the approximate value for the margin of error will be \[Z_{1-(1-p)/2}\frac{\sigma}{\sqrt{n}} = 1.64\frac{2}{\sqrt{n}}.\] Thus, we solve \[\begin{align*} 1.64\frac{2}{\sqrt{n}} &\leq 0.1 \\ \frac{3.28}{0.1} \leq \sqrt{n} \\ 32.8^2 \leq n \\ 1075.84 \leq n. \end{align*}\] Therefore, Charles would need to buy at least \(1076\) balls of yarn to have the level of confidence desired in the tensile strength.
To be within \(1\), we solve the same expression as in (a) but for a margin of error of \(1\). This results in \(n \geq 3.28^2 = 10.8\), and so in this case Charles would need to purchase \(11\) balls of yarn.
Here, we wish to estimate a proportion within \(0.01\) of the truth. To do so, recognize that we can take the normal approximation for the confidence interval, which gives a margin of error equal to \[Z_{1-(1-p)/2}\sqrt{\frac{p(1-p)}{n}}.\] In principle, we can solve for this equal to \(0.01\). However, it is important to note that we should take a conservative approach to \(p\) here. The most variability will come when \(p=0.5\), and so we should estimate the population variance to be based on \(p=0.5\). This gives \[\begin{align*} Z_{0.975}\sqrt{\frac{0.5^2}{n}} &\leq 0.01 \\ \frac{1.96(0.5)}{0.01} &\leq \sqrt{n} \\ 9.8^2 &\leq n \\ 96.04 \leq n. \end{align*}\] There should be at least \(97\) pairs of balls of yarn purchased to have this degree of confidence in the proportion estimate.

14.7 Confidence Intervals in R

The primary utility of R in terms of forming confidence intervals is in making it easy to determine the critical values for the required distribution. Recall from earlier chapters that the critical values for distributions are typically given by q{distname}, where distname is the relevant distribution. That is, should you want the critical values for a normally distributed population, you can use qnorm(1 - (1-p)/2), where p is the relevant level. This can then be supplemented with relevant calls to mean, sd, and var to form the various intervals.

In the coming modules we will see ways of calculating confidence intervals less directly, however, these procedures are designed for alternative purposes and thus are unnecessary when all that is desired is a confidence interval.

Self-Assessment

Note: the following questions are still experimental. Please contact me if you have any issues with these components. This can be if there are incorrect answers, or if there are any technical concerns. Each question currently has an ID with it, randomized for each version. If you have issues, reporting the specific ID will allow for easier checking!

For each question, you can check your answer using the checkmark button. You can cycle through variants of the question by pressing the arrow icon.

Self Assessment 14.1

The following are confidence intervals for \(\mu\), the true mean for a population: \(A = (116.98, 125.02)\) and \(B = (118.8, 123.2)\). These intervals are standard, symmetric confidence intervals for the mean. One is a \(90\%\) confidence interval, while the other is an \(87\%\) confidence interval.

What is the sample mean?
What is the confidence level for \(A\)?
What is the confidence level for \(B\)?

(Question ID: 0644607937)

The following are confidence intervals for \(\mu\), the true mean for a population: \(A = (-92.15, -87.85)\) and \(B = (-97.53, -82.47)\). These intervals are standard, symmetric confidence intervals for the mean. One is a \(97\%\) confidence interval, while the other is an \(81\%\) confidence interval.

What is the sample mean?
What is the confidence level for \(A\)?
What is the confidence level for \(B\)?

(Question ID: 0559282069)

The following are confidence intervals for \(\mu\), the true mean for a population: \(A = (135.76, 148.24)\) and \(B = (138.88, 145.12)\). These intervals are standard, symmetric confidence intervals for the mean. One is a \(94\%\) confidence interval, while the other is an \(86\%\) confidence interval.

What is the sample mean?
What is the confidence level for \(A\)?
What is the confidence level for \(B\)?

(Question ID: 0075755065)

The following are confidence intervals for \(\mu\), the true mean for a population: \(A = (-25.34, -22.66)\) and \(B = (-26.88, -21.12)\). These intervals are standard, symmetric confidence intervals for the mean. One is a \(99\%\) confidence interval, while the other is an \(86\%\) confidence interval.

What is the sample mean?
What is the confidence level for \(A\)?
What is the confidence level for \(B\)?

(Question ID: 0008624060)

The following are confidence intervals for \(\mu\), the true mean for a population: \(A = (43.2, 48.8)\) and \(B = (37.3, 54.7)\). These intervals are standard, symmetric confidence intervals for the mean. One is a \(90\%\) confidence interval, while the other is an \(85\%\) confidence interval.

What is the sample mean?
What is the confidence level for \(A\)?
What is the confidence level for \(B\)?

(Question ID: 0870013095)

The following are confidence intervals for \(\mu\), the true mean for a population: \(A = (-146.29, -137.71)\) and \(B = (-145.57, -138.43)\). These intervals are standard, symmetric confidence intervals for the mean. One is an \(85\%\) confidence interval, while the other is an \(84\%\) confidence interval.

What is the sample mean?
What is the confidence level for \(A\)?
What is the confidence level for \(B\)?

(Question ID: 0976628047)

The following are confidence intervals for \(\mu\), the true mean for a population: \(A = (6.7, 15.3)\) and \(B = (4.79, 17.21)\). These intervals are standard, symmetric confidence intervals for the mean. One is a \(92\%\) confidence interval, while the other is an \(89\%\) confidence interval.

What is the sample mean?
What is the confidence level for \(A\)?
What is the confidence level for \(B\)?

(Question ID: 0392640888)

The following are confidence intervals for \(\mu\), the true mean for a population: \(A = (-85.15, -80.85)\) and \(B = (-92.9, -73.1)\). These intervals are standard, symmetric confidence intervals for the mean. One is a \(93\%\) confidence interval, while the other is a \(90\%\) confidence interval.

What is the sample mean?
What is the confidence level for \(A\)?
What is the confidence level for \(B\)?

(Question ID: 0060529340)

The following are confidence intervals for \(\mu\), the true mean for a population: \(A = (-11.85, 1.85)\) and \(B = (-8.9, -1.1)\). These intervals are standard, symmetric confidence intervals for the mean. One is an \(87\%\) confidence interval, while the other is an \(86\%\) confidence interval.

What is the sample mean?
What is the confidence level for \(A\)?
What is the confidence level for \(B\)?

(Question ID: 0756353183)

The following are confidence intervals for \(\mu\), the true mean for a population: \(A = (-125.46, -114.54)\) and \(B = (-128.99, -111.01)\). These intervals are standard, symmetric confidence intervals for the mean. One is a \(98\%\) confidence interval, while the other is an \(82\%\) confidence interval.

What is the sample mean?
What is the confidence level for \(A\)?
What is the confidence level for \(B\)?

(Question ID: 0677848547)

The following are confidence intervals for \(\mu\), the true mean for a population: \(A = (36.25, 43.75)\) and \(B = (31.34, 48.66)\). These intervals are standard, symmetric confidence intervals for the mean. One is an \(84\%\) confidence interval, while the other is an \(80\%\) confidence interval.

What is the sample mean?
What is the confidence level for \(A\)?
What is the confidence level for \(B\)?

(Question ID: 0722394021)

The following are confidence intervals for \(\mu\), the true mean for a population: \(A = (-30.03, -25.97)\) and \(B = (-33.11, -22.89)\). These intervals are standard, symmetric confidence intervals for the mean. One is an \(87\%\) confidence interval, while the other is an \(81\%\) confidence interval.

What is the sample mean?
What is the confidence level for \(A\)?
What is the confidence level for \(B\)?

(Question ID: 0303001117)

The following are confidence intervals for \(\mu\), the true mean for a population: \(A = (120.45, 131.55)\) and \(B = (117.08, 134.92)\). These intervals are standard, symmetric confidence intervals for the mean. One is a \(90\%\) confidence interval, while the other is an \(83\%\) confidence interval.

What is the sample mean?
What is the confidence level for \(A\)?
What is the confidence level for \(B\)?

(Question ID: 0046200067)

The following are confidence intervals for \(\mu\), the true mean for a population: \(A = (11.63, 20.37)\) and \(B = (14.51, 17.49)\). These intervals are standard, symmetric confidence intervals for the mean. One is a \(96\%\) confidence interval, while the other is an \(83\%\) confidence interval.

What is the sample mean?
What is the confidence level for \(A\)?
What is the confidence level for \(B\)?

(Question ID: 0025870811)

The following are confidence intervals for \(\mu\), the true mean for a population: \(A = (180.16, 189.84)\) and \(B = (181.37, 188.63)\). These intervals are standard, symmetric confidence intervals for the mean. One is an \(87\%\) confidence interval, while the other is an \(83\%\) confidence interval.

What is the sample mean?
What is the confidence level for \(A\)?
What is the confidence level for \(B\)?

(Question ID: 0528674136)

The following are confidence intervals for \(\mu\), the true mean for a population: \(A = (-44.19, -33.81)\) and \(B = (-47.62, -30.38)\). These intervals are standard, symmetric confidence intervals for the mean. One is a \(94\%\) confidence interval, while the other is an \(85\%\) confidence interval.

What is the sample mean?
What is the confidence level for \(A\)?
What is the confidence level for \(B\)?

(Question ID: 0409310245)

The following are confidence intervals for \(\mu\), the true mean for a population: \(A = (-68.08, -53.92)\) and \(B = (-66.18, -55.82)\). These intervals are standard, symmetric confidence intervals for the mean. One is a \(97\%\) confidence interval, while the other is a \(90\%\) confidence interval.

What is the sample mean?
What is the confidence level for \(A\)?
What is the confidence level for \(B\)?

(Question ID: 0701795618)

The following are confidence intervals for \(\mu\), the true mean for a population: \(A = (-67.65, -50.35)\) and \(B = (-66.09, -51.91)\). These intervals are standard, symmetric confidence intervals for the mean. One is a \(91\%\) confidence interval, while the other is an \(82\%\) confidence interval.

What is the sample mean?
What is the confidence level for \(A\)?
What is the confidence level for \(B\)?

(Question ID: 0050322632)

The following are confidence intervals for \(\mu\), the true mean for a population: \(A = (-146.77, -129.23)\) and \(B = (-147.69, -128.31)\). These intervals are standard, symmetric confidence intervals for the mean. One is a \(94\%\) confidence interval, while the other is an \(82\%\) confidence interval.

What is the sample mean?
What is the confidence level for \(A\)?
What is the confidence level for \(B\)?

(Question ID: 0421247229)

The following are confidence intervals for \(\mu\), the true mean for a population: \(A = (122.41, 125.59)\) and \(B = (122.59, 125.41)\). These intervals are standard, symmetric confidence intervals for the mean. One is a \(95\%\) confidence interval, while the other is a \(92\%\) confidence interval.

What is the sample mean?
What is the confidence level for \(A\)?
What is the confidence level for \(B\)?

(Question ID: 0165475197)

The following are confidence intervals for \(\mu\), the true mean for a population: \(A = (-30.9, -27.1)\) and \(B = (-33.5, -24.5)\). These intervals are standard, symmetric confidence intervals for the mean. One is a \(91\%\) confidence interval, while the other is an \(84\%\) confidence interval.

What is the sample mean?
What is the confidence level for \(A\)?
What is the confidence level for \(B\)?

(Question ID: 0038799228)

The following are confidence intervals for \(\mu\), the true mean for a population: \(A = (-129.97, -112.03)\) and \(B = (-122.52, -119.48)\). These intervals are standard, symmetric confidence intervals for the mean. One is an \(85\%\) confidence interval, while the other is an \(83\%\) confidence interval.

What is the sample mean?
What is the confidence level for \(A\)?
What is the confidence level for \(B\)?

(Question ID: 0272242421)

The following are confidence intervals for \(\mu\), the true mean for a population: \(A = (-89.83, -82.17)\) and \(B = (-92.84, -79.16)\). These intervals are standard, symmetric confidence intervals for the mean. One is a \(97\%\) confidence interval, while the other is an \(82\%\) confidence interval.

What is the sample mean?
What is the confidence level for \(A\)?
What is the confidence level for \(B\)?

(Question ID: 0315497795)

The following are confidence intervals for \(\mu\), the true mean for a population: \(A = (-10.79, -7.21)\) and \(B = (-16.98, -1.02)\). These intervals are standard, symmetric confidence intervals for the mean. One is a \(92\%\) confidence interval, while the other is an \(84\%\) confidence interval.

What is the sample mean?
What is the confidence level for \(A\)?
What is the confidence level for \(B\)?

(Question ID: 0996176265)

The following are confidence intervals for \(\mu\), the true mean for a population: \(A = (-51.13, -32.87)\) and \(B = (-46.91, -37.09)\). These intervals are standard, symmetric confidence intervals for the mean. One is a \(94\%\) confidence interval, while the other is an \(84\%\) confidence interval.

What is the sample mean?
What is the confidence level for \(A\)?
What is the confidence level for \(B\)?

(Question ID: 0625297772)

The following are confidence intervals for \(\mu\), the true mean for a population: \(A = (-200.29, -197.71)\) and \(B = (-201.1, -196.9)\). These intervals are standard, symmetric confidence intervals for the mean. One is an \(86\%\) confidence interval, while the other is an \(85\%\) confidence interval.

What is the sample mean?
What is the confidence level for \(A\)?
What is the confidence level for \(B\)?

(Question ID: 0171696837)

The following are confidence intervals for \(\mu\), the true mean for a population: \(A = (179, 199)\) and \(B = (179.91, 198.09)\). These intervals are standard, symmetric confidence intervals for the mean. One is a \(96\%\) confidence interval, while the other is a \(93\%\) confidence interval.

What is the sample mean?
What is the confidence level for \(A\)?
What is the confidence level for \(B\)?

(Question ID: 0460606047)

The following are confidence intervals for \(\mu\), the true mean for a population: \(A = (76.96, 83.04)\) and \(B = (70.37, 89.63)\). These intervals are standard, symmetric confidence intervals for the mean. One is a \(95\%\) confidence interval, while the other is a \(94\%\) confidence interval.

What is the sample mean?
What is the confidence level for \(A\)?
What is the confidence level for \(B\)?

(Question ID: 0505507992)

The following are confidence intervals for \(\mu\), the true mean for a population: \(A = (-13.05, 1.05)\) and \(B = (-15.16, 3.16)\). These intervals are standard, symmetric confidence intervals for the mean. One is a \(99\%\) confidence interval, while the other is a \(95\%\) confidence interval.

What is the sample mean?
What is the confidence level for \(A\)?
What is the confidence level for \(B\)?

(Question ID: 0360817529)

The following are confidence intervals for \(\mu\), the true mean for a population: \(A = (-90.44, -85.56)\) and \(B = (-90.58, -85.42)\). These intervals are standard, symmetric confidence intervals for the mean. One is a \(95\%\) confidence interval, while the other is a \(94\%\) confidence interval.

What is the sample mean?
What is the confidence level for \(A\)?
What is the confidence level for \(B\)?

(Question ID: 0979790030)

The following are confidence intervals for \(\mu\), the true mean for a population: \(A = (62.39, 79.61)\) and \(B = (69.59, 72.41)\). These intervals are standard, symmetric confidence intervals for the mean. One is a \(92\%\) confidence interval, while the other is an \(82\%\) confidence interval.

What is the sample mean?
What is the confidence level for \(A\)?
What is the confidence level for \(B\)?

(Question ID: 0800045695)

The following are confidence intervals for \(\mu\), the true mean for a population: \(A = (-110.27, -103.73)\) and \(B = (-110.72, -103.28)\). These intervals are standard, symmetric confidence intervals for the mean. One is a \(99\%\) confidence interval, while the other is an \(83\%\) confidence interval.

What is the sample mean?
What is the confidence level for \(A\)?
What is the confidence level for \(B\)?

(Question ID: 0323323644)

The following are confidence intervals for \(\mu\), the true mean for a population: \(A = (124.23, 127.77)\) and \(B = (116.66, 135.34)\). These intervals are standard, symmetric confidence intervals for the mean. One is an \(89\%\) confidence interval, while the other is an \(84\%\) confidence interval.

What is the sample mean?
What is the confidence level for \(A\)?
What is the confidence level for \(B\)?

(Question ID: 0362526296)

The following are confidence intervals for \(\mu\), the true mean for a population: \(A = (-92.44, -73.56)\) and \(B = (-88.77, -77.23)\). These intervals are standard, symmetric confidence intervals for the mean. One is an \(87\%\) confidence interval, while the other is an \(82\%\) confidence interval.

What is the sample mean?
What is the confidence level for \(A\)?
What is the confidence level for \(B\)?

(Question ID: 0222888025)

The following are confidence intervals for \(\mu\), the true mean for a population: \(A = (-15.63, -2.37)\) and \(B = (-16.89, -1.11)\). These intervals are standard, symmetric confidence intervals for the mean. One is a \(99\%\) confidence interval, while the other is a \(94\%\) confidence interval.

What is the sample mean?
What is the confidence level for \(A\)?
What is the confidence level for \(B\)?

(Question ID: 0211916989)

The following are confidence intervals for \(\mu\), the true mean for a population: \(A = (-149.27, -138.73)\) and \(B = (-153.72, -134.28)\). These intervals are standard, symmetric confidence intervals for the mean. One is a \(93\%\) confidence interval, while the other is an \(85\%\) confidence interval.

What is the sample mean?
What is the confidence level for \(A\)?
What is the confidence level for \(B\)?

(Question ID: 0336431613)

The following are confidence intervals for \(\mu\), the true mean for a population: \(A = (-28.11, -25.89)\) and \(B = (-31.51, -22.49)\). These intervals are standard, symmetric confidence intervals for the mean. One is a \(91\%\) confidence interval, while the other is an \(88\%\) confidence interval.

What is the sample mean?
What is the confidence level for \(A\)?
What is the confidence level for \(B\)?

(Question ID: 0469527961)

The following are confidence intervals for \(\mu\), the true mean for a population: \(A = (-154, -150)\) and \(B = (-160.52, -143.48)\). These intervals are standard, symmetric confidence intervals for the mean. One is a \(96\%\) confidence interval, while the other is a \(90\%\) confidence interval.

What is the sample mean?
What is the confidence level for \(A\)?
What is the confidence level for \(B\)?

(Question ID: 0783212833)

The following are confidence intervals for \(\mu\), the true mean for a population: \(A = (129.82, 136.18)\) and \(B = (123.79, 142.21)\). These intervals are standard, symmetric confidence intervals for the mean. One is a \(97\%\) confidence interval, while the other is a \(95\%\) confidence interval.

What is the sample mean?
What is the confidence level for \(A\)?
What is the confidence level for \(B\)?

(Question ID: 0922031085)

The following are confidence intervals for \(\mu\), the true mean for a population: \(A = (-101.92, -94.08)\) and \(B = (-102.72, -93.28)\). These intervals are standard, symmetric confidence intervals for the mean. One is a \(98\%\) confidence interval, while the other is a \(93\%\) confidence interval.

What is the sample mean?
What is the confidence level for \(A\)?
What is the confidence level for \(B\)?

(Question ID: 0817157071)

The following are confidence intervals for \(\mu\), the true mean for a population: \(A = (-52.5, -43.5)\) and \(B = (-57.04, -38.96)\). These intervals are standard, symmetric confidence intervals for the mean. One is a \(98\%\) confidence interval, while the other is an \(87\%\) confidence interval.

What is the sample mean?
What is the confidence level for \(A\)?
What is the confidence level for \(B\)?

(Question ID: 0574181227)

The following are confidence intervals for \(\mu\), the true mean for a population: \(A = (174.24, 183.76)\) and \(B = (171.02, 186.98)\). These intervals are standard, symmetric confidence intervals for the mean. One is a \(90\%\) confidence interval, while the other is an \(87\%\) confidence interval.

What is the sample mean?
What is the confidence level for \(A\)?
What is the confidence level for \(B\)?

(Question ID: 0888263175)

The following are confidence intervals for \(\mu\), the true mean for a population: \(A = (-137.86, -122.14)\) and \(B = (-138.59, -121.41)\). These intervals are standard, symmetric confidence intervals for the mean. One is a \(96\%\) confidence interval, while the other is a \(95\%\) confidence interval.

What is the sample mean?
What is the confidence level for \(A\)?
What is the confidence level for \(B\)?

(Question ID: 0609965474)

The following are confidence intervals for \(\mu\), the true mean for a population: \(A = (59.91, 76.09)\) and \(B = (64.91, 71.09)\). These intervals are standard, symmetric confidence intervals for the mean. One is a \(94\%\) confidence interval, while the other is a \(93\%\) confidence interval.

What is the sample mean?
What is the confidence level for \(A\)?
What is the confidence level for \(B\)?

(Question ID: 0521537717)

The following are confidence intervals for \(\mu\), the true mean for a population: \(A = (-197.17, -182.83)\) and \(B = (-199.62, -180.38)\). These intervals are standard, symmetric confidence intervals for the mean. One is a \(98\%\) confidence interval, while the other is an \(80\%\) confidence interval.

What is the sample mean?
What is the confidence level for \(A\)?
What is the confidence level for \(B\)?

(Question ID: 0551262246)

The following are confidence intervals for \(\mu\), the true mean for a population: \(A = (-39.98, -30.02)\) and \(B = (-44.75, -25.25)\). These intervals are standard, symmetric confidence intervals for the mean. One is a \(91\%\) confidence interval, while the other is an \(88\%\) confidence interval.

What is the sample mean?
What is the confidence level for \(A\)?
What is the confidence level for \(B\)?

(Question ID: 0796216662)

The following are confidence intervals for \(\mu\), the true mean for a population: \(A = (83.83, 86.17)\) and \(B = (81.73, 88.27)\). These intervals are standard, symmetric confidence intervals for the mean. One is a \(92\%\) confidence interval, while the other is a \(90\%\) confidence interval.

What is the sample mean?
What is the confidence level for \(A\)?
What is the confidence level for \(B\)?

(Question ID: 0270748397)

The following are confidence intervals for \(\mu\), the true mean for a population: \(A = (-118.26, -113.74)\) and \(B = (-121.56, -110.44)\). These intervals are standard, symmetric confidence intervals for the mean. One is a \(94\%\) confidence interval, while the other is an \(80\%\) confidence interval.

What is the sample mean?
What is the confidence level for \(A\)?
What is the confidence level for \(B\)?

(Question ID: 0385306859)

The following are confidence intervals for \(\mu\), the true mean for a population: \(A = (189, 205)\) and \(B = (195.89, 198.11)\). These intervals are standard, symmetric confidence intervals for the mean. One is a \(98\%\) confidence interval, while the other is a \(95\%\) confidence interval.

What is the sample mean?
What is the confidence level for \(A\)?
What is the confidence level for \(B\)?

(Question ID: 0666284501)

The following are confidence intervals for \(\mu\), the true mean for a population: \(A = (-7.75, 5.75)\) and \(B = (-5.71, 3.71)\). These intervals are standard, symmetric confidence intervals for the mean. One is a \(99\%\) confidence interval, while the other is a \(95\%\) confidence interval.

What is the sample mean?
What is the confidence level for \(A\)?
What is the confidence level for \(B\)?

(Question ID: 0735859733)

Self Assessment 14.2

Suppose that an \(89\%\) confidence interval is going to be formed for some parameter, \(\theta\). The interval estimator is expressed as \((\widehat{L}, \widehat{U})\).

Prior to taking the sample, what is the probability that \(\theta\) is contained in \((\widehat{L}, \widehat{U})\) ?
Suppose that \(\theta = 5.16\). If the interval is computed as \((-5.89, 3.09)\), what is the probability that the interval contains \(\theta\)?

(Question ID: 0999267168)

Suppose that an \(80\%\) confidence interval is going to be formed for some parameter, \(\theta\). The interval estimator is expressed as \((\widehat{L}, \widehat{U})\).

Prior to taking the sample, what is the probability that \(\theta\) is contained in \((\widehat{L}, \widehat{U})\) ?
Suppose that \(\theta = -9.74\). If the interval is computed as \((-19.43, -10.36)\), what is the probability that the interval contains \(\theta\)?

(Question ID: 0462598950)

Suppose that a \(94\%\) confidence interval is going to be formed for some parameter, \(\theta\). The interval estimator is expressed as \((\widehat{L}, \widehat{U})\).

Prior to taking the sample, what is the probability that \(\theta\) is contained in \((\widehat{L}, \widehat{U})\) ?
Suppose that \(\theta = 8.96\). If the interval is computed as \((-2.04, 7.41)\), what is the probability that the interval contains \(\theta\)?

(Question ID: 0942255388)

Suppose that a \(71\%\) confidence interval is going to be formed for some parameter, \(\theta\). The interval estimator is expressed as \((\widehat{L}, \widehat{U})\).

Prior to taking the sample, what is the probability that \(\theta\) is contained in \((\widehat{L}, \widehat{U})\) ?
Suppose that \(\theta = 1.89\). If the interval is computed as \((-5.51, 0.86)\), what is the probability that the interval contains \(\theta\)?

(Question ID: 0072294476)

Suppose that a \(75\%\) confidence interval is going to be formed for some parameter, \(\theta\). The interval estimator is expressed as \((\widehat{L}, \widehat{U})\).

Prior to taking the sample, what is the probability that \(\theta\) is contained in \((\widehat{L}, \widehat{U})\) ?
Suppose that \(\theta = 6.37\). If the interval is computed as \((4.01, 9.73)\), what is the probability that the interval contains \(\theta\)?

(Question ID: 0702783359)

Suppose that an \(86\%\) confidence interval is going to be formed for some parameter, \(\theta\). The interval estimator is expressed as \((\widehat{L}, \widehat{U})\).

Prior to taking the sample, what is the probability that \(\theta\) is contained in \((\widehat{L}, \widehat{U})\) ?
Suppose that \(\theta = -0.07\). If the interval is computed as \((-2.28, -0.64)\), what is the probability that the interval contains \(\theta\)?

(Question ID: 0299464345)

Suppose that a \(76\%\) confidence interval is going to be formed for some parameter, \(\theta\). The interval estimator is expressed as \((\widehat{L}, \widehat{U})\).

Prior to taking the sample, what is the probability that \(\theta\) is contained in \((\widehat{L}, \widehat{U})\) ?
Suppose that \(\theta = -5.95\). If the interval is computed as \((-7.96, 0.99)\), what is the probability that the interval contains \(\theta\)?

(Question ID: 0148401362)

Suppose that a \(65\%\) confidence interval is going to be formed for some parameter, \(\theta\). The interval estimator is expressed as \((\widehat{L}, \widehat{U})\).

Prior to taking the sample, what is the probability that \(\theta\) is contained in \((\widehat{L}, \widehat{U})\) ?
Suppose that \(\theta = -5.65\). If the interval is computed as \((-6.76, -2.69)\), what is the probability that the interval contains \(\theta\)?

(Question ID: 0088140347)

Suppose that an \(83\%\) confidence interval is going to be formed for some parameter, \(\theta\). The interval estimator is expressed as \((\widehat{L}, \widehat{U})\).

Prior to taking the sample, what is the probability that \(\theta\) is contained in \((\widehat{L}, \widehat{U})\) ?
Suppose that \(\theta = 4.44\). If the interval is computed as \((-3.75, 3.43)\), what is the probability that the interval contains \(\theta\)?

(Question ID: 0116559057)

Suppose that an \(86\%\) confidence interval is going to be formed for some parameter, \(\theta\). The interval estimator is expressed as \((\widehat{L}, \widehat{U})\).

Prior to taking the sample, what is the probability that \(\theta\) is contained in \((\widehat{L}, \widehat{U})\) ?
Suppose that \(\theta = 5.18\). If the interval is computed as \((3.85, 10.8)\), what is the probability that the interval contains \(\theta\)?

(Question ID: 0441143437)

Suppose that an \(86\%\) confidence interval is going to be formed for some parameter, \(\theta\). The interval estimator is expressed as \((\widehat{L}, \widehat{U})\).

Prior to taking the sample, what is the probability that \(\theta\) is contained in \((\widehat{L}, \widehat{U})\) ?
Suppose that \(\theta = 7.15\). If the interval is computed as \((5.71, 9.37)\), what is the probability that the interval contains \(\theta\)?

(Question ID: 0033288154)

Suppose that a \(65\%\) confidence interval is going to be formed for some parameter, \(\theta\). The interval estimator is expressed as \((\widehat{L}, \widehat{U})\).

Prior to taking the sample, what is the probability that \(\theta\) is contained in \((\widehat{L}, \widehat{U})\) ?
Suppose that \(\theta = -6.54\). If the interval is computed as \((-14.39, -7.52)\), what is the probability that the interval contains \(\theta\)?

(Question ID: 0023282731)

Suppose that an \(87\%\) confidence interval is going to be formed for some parameter, \(\theta\). The interval estimator is expressed as \((\widehat{L}, \widehat{U})\).

Prior to taking the sample, what is the probability that \(\theta\) is contained in \((\widehat{L}, \widehat{U})\) ?
Suppose that \(\theta = -7.35\). If the interval is computed as \((-18.48, -9.82)\), what is the probability that the interval contains \(\theta\)?

(Question ID: 0530341210)

Suppose that an \(80\%\) confidence interval is going to be formed for some parameter, \(\theta\). The interval estimator is expressed as \((\widehat{L}, \widehat{U})\).

Prior to taking the sample, what is the probability that \(\theta\) is contained in \((\widehat{L}, \widehat{U})\) ?
Suppose that \(\theta = -0.37\). If the interval is computed as \((-11.14, -1.6)\), what is the probability that the interval contains \(\theta\)?

(Question ID: 0742870365)

Suppose that a \(91\%\) confidence interval is going to be formed for some parameter, \(\theta\). The interval estimator is expressed as \((\widehat{L}, \widehat{U})\).

Prior to taking the sample, what is the probability that \(\theta\) is contained in \((\widehat{L}, \widehat{U})\) ?
Suppose that \(\theta = -4.28\). If the interval is computed as \((-12.34, -6.09)\), what is the probability that the interval contains \(\theta\)?

(Question ID: 0155470949)

Suppose that a \(65\%\) confidence interval is going to be formed for some parameter, \(\theta\). The interval estimator is expressed as \((\widehat{L}, \widehat{U})\).

Prior to taking the sample, what is the probability that \(\theta\) is contained in \((\widehat{L}, \widehat{U})\) ?
Suppose that \(\theta = 3.13\). If the interval is computed as \((-8.16, 0.8)\), what is the probability that the interval contains \(\theta\)?

(Question ID: 0660815617)

Suppose that a \(60\%\) confidence interval is going to be formed for some parameter, \(\theta\). The interval estimator is expressed as \((\widehat{L}, \widehat{U})\).

Prior to taking the sample, what is the probability that \(\theta\) is contained in \((\widehat{L}, \widehat{U})\) ?
Suppose that \(\theta = -7.81\). If the interval is computed as \((-8.82, 0.75)\), what is the probability that the interval contains \(\theta\)?

(Question ID: 0213078827)

Suppose that an \(81\%\) confidence interval is going to be formed for some parameter, \(\theta\). The interval estimator is expressed as \((\widehat{L}, \widehat{U})\).

Prior to taking the sample, what is the probability that \(\theta\) is contained in \((\widehat{L}, \widehat{U})\) ?
Suppose that \(\theta = 8.51\). If the interval is computed as \((1.75, 7.09)\), what is the probability that the interval contains \(\theta\)?

(Question ID: 0730322196)

Suppose that a \(67\%\) confidence interval is going to be formed for some parameter, \(\theta\). The interval estimator is expressed as \((\widehat{L}, \widehat{U})\).

Prior to taking the sample, what is the probability that \(\theta\) is contained in \((\widehat{L}, \widehat{U})\) ?
Suppose that \(\theta = 7.38\). If the interval is computed as \((0.27, 5.6)\), what is the probability that the interval contains \(\theta\)?

(Question ID: 0023800386)

Suppose that a \(73\%\) confidence interval is going to be formed for some parameter, \(\theta\). The interval estimator is expressed as \((\widehat{L}, \widehat{U})\).

Prior to taking the sample, what is the probability that \(\theta\) is contained in \((\widehat{L}, \widehat{U})\) ?
Suppose that \(\theta = -2.86\). If the interval is computed as \((-4, 4.27)\), what is the probability that the interval contains \(\theta\)?

(Question ID: 0247609104)

Suppose that an \(80\%\) confidence interval is going to be formed for some parameter, \(\theta\). The interval estimator is expressed as \((\widehat{L}, \widehat{U})\).

Prior to taking the sample, what is the probability that \(\theta\) is contained in \((\widehat{L}, \widehat{U})\) ?
Suppose that \(\theta = -2.47\). If the interval is computed as \((-10.34, -3.83)\), what is the probability that the interval contains \(\theta\)?

(Question ID: 0971156526)

Suppose that an \(84\%\) confidence interval is going to be formed for some parameter, \(\theta\). The interval estimator is expressed as \((\widehat{L}, \widehat{U})\).

Prior to taking the sample, what is the probability that \(\theta\) is contained in \((\widehat{L}, \widehat{U})\) ?
Suppose that \(\theta = -5.9\). If the interval is computed as \((-6.72, -4.59)\), what is the probability that the interval contains \(\theta\)?

(Question ID: 0953594570)

Suppose that a \(94\%\) confidence interval is going to be formed for some parameter, \(\theta\). The interval estimator is expressed as \((\widehat{L}, \widehat{U})\).

Prior to taking the sample, what is the probability that \(\theta\) is contained in \((\widehat{L}, \widehat{U})\) ?
Suppose that \(\theta = -1.47\). If the interval is computed as \((-9.87, -3.47)\), what is the probability that the interval contains \(\theta\)?

(Question ID: 0587644042)

Suppose that an \(81\%\) confidence interval is going to be formed for some parameter, \(\theta\). The interval estimator is expressed as \((\widehat{L}, \widehat{U})\).

Prior to taking the sample, what is the probability that \(\theta\) is contained in \((\widehat{L}, \widehat{U})\) ?
Suppose that \(\theta = -3.64\). If the interval is computed as \((-14.61, -4.79)\), what is the probability that the interval contains \(\theta\)?

(Question ID: 0581104406)

Suppose that a \(74\%\) confidence interval is going to be formed for some parameter, \(\theta\). The interval estimator is expressed as \((\widehat{L}, \widehat{U})\).

Prior to taking the sample, what is the probability that \(\theta\) is contained in \((\widehat{L}, \widehat{U})\) ?
Suppose that \(\theta = -2.68\). If the interval is computed as \((-9.81, -3.26)\), what is the probability that the interval contains \(\theta\)?

(Question ID: 0065053724)

Suppose that an \(81\%\) confidence interval is going to be formed for some parameter, \(\theta\). The interval estimator is expressed as \((\widehat{L}, \widehat{U})\).

Prior to taking the sample, what is the probability that \(\theta\) is contained in \((\widehat{L}, \widehat{U})\) ?
Suppose that \(\theta = -6.43\). If the interval is computed as \((-8.32, -1.67)\), what is the probability that the interval contains \(\theta\)?

(Question ID: 0213054064)

Suppose that a \(78\%\) confidence interval is going to be formed for some parameter, \(\theta\). The interval estimator is expressed as \((\widehat{L}, \widehat{U})\).

Prior to taking the sample, what is the probability that \(\theta\) is contained in \((\widehat{L}, \widehat{U})\) ?
Suppose that \(\theta = -2.01\). If the interval is computed as \((-8.29, -3.23)\), what is the probability that the interval contains \(\theta\)?

(Question ID: 0458877327)

Suppose that a \(97\%\) confidence interval is going to be formed for some parameter, \(\theta\). The interval estimator is expressed as \((\widehat{L}, \widehat{U})\).

Prior to taking the sample, what is the probability that \(\theta\) is contained in \((\widehat{L}, \widehat{U})\) ?
Suppose that \(\theta = 5.62\). If the interval is computed as \((3.64, 5.06)\), what is the probability that the interval contains \(\theta\)?

(Question ID: 0938548562)

Suppose that a \(70\%\) confidence interval is going to be formed for some parameter, \(\theta\). The interval estimator is expressed as \((\widehat{L}, \widehat{U})\).

Prior to taking the sample, what is the probability that \(\theta\) is contained in \((\widehat{L}, \widehat{U})\) ?
Suppose that \(\theta = 8.75\). If the interval is computed as \((2.03, 7.95)\), what is the probability that the interval contains \(\theta\)?

(Question ID: 0018582521)

Suppose that a \(94\%\) confidence interval is going to be formed for some parameter, \(\theta\). The interval estimator is expressed as \((\widehat{L}, \widehat{U})\).

Prior to taking the sample, what is the probability that \(\theta\) is contained in \((\widehat{L}, \widehat{U})\) ?
Suppose that \(\theta = -3\). If the interval is computed as \((-12.56, -4.92)\), what is the probability that the interval contains \(\theta\)?

(Question ID: 0291401938)

Suppose that a \(67\%\) confidence interval is going to be formed for some parameter, \(\theta\). The interval estimator is expressed as \((\widehat{L}, \widehat{U})\).

Prior to taking the sample, what is the probability that \(\theta\) is contained in \((\widehat{L}, \widehat{U})\) ?
Suppose that \(\theta = 4.46\). If the interval is computed as \((-4.94, 3.23)\), what is the probability that the interval contains \(\theta\)?

(Question ID: 0430688812)

Suppose that a \(75\%\) confidence interval is going to be formed for some parameter, \(\theta\). The interval estimator is expressed as \((\widehat{L}, \widehat{U})\).

Prior to taking the sample, what is the probability that \(\theta\) is contained in \((\widehat{L}, \widehat{U})\) ?
Suppose that \(\theta = 3.03\). If the interval is computed as \((1.95, 6.48)\), what is the probability that the interval contains \(\theta\)?

(Question ID: 0637359245)

Suppose that an \(89\%\) confidence interval is going to be formed for some parameter, \(\theta\). The interval estimator is expressed as \((\widehat{L}, \widehat{U})\).

Prior to taking the sample, what is the probability that \(\theta\) is contained in \((\widehat{L}, \widehat{U})\) ?
Suppose that \(\theta = 0.16\). If the interval is computed as \((-1.08, 4.86)\), what is the probability that the interval contains \(\theta\)?

(Question ID: 0203039819)

Suppose that a \(71\%\) confidence interval is going to be formed for some parameter, \(\theta\). The interval estimator is expressed as \((\widehat{L}, \widehat{U})\).

Prior to taking the sample, what is the probability that \(\theta\) is contained in \((\widehat{L}, \widehat{U})\) ?
Suppose that \(\theta = -2.88\). If the interval is computed as \((-12.44, -4.51)\), what is the probability that the interval contains \(\theta\)?

(Question ID: 0274299685)

Suppose that a \(94\%\) confidence interval is going to be formed for some parameter, \(\theta\). The interval estimator is expressed as \((\widehat{L}, \widehat{U})\).

Prior to taking the sample, what is the probability that \(\theta\) is contained in \((\widehat{L}, \widehat{U})\) ?
Suppose that \(\theta = -5.99\). If the interval is computed as \((-11.88, -6.84)\), what is the probability that the interval contains \(\theta\)?

(Question ID: 0108732007)

Suppose that a \(63\%\) confidence interval is going to be formed for some parameter, \(\theta\). The interval estimator is expressed as \((\widehat{L}, \widehat{U})\).

Prior to taking the sample, what is the probability that \(\theta\) is contained in \((\widehat{L}, \widehat{U})\) ?
Suppose that \(\theta = 2.37\). If the interval is computed as \((-8.48, -0.02)\), what is the probability that the interval contains \(\theta\)?

(Question ID: 0901291310)

Suppose that a \(63\%\) confidence interval is going to be formed for some parameter, \(\theta\). The interval estimator is expressed as \((\widehat{L}, \widehat{U})\).

Prior to taking the sample, what is the probability that \(\theta\) is contained in \((\widehat{L}, \widehat{U})\) ?
Suppose that \(\theta = 9.66\). If the interval is computed as \((7.45, 12.05)\), what is the probability that the interval contains \(\theta\)?

(Question ID: 0116568257)

Suppose that a \(79\%\) confidence interval is going to be formed for some parameter, \(\theta\). The interval estimator is expressed as \((\widehat{L}, \widehat{U})\).

Prior to taking the sample, what is the probability that \(\theta\) is contained in \((\widehat{L}, \widehat{U})\) ?
Suppose that \(\theta = 6.38\). If the interval is computed as \((-3.42, 5.08)\), what is the probability that the interval contains \(\theta\)?

(Question ID: 0412403743)

Suppose that an \(81\%\) confidence interval is going to be formed for some parameter, \(\theta\). The interval estimator is expressed as \((\widehat{L}, \widehat{U})\).

Prior to taking the sample, what is the probability that \(\theta\) is contained in \((\widehat{L}, \widehat{U})\) ?
Suppose that \(\theta = 1.41\). If the interval is computed as \((-4.37, 0.75)\), what is the probability that the interval contains \(\theta\)?

(Question ID: 0015281139)

Suppose that a \(74\%\) confidence interval is going to be formed for some parameter, \(\theta\). The interval estimator is expressed as \((\widehat{L}, \widehat{U})\).

Prior to taking the sample, what is the probability that \(\theta\) is contained in \((\widehat{L}, \widehat{U})\) ?
Suppose that \(\theta = -1.95\). If the interval is computed as \((-9.2, -2.91)\), what is the probability that the interval contains \(\theta\)?

(Question ID: 0781574857)

Suppose that a \(72\%\) confidence interval is going to be formed for some parameter, \(\theta\). The interval estimator is expressed as \((\widehat{L}, \widehat{U})\).

Prior to taking the sample, what is the probability that \(\theta\) is contained in \((\widehat{L}, \widehat{U})\) ?
Suppose that \(\theta = 8.88\). If the interval is computed as \((-0.51, 7.75)\), what is the probability that the interval contains \(\theta\)?

(Question ID: 0880575605)

Suppose that an \(83\%\) confidence interval is going to be formed for some parameter, \(\theta\). The interval estimator is expressed as \((\widehat{L}, \widehat{U})\).

Prior to taking the sample, what is the probability that \(\theta\) is contained in \((\widehat{L}, \widehat{U})\) ?
Suppose that \(\theta = 6.76\). If the interval is computed as \((4.59, 10.75)\), what is the probability that the interval contains \(\theta\)?

(Question ID: 0105771626)

Suppose that an \(84\%\) confidence interval is going to be formed for some parameter, \(\theta\). The interval estimator is expressed as \((\widehat{L}, \widehat{U})\).

Prior to taking the sample, what is the probability that \(\theta\) is contained in \((\widehat{L}, \widehat{U})\) ?
Suppose that \(\theta = 5.39\). If the interval is computed as \((4.05, 10.01)\), what is the probability that the interval contains \(\theta\)?

(Question ID: 0892360035)

Suppose that a \(79\%\) confidence interval is going to be formed for some parameter, \(\theta\). The interval estimator is expressed as \((\widehat{L}, \widehat{U})\).

Prior to taking the sample, what is the probability that \(\theta\) is contained in \((\widehat{L}, \widehat{U})\) ?
Suppose that \(\theta = -3.69\). If the interval is computed as \((-13.46, -5.3)\), what is the probability that the interval contains \(\theta\)?

(Question ID: 0076486700)

Suppose that a \(79\%\) confidence interval is going to be formed for some parameter, \(\theta\). The interval estimator is expressed as \((\widehat{L}, \widehat{U})\).

Prior to taking the sample, what is the probability that \(\theta\) is contained in \((\widehat{L}, \widehat{U})\) ?
Suppose that \(\theta = -4.19\). If the interval is computed as \((-4.86, 3.43)\), what is the probability that the interval contains \(\theta\)?

(Question ID: 0187172686)

Suppose that an \(81\%\) confidence interval is going to be formed for some parameter, \(\theta\). The interval estimator is expressed as \((\widehat{L}, \widehat{U})\).

Prior to taking the sample, what is the probability that \(\theta\) is contained in \((\widehat{L}, \widehat{U})\) ?
Suppose that \(\theta = -7.35\). If the interval is computed as \((-14.68, -8.59)\), what is the probability that the interval contains \(\theta\)?

(Question ID: 0271785822)

Suppose that a \(74\%\) confidence interval is going to be formed for some parameter, \(\theta\). The interval estimator is expressed as \((\widehat{L}, \widehat{U})\).

Prior to taking the sample, what is the probability that \(\theta\) is contained in \((\widehat{L}, \widehat{U})\) ?
Suppose that \(\theta = 6.84\). If the interval is computed as \((1.91, 6.18)\), what is the probability that the interval contains \(\theta\)?

(Question ID: 0001361110)

Suppose that a \(78\%\) confidence interval is going to be formed for some parameter, \(\theta\). The interval estimator is expressed as \((\widehat{L}, \widehat{U})\).

Prior to taking the sample, what is the probability that \(\theta\) is contained in \((\widehat{L}, \widehat{U})\) ?
Suppose that \(\theta = 2.55\). If the interval is computed as \((-3.48, 1.43)\), what is the probability that the interval contains \(\theta\)?

(Question ID: 0629411982)

Suppose that a \(66\%\) confidence interval is going to be formed for some parameter, \(\theta\). The interval estimator is expressed as \((\widehat{L}, \widehat{U})\).

Prior to taking the sample, what is the probability that \(\theta\) is contained in \((\widehat{L}, \widehat{U})\) ?
Suppose that \(\theta = 5.62\). If the interval is computed as \((4.72, 11.19)\), what is the probability that the interval contains \(\theta\)?

(Question ID: 0378150907)

Suppose that an \(88\%\) confidence interval is going to be formed for some parameter, \(\theta\). The interval estimator is expressed as \((\widehat{L}, \widehat{U})\).

Prior to taking the sample, what is the probability that \(\theta\) is contained in \((\widehat{L}, \widehat{U})\) ?
Suppose that \(\theta = 8.31\). If the interval is computed as \((6.31, 14.02)\), what is the probability that the interval contains \(\theta\)?

(Question ID: 0168145057)

Self Assessment 14.3

Consider forming a confidence interval for a population mean. Suppose that the confidence interval is computed based on a sample of size 93, at the \(80\%\) level of confidence.

Suppose that a second confidence interval is computed, using the same known population variance, at the same level of confidence, with a sample size that is 8.6 times as large as the original sample. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same estimated variance, at the same level of confidence, with a sample size of 208. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same known population variance, with the same sample size, but with a confidence level of \(75\%\). What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same sample size and confidence level, but with an estimated variance that is 3.1 times as large as the original. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same estimated variance, the same sample size, and the same confidence level, but where the sample mean is 899.6 higher than the initial sample mean. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).

(Question ID: 0083330920)

Consider forming a confidence interval for a population mean. Suppose that the confidence interval is computed based on a sample of size 60, at the \(76\%\) level of confidence.

Suppose that a second confidence interval is computed, using the same known population variance, at the same level of confidence, with a sample size that is 2.8 times as large as the original sample. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same estimated variance, at the same level of confidence, with a sample size of 180. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same known population variance, with the same sample size, but with a confidence level of \(84\%\). What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same sample size and confidence level, but with an estimated variance that is 3.3 times as large as the original. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same estimated variance, the same sample size, and the same confidence level, but where the sample mean is 173.6 higher than the initial sample mean. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).

(Question ID: 0072958506)

Consider forming a confidence interval for a population mean. Suppose that the confidence interval is computed based on a sample of size 44, at the \(97\%\) level of confidence.

Suppose that a second confidence interval is computed, using the same known population variance, at the same level of confidence, with a sample size that is 8.8 times as large as the original sample. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same estimated variance, at the same level of confidence, with a sample size of 133. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same known population variance, with the same sample size, but with a confidence level of \(73\%\). What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same sample size and confidence level, but with an estimated variance that is 3.5 times as large as the original. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same estimated variance, the same sample size, and the same confidence level, but where the sample mean is 430.2 higher than the initial sample mean. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).

(Question ID: 0835551870)

Consider forming a confidence interval for a population mean. Suppose that the confidence interval is computed based on a sample of size 38, at the \(96\%\) level of confidence.

Suppose that a second confidence interval is computed, using the same known population variance, at the same level of confidence, with a sample size that is 6.2 times as large as the original sample. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same estimated variance, at the same level of confidence, with a sample size of 430. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same known population variance, with the same sample size, but with a confidence level of \(89\%\). What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same sample size and confidence level, but with an estimated variance that is 9.2 times as large as the original. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same estimated variance, the same sample size, and the same confidence level, but where the sample mean is 972.2 higher than the initial sample mean. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).

(Question ID: 0571907673)

Consider forming a confidence interval for a population mean. Suppose that the confidence interval is computed based on a sample of size 91, at the \(72\%\) level of confidence.

Suppose that a second confidence interval is computed, using the same known population variance, at the same level of confidence, with a sample size that is 7 times as large as the original sample. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same estimated variance, at the same level of confidence, with a sample size of 310. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same known population variance, with the same sample size, but with a confidence level of \(93\%\). What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same sample size and confidence level, but with an estimated variance that is 8.8 times as large as the original. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same estimated variance, the same sample size, and the same confidence level, but where the sample mean is 635.7 higher than the initial sample mean. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).

(Question ID: 0000373466)

Consider forming a confidence interval for a population mean. Suppose that the confidence interval is computed based on a sample of size 72, at the \(95\%\) level of confidence.

Suppose that a second confidence interval is computed, using the same known population variance, at the same level of confidence, with a sample size that is 9.7 times as large as the original sample. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same estimated variance, at the same level of confidence, with a sample size of 147. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same known population variance, with the same sample size, but with a confidence level of \(75\%\). What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same sample size and confidence level, but with an estimated variance that is 9.2 times as large as the original. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same estimated variance, the same sample size, and the same confidence level, but where the sample mean is 825.6 higher than the initial sample mean. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).

(Question ID: 0113295907)

Consider forming a confidence interval for a population mean. Suppose that the confidence interval is computed based on a sample of size 88, at the \(79\%\) level of confidence.

Suppose that a second confidence interval is computed, using the same known population variance, at the same level of confidence, with a sample size that is 8.2 times as large as the original sample. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same estimated variance, at the same level of confidence, with a sample size of 317. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same known population variance, with the same sample size, but with a confidence level of \(91\%\). What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same sample size and confidence level, but with an estimated variance that is 5.6 times as large as the original. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same estimated variance, the same sample size, and the same confidence level, but where the sample mean is 52 higher than the initial sample mean. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).

(Question ID: 0920114525)

Consider forming a confidence interval for a population mean. Suppose that the confidence interval is computed based on a sample of size 56, at the \(86\%\) level of confidence.

Suppose that a second confidence interval is computed, using the same known population variance, at the same level of confidence, with a sample size that is 1.9 times as large as the original sample. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same estimated variance, at the same level of confidence, with a sample size of 163. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same known population variance, with the same sample size, but with a confidence level of \(77\%\). What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same sample size and confidence level, but with an estimated variance that is 5.6 times as large as the original. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same estimated variance, the same sample size, and the same confidence level, but where the sample mean is 247.6 higher than the initial sample mean. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).

(Question ID: 0166548917)

Consider forming a confidence interval for a population mean. Suppose that the confidence interval is computed based on a sample of size 43, at the \(91\%\) level of confidence.

Suppose that a second confidence interval is computed, using the same known population variance, at the same level of confidence, with a sample size that is 3.8 times as large as the original sample. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same estimated variance, at the same level of confidence, with a sample size of 321. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same known population variance, with the same sample size, but with a confidence level of \(71\%\). What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same sample size and confidence level, but with an estimated variance that is 1.5 times as large as the original. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same estimated variance, the same sample size, and the same confidence level, but where the sample mean is 531 higher than the initial sample mean. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).

(Question ID: 0668047410)

Consider forming a confidence interval for a population mean. Suppose that the confidence interval is computed based on a sample of size 75, at the \(95\%\) level of confidence.

Suppose that a second confidence interval is computed, using the same known population variance, at the same level of confidence, with a sample size that is 9.7 times as large as the original sample. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same estimated variance, at the same level of confidence, with a sample size of 288. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same known population variance, with the same sample size, but with a confidence level of \(82\%\). What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same sample size and confidence level, but with an estimated variance that is 2.4 times as large as the original. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same estimated variance, the same sample size, and the same confidence level, but where the sample mean is 74 higher than the initial sample mean. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).

(Question ID: 0794606368)

Consider forming a confidence interval for a population mean. Suppose that the confidence interval is computed based on a sample of size 87, at the \(87\%\) level of confidence.

Suppose that a second confidence interval is computed, using the same known population variance, at the same level of confidence, with a sample size that is 9.3 times as large as the original sample. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same estimated variance, at the same level of confidence, with a sample size of 268. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same known population variance, with the same sample size, but with a confidence level of \(70\%\). What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same sample size and confidence level, but with an estimated variance that is 8.1 times as large as the original. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same estimated variance, the same sample size, and the same confidence level, but where the sample mean is 149.4 higher than the initial sample mean. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).

(Question ID: 0508199482)

Consider forming a confidence interval for a population mean. Suppose that the confidence interval is computed based on a sample of size 64, at the \(86\%\) level of confidence.

Suppose that a second confidence interval is computed, using the same known population variance, at the same level of confidence, with a sample size that is 4.4 times as large as the original sample. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same estimated variance, at the same level of confidence, with a sample size of 296. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same known population variance, with the same sample size, but with a confidence level of \(71\%\). What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same sample size and confidence level, but with an estimated variance that is 8.2 times as large as the original. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same estimated variance, the same sample size, and the same confidence level, but where the sample mean is 203.9 higher than the initial sample mean. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).

(Question ID: 0105950044)

Consider forming a confidence interval for a population mean. Suppose that the confidence interval is computed based on a sample of size 62, at the \(74\%\) level of confidence.

Suppose that a second confidence interval is computed, using the same known population variance, at the same level of confidence, with a sample size that is 6.7 times as large as the original sample. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same estimated variance, at the same level of confidence, with a sample size of 499. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same known population variance, with the same sample size, but with a confidence level of \(78\%\). What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same sample size and confidence level, but with an estimated variance that is 6.8 times as large as the original. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same estimated variance, the same sample size, and the same confidence level, but where the sample mean is 922.5 higher than the initial sample mean. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).

(Question ID: 0760890704)

Consider forming a confidence interval for a population mean. Suppose that the confidence interval is computed based on a sample of size 62, at the \(81\%\) level of confidence.

Suppose that a second confidence interval is computed, using the same known population variance, at the same level of confidence, with a sample size that is 4.2 times as large as the original sample. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same estimated variance, at the same level of confidence, with a sample size of 411. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same known population variance, with the same sample size, but with a confidence level of \(88\%\). What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same sample size and confidence level, but with an estimated variance that is 5.5 times as large as the original. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same estimated variance, the same sample size, and the same confidence level, but where the sample mean is 415.7 higher than the initial sample mean. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).

(Question ID: 0641008091)

Consider forming a confidence interval for a population mean. Suppose that the confidence interval is computed based on a sample of size 42, at the \(80\%\) level of confidence.

Suppose that a second confidence interval is computed, using the same known population variance, at the same level of confidence, with a sample size that is 9.6 times as large as the original sample. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same estimated variance, at the same level of confidence, with a sample size of 93. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same known population variance, with the same sample size, but with a confidence level of \(73\%\). What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same sample size and confidence level, but with an estimated variance that is 1.5 times as large as the original. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same estimated variance, the same sample size, and the same confidence level, but where the sample mean is 575.9 higher than the initial sample mean. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).

(Question ID: 0276074651)

Consider forming a confidence interval for a population mean. Suppose that the confidence interval is computed based on a sample of size 43, at the \(73\%\) level of confidence.

Suppose that a second confidence interval is computed, using the same known population variance, at the same level of confidence, with a sample size that is 7.9 times as large as the original sample. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same estimated variance, at the same level of confidence, with a sample size of 397. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same known population variance, with the same sample size, but with a confidence level of \(89\%\). What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same sample size and confidence level, but with an estimated variance that is 9.3 times as large as the original. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same estimated variance, the same sample size, and the same confidence level, but where the sample mean is 394.2 higher than the initial sample mean. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).

(Question ID: 0758220795)

Consider forming a confidence interval for a population mean. Suppose that the confidence interval is computed based on a sample of size 91, at the \(90\%\) level of confidence.

Suppose that a second confidence interval is computed, using the same known population variance, at the same level of confidence, with a sample size that is 3.9 times as large as the original sample. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same estimated variance, at the same level of confidence, with a sample size of 110. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same known population variance, with the same sample size, but with a confidence level of \(84\%\). What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same sample size and confidence level, but with an estimated variance that is 3.1 times as large as the original. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same estimated variance, the same sample size, and the same confidence level, but where the sample mean is 508 higher than the initial sample mean. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).

(Question ID: 0640213893)

Consider forming a confidence interval for a population mean. Suppose that the confidence interval is computed based on a sample of size 83, at the \(80\%\) level of confidence.

Suppose that a second confidence interval is computed, using the same known population variance, at the same level of confidence, with a sample size that is 6.9 times as large as the original sample. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same estimated variance, at the same level of confidence, with a sample size of 450. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same known population variance, with the same sample size, but with a confidence level of \(87\%\). What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same sample size and confidence level, but with an estimated variance that is 8.2 times as large as the original. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same estimated variance, the same sample size, and the same confidence level, but where the sample mean is 412.2 higher than the initial sample mean. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).

(Question ID: 0088516848)

Consider forming a confidence interval for a population mean. Suppose that the confidence interval is computed based on a sample of size 95, at the \(94\%\) level of confidence.

Suppose that a second confidence interval is computed, using the same known population variance, at the same level of confidence, with a sample size that is 1.3 times as large as the original sample. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same estimated variance, at the same level of confidence, with a sample size of 124. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same known population variance, with the same sample size, but with a confidence level of \(75\%\). What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same sample size and confidence level, but with an estimated variance that is 1.1 times as large as the original. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same estimated variance, the same sample size, and the same confidence level, but where the sample mean is 144.9 higher than the initial sample mean. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).

(Question ID: 0181585348)

Consider forming a confidence interval for a population mean. Suppose that the confidence interval is computed based on a sample of size 52, at the \(81\%\) level of confidence.

Suppose that a second confidence interval is computed, using the same known population variance, at the same level of confidence, with a sample size that is 2.4 times as large as the original sample. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same estimated variance, at the same level of confidence, with a sample size of 145. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same known population variance, with the same sample size, but with a confidence level of \(75\%\). What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same sample size and confidence level, but with an estimated variance that is 0.9 times as large as the original. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same estimated variance, the same sample size, and the same confidence level, but where the sample mean is 45.3 higher than the initial sample mean. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).

(Question ID: 0091934752)

Consider forming a confidence interval for a population mean. Suppose that the confidence interval is computed based on a sample of size 71, at the \(78\%\) level of confidence.

Suppose that a second confidence interval is computed, using the same known population variance, at the same level of confidence, with a sample size that is 5.1 times as large as the original sample. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same estimated variance, at the same level of confidence, with a sample size of 338. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same known population variance, with the same sample size, but with a confidence level of \(98\%\). What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same sample size and confidence level, but with an estimated variance that is 0.7 times as large as the original. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same estimated variance, the same sample size, and the same confidence level, but where the sample mean is 276 higher than the initial sample mean. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).

(Question ID: 0864520655)

Consider forming a confidence interval for a population mean. Suppose that the confidence interval is computed based on a sample of size 76, at the \(97\%\) level of confidence.

Suppose that a second confidence interval is computed, using the same known population variance, at the same level of confidence, with a sample size that is 10 times as large as the original sample. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same estimated variance, at the same level of confidence, with a sample size of 311. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same known population variance, with the same sample size, but with a confidence level of \(76\%\). What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same sample size and confidence level, but with an estimated variance that is 4.4 times as large as the original. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same estimated variance, the same sample size, and the same confidence level, but where the sample mean is 874.4 higher than the initial sample mean. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).

(Question ID: 0590630894)

Consider forming a confidence interval for a population mean. Suppose that the confidence interval is computed based on a sample of size 76, at the \(76\%\) level of confidence.

Suppose that a second confidence interval is computed, using the same known population variance, at the same level of confidence, with a sample size that is 4.3 times as large as the original sample. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same estimated variance, at the same level of confidence, with a sample size of 403. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same known population variance, with the same sample size, but with a confidence level of \(84\%\). What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same sample size and confidence level, but with an estimated variance that is 8.4 times as large as the original. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same estimated variance, the same sample size, and the same confidence level, but where the sample mean is 258.7 higher than the initial sample mean. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).

(Question ID: 0054461231)

Consider forming a confidence interval for a population mean. Suppose that the confidence interval is computed based on a sample of size 83, at the \(77\%\) level of confidence.

Suppose that a second confidence interval is computed, using the same known population variance, at the same level of confidence, with a sample size that is 3.7 times as large as the original sample. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same estimated variance, at the same level of confidence, with a sample size of 147. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same known population variance, with the same sample size, but with a confidence level of \(92\%\). What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same sample size and confidence level, but with an estimated variance that is 8.6 times as large as the original. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same estimated variance, the same sample size, and the same confidence level, but where the sample mean is 324.7 higher than the initial sample mean. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).

(Question ID: 0781142314)

Consider forming a confidence interval for a population mean. Suppose that the confidence interval is computed based on a sample of size 46, at the \(77\%\) level of confidence.

Suppose that a second confidence interval is computed, using the same known population variance, at the same level of confidence, with a sample size that is 6.6 times as large as the original sample. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same estimated variance, at the same level of confidence, with a sample size of 409. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same known population variance, with the same sample size, but with a confidence level of \(84\%\). What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same sample size and confidence level, but with an estimated variance that is 2.3 times as large as the original. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same estimated variance, the same sample size, and the same confidence level, but where the sample mean is 821.5 higher than the initial sample mean. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).

(Question ID: 0115054122)

Consider forming a confidence interval for a population mean. Suppose that the confidence interval is computed based on a sample of size 66, at the \(97\%\) level of confidence.

Suppose that a second confidence interval is computed, using the same known population variance, at the same level of confidence, with a sample size that is 8.4 times as large as the original sample. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same estimated variance, at the same level of confidence, with a sample size of 135. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same known population variance, with the same sample size, but with a confidence level of \(85\%\). What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same sample size and confidence level, but with an estimated variance that is 9 times as large as the original. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same estimated variance, the same sample size, and the same confidence level, but where the sample mean is 884.5 higher than the initial sample mean. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).

(Question ID: 0020206421)

Consider forming a confidence interval for a population mean. Suppose that the confidence interval is computed based on a sample of size 60, at the \(79\%\) level of confidence.

Suppose that a second confidence interval is computed, using the same known population variance, at the same level of confidence, with a sample size that is 5.4 times as large as the original sample. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same estimated variance, at the same level of confidence, with a sample size of 311. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same known population variance, with the same sample size, but with a confidence level of \(98\%\). What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same sample size and confidence level, but with an estimated variance that is 7.4 times as large as the original. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same estimated variance, the same sample size, and the same confidence level, but where the sample mean is 325.7 higher than the initial sample mean. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).

(Question ID: 0401650822)

Consider forming a confidence interval for a population mean. Suppose that the confidence interval is computed based on a sample of size 35, at the \(85\%\) level of confidence.

Suppose that a second confidence interval is computed, using the same known population variance, at the same level of confidence, with a sample size that is 4.7 times as large as the original sample. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same estimated variance, at the same level of confidence, with a sample size of 121. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same known population variance, with the same sample size, but with a confidence level of \(91\%\). What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same sample size and confidence level, but with an estimated variance that is 2.6 times as large as the original. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same estimated variance, the same sample size, and the same confidence level, but where the sample mean is 457.4 higher than the initial sample mean. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).

(Question ID: 0648484598)

Consider forming a confidence interval for a population mean. Suppose that the confidence interval is computed based on a sample of size 61, at the \(76\%\) level of confidence.

Suppose that a second confidence interval is computed, using the same known population variance, at the same level of confidence, with a sample size that is 2.9 times as large as the original sample. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same estimated variance, at the same level of confidence, with a sample size of 112. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same known population variance, with the same sample size, but with a confidence level of \(97\%\). What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same sample size and confidence level, but with an estimated variance that is 5.2 times as large as the original. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same estimated variance, the same sample size, and the same confidence level, but where the sample mean is 488.8 higher than the initial sample mean. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).

(Question ID: 0091854718)

Consider forming a confidence interval for a population mean. Suppose that the confidence interval is computed based on a sample of size 47, at the \(83\%\) level of confidence.

Suppose that a second confidence interval is computed, using the same known population variance, at the same level of confidence, with a sample size that is 7.4 times as large as the original sample. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same estimated variance, at the same level of confidence, with a sample size of 277. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same known population variance, with the same sample size, but with a confidence level of \(74\%\). What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same sample size and confidence level, but with an estimated variance that is 5.8 times as large as the original. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same estimated variance, the same sample size, and the same confidence level, but where the sample mean is 177.8 higher than the initial sample mean. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).

(Question ID: 0310474625)

Consider forming a confidence interval for a population mean. Suppose that the confidence interval is computed based on a sample of size 65, at the \(80\%\) level of confidence.

Suppose that a second confidence interval is computed, using the same known population variance, at the same level of confidence, with a sample size that is 6.4 times as large as the original sample. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same estimated variance, at the same level of confidence, with a sample size of 170. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same known population variance, with the same sample size, but with a confidence level of \(84\%\). What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same sample size and confidence level, but with an estimated variance that is 7.9 times as large as the original. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same estimated variance, the same sample size, and the same confidence level, but where the sample mean is 887.2 higher than the initial sample mean. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).

(Question ID: 0731017474)

Consider forming a confidence interval for a population mean. Suppose that the confidence interval is computed based on a sample of size 67, at the \(86\%\) level of confidence.

Suppose that a second confidence interval is computed, using the same known population variance, at the same level of confidence, with a sample size that is 7.7 times as large as the original sample. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same estimated variance, at the same level of confidence, with a sample size of 176. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same known population variance, with the same sample size, but with a confidence level of \(70\%\). What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same sample size and confidence level, but with an estimated variance that is 5.8 times as large as the original. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same estimated variance, the same sample size, and the same confidence level, but where the sample mean is 621.9 higher than the initial sample mean. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).

(Question ID: 0316381104)

Consider forming a confidence interval for a population mean. Suppose that the confidence interval is computed based on a sample of size 47, at the \(90\%\) level of confidence.

Suppose that a second confidence interval is computed, using the same known population variance, at the same level of confidence, with a sample size that is 4.3 times as large as the original sample. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same estimated variance, at the same level of confidence, with a sample size of 75. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same known population variance, with the same sample size, but with a confidence level of \(94\%\). What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same sample size and confidence level, but with an estimated variance that is 8.1 times as large as the original. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same estimated variance, the same sample size, and the same confidence level, but where the sample mean is 183.9 higher than the initial sample mean. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).

(Question ID: 0612936733)

Consider forming a confidence interval for a population mean. Suppose that the confidence interval is computed based on a sample of size 45, at the \(80\%\) level of confidence.

Suppose that a second confidence interval is computed, using the same known population variance, at the same level of confidence, with a sample size that is 7.1 times as large as the original sample. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same estimated variance, at the same level of confidence, with a sample size of 439. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same known population variance, with the same sample size, but with a confidence level of \(97\%\). What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same sample size and confidence level, but with an estimated variance that is 9.2 times as large as the original. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same estimated variance, the same sample size, and the same confidence level, but where the sample mean is 419.9 higher than the initial sample mean. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).

(Question ID: 0705120144)

Consider forming a confidence interval for a population mean. Suppose that the confidence interval is computed based on a sample of size 84, at the \(96\%\) level of confidence.

Suppose that a second confidence interval is computed, using the same known population variance, at the same level of confidence, with a sample size that is 1.1 times as large as the original sample. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same estimated variance, at the same level of confidence, with a sample size of 286. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same known population variance, with the same sample size, but with a confidence level of \(77\%\). What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same sample size and confidence level, but with an estimated variance that is 8.9 times as large as the original. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same estimated variance, the same sample size, and the same confidence level, but where the sample mean is 787.5 higher than the initial sample mean. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).

(Question ID: 0067186262)

Consider forming a confidence interval for a population mean. Suppose that the confidence interval is computed based on a sample of size 41, at the \(95\%\) level of confidence.

Suppose that a second confidence interval is computed, using the same known population variance, at the same level of confidence, with a sample size that is 9 times as large as the original sample. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same estimated variance, at the same level of confidence, with a sample size of 444. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same known population variance, with the same sample size, but with a confidence level of \(81\%\). What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same sample size and confidence level, but with an estimated variance that is 1.1 times as large as the original. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same estimated variance, the same sample size, and the same confidence level, but where the sample mean is 7.8 higher than the initial sample mean. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).

(Question ID: 0361457613)

Consider forming a confidence interval for a population mean. Suppose that the confidence interval is computed based on a sample of size 53, at the \(90\%\) level of confidence.

Suppose that a second confidence interval is computed, using the same known population variance, at the same level of confidence, with a sample size that is 4.1 times as large as the original sample. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same estimated variance, at the same level of confidence, with a sample size of 171. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same known population variance, with the same sample size, but with a confidence level of \(72\%\). What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same sample size and confidence level, but with an estimated variance that is 3.7 times as large as the original. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same estimated variance, the same sample size, and the same confidence level, but where the sample mean is 257.3 higher than the initial sample mean. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).

(Question ID: 0268244531)

Consider forming a confidence interval for a population mean. Suppose that the confidence interval is computed based on a sample of size 39, at the \(93\%\) level of confidence.

Suppose that a second confidence interval is computed, using the same known population variance, at the same level of confidence, with a sample size that is 8.3 times as large as the original sample. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same estimated variance, at the same level of confidence, with a sample size of 377. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same known population variance, with the same sample size, but with a confidence level of \(75\%\). What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same sample size and confidence level, but with an estimated variance that is 9.1 times as large as the original. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same estimated variance, the same sample size, and the same confidence level, but where the sample mean is 165.7 higher than the initial sample mean. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).

(Question ID: 0319050189)

Consider forming a confidence interval for a population mean. Suppose that the confidence interval is computed based on a sample of size 63, at the \(99\%\) level of confidence.

Suppose that a second confidence interval is computed, using the same known population variance, at the same level of confidence, with a sample size that is 9.3 times as large as the original sample. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same estimated variance, at the same level of confidence, with a sample size of 263. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same known population variance, with the same sample size, but with a confidence level of \(91\%\). What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same sample size and confidence level, but with an estimated variance that is 6.8 times as large as the original. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same estimated variance, the same sample size, and the same confidence level, but where the sample mean is 236.5 higher than the initial sample mean. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).

(Question ID: 0488434201)

Consider forming a confidence interval for a population mean. Suppose that the confidence interval is computed based on a sample of size 56, at the \(78\%\) level of confidence.

Suppose that a second confidence interval is computed, using the same known population variance, at the same level of confidence, with a sample size that is 9.2 times as large as the original sample. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same estimated variance, at the same level of confidence, with a sample size of 123. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same known population variance, with the same sample size, but with a confidence level of \(95\%\). What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same sample size and confidence level, but with an estimated variance that is 2.4 times as large as the original. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same estimated variance, the same sample size, and the same confidence level, but where the sample mean is 753.4 higher than the initial sample mean. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).

(Question ID: 0555590573)

Consider forming a confidence interval for a population mean. Suppose that the confidence interval is computed based on a sample of size 38, at the \(83\%\) level of confidence.

Suppose that a second confidence interval is computed, using the same known population variance, at the same level of confidence, with a sample size that is 6.4 times as large as the original sample. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same estimated variance, at the same level of confidence, with a sample size of 315. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same known population variance, with the same sample size, but with a confidence level of \(78\%\). What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same sample size and confidence level, but with an estimated variance that is 3.7 times as large as the original. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same estimated variance, the same sample size, and the same confidence level, but where the sample mean is 958.5 higher than the initial sample mean. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).

(Question ID: 0811569378)

Consider forming a confidence interval for a population mean. Suppose that the confidence interval is computed based on a sample of size 83, at the \(94\%\) level of confidence.

Suppose that a second confidence interval is computed, using the same known population variance, at the same level of confidence, with a sample size that is 6.7 times as large as the original sample. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same estimated variance, at the same level of confidence, with a sample size of 248. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same known population variance, with the same sample size, but with a confidence level of \(79\%\). What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same sample size and confidence level, but with an estimated variance that is 5 times as large as the original. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same estimated variance, the same sample size, and the same confidence level, but where the sample mean is 842.5 higher than the initial sample mean. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).

(Question ID: 0176022555)

Consider forming a confidence interval for a population mean. Suppose that the confidence interval is computed based on a sample of size 77, at the \(76\%\) level of confidence.

Suppose that a second confidence interval is computed, using the same known population variance, at the same level of confidence, with a sample size that is 2.4 times as large as the original sample. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same estimated variance, at the same level of confidence, with a sample size of 124. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same known population variance, with the same sample size, but with a confidence level of \(87\%\). What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same sample size and confidence level, but with an estimated variance that is 8.6 times as large as the original. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same estimated variance, the same sample size, and the same confidence level, but where the sample mean is 723.8 higher than the initial sample mean. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).

(Question ID: 0318699686)

Consider forming a confidence interval for a population mean. Suppose that the confidence interval is computed based on a sample of size 71, at the \(71\%\) level of confidence.

Suppose that a second confidence interval is computed, using the same known population variance, at the same level of confidence, with a sample size that is 4.2 times as large as the original sample. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same estimated variance, at the same level of confidence, with a sample size of 485. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same known population variance, with the same sample size, but with a confidence level of \(90\%\). What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same sample size and confidence level, but with an estimated variance that is 4.8 times as large as the original. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same estimated variance, the same sample size, and the same confidence level, but where the sample mean is 895.7 higher than the initial sample mean. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).

(Question ID: 0472524562)

Consider forming a confidence interval for a population mean. Suppose that the confidence interval is computed based on a sample of size 45, at the \(79\%\) level of confidence.

Suppose that a second confidence interval is computed, using the same known population variance, at the same level of confidence, with a sample size that is 6.2 times as large as the original sample. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same estimated variance, at the same level of confidence, with a sample size of 209. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same known population variance, with the same sample size, but with a confidence level of \(91\%\). What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same sample size and confidence level, but with an estimated variance that is 8.9 times as large as the original. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same estimated variance, the same sample size, and the same confidence level, but where the sample mean is 525.3 higher than the initial sample mean. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).

(Question ID: 0951700812)

Consider forming a confidence interval for a population mean. Suppose that the confidence interval is computed based on a sample of size 97, at the \(76\%\) level of confidence.

Suppose that a second confidence interval is computed, using the same known population variance, at the same level of confidence, with a sample size that is 2.6 times as large as the original sample. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same estimated variance, at the same level of confidence, with a sample size of 494. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same known population variance, with the same sample size, but with a confidence level of \(82\%\). What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same sample size and confidence level, but with an estimated variance that is 3.8 times as large as the original. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same estimated variance, the same sample size, and the same confidence level, but where the sample mean is 490.9 higher than the initial sample mean. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).

(Question ID: 0278074922)

Consider forming a confidence interval for a population mean. Suppose that the confidence interval is computed based on a sample of size 98, at the \(80\%\) level of confidence.

Suppose that a second confidence interval is computed, using the same known population variance, at the same level of confidence, with a sample size that is 8.2 times as large as the original sample. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same estimated variance, at the same level of confidence, with a sample size of 355. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same known population variance, with the same sample size, but with a confidence level of \(96\%\). What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same sample size and confidence level, but with an estimated variance that is 0.7 times as large as the original. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same estimated variance, the same sample size, and the same confidence level, but where the sample mean is 555.7 higher than the initial sample mean. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).

(Question ID: 0217379823)

Consider forming a confidence interval for a population mean. Suppose that the confidence interval is computed based on a sample of size 76, at the \(71\%\) level of confidence.

Suppose that a second confidence interval is computed, using the same known population variance, at the same level of confidence, with a sample size that is 5.4 times as large as the original sample. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same estimated variance, at the same level of confidence, with a sample size of 130. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same known population variance, with the same sample size, but with a confidence level of \(79\%\). What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same sample size and confidence level, but with an estimated variance that is 5.9 times as large as the original. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same estimated variance, the same sample size, and the same confidence level, but where the sample mean is 811.7 higher than the initial sample mean. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).

(Question ID: 0227274256)

Consider forming a confidence interval for a population mean. Suppose that the confidence interval is computed based on a sample of size 50, at the \(99\%\) level of confidence.

Suppose that a second confidence interval is computed, using the same known population variance, at the same level of confidence, with a sample size that is 3.5 times as large as the original sample. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same estimated variance, at the same level of confidence, with a sample size of 444. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same known population variance, with the same sample size, but with a confidence level of \(83\%\). What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same sample size and confidence level, but with an estimated variance that is 3.5 times as large as the original. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same estimated variance, the same sample size, and the same confidence level, but where the sample mean is 371.1 higher than the initial sample mean. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).

(Question ID: 0954734289)

Consider forming a confidence interval for a population mean. Suppose that the confidence interval is computed based on a sample of size 91, at the \(71\%\) level of confidence.

Suppose that a second confidence interval is computed, using the same known population variance, at the same level of confidence, with a sample size that is 6.6 times as large as the original sample. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same estimated variance, at the same level of confidence, with a sample size of 331. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same known population variance, with the same sample size, but with a confidence level of \(75\%\). What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same sample size and confidence level, but with an estimated variance that is 6.7 times as large as the original. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).
Suppose that a second confidence interval is computed, using the same estimated variance, the same sample size, and the same confidence level, but where the sample mean is 816.3 higher than the initial sample mean. What is the ratio of the length of the confidence intervals (original divided by newly formed)? (If there is insufficient information, answer -1).

(Question ID: 0197127781)

Self Assessment 14.5

Suppose that the reaction times of individuals in a population have an unknown population distribution.

In a sample of size 358, the mean reaction time was found to be 251.97.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

If the population standard deviation is known to be 53.77, give an \(86\%\) symmetric confidence interval for the mean.

Lower Bound: Upper Bound:

If the sample standard deviation is found to be 53.77, give an \(86\%\) symmetric confidence interval for the mean.

Lower Bound: Upper Bound:

If the population standard deviation is known to be 53.77, give an \(86\%\) upper bound for the mean.
If the sample standard deviation is found to be 53.77, give an \(86\%\) upper bound for the mean.

(Question ID: 0142823925)

Suppose that the heights of individuals in a population have an unknown population distribution.

In a sample of size 252, the mean height was found to be 168.28.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

If the population standard deviation is known to be 7.02, give a \(96\%\) symmetric confidence interval for the mean.

Lower Bound: Upper Bound:

If the sample standard deviation is found to be 7.02, give a \(96\%\) symmetric confidence interval for the mean.

Lower Bound: Upper Bound:

If the population standard deviation is known to be 7.02, give a \(96\%\) upper bound for the mean.
If the sample standard deviation is found to be 7.02, give a \(96\%\) upper bound for the mean.

(Question ID: 0967717275)

Suppose that the serum cholesterol levels of individuals in a population are approximately normally distributed.

In a sample of size 32, the mean serum cholesterol level was found to be 190.76.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

If the population standard deviation is known to be 43.1, give a \(99\%\) symmetric confidence interval for the mean.

Lower Bound: Upper Bound:

If the sample standard deviation is found to be 43.1, give a \(99\%\) symmetric confidence interval for the mean.

Lower Bound: Upper Bound:

If the population standard deviation is known to be 43.1, give a \(99\%\) upper bound for the mean.
If the sample standard deviation is found to be 43.1, give a \(99\%\) upper bound for the mean.

(Question ID: 0600786088)

Suppose that the serum cholesterol levels of individuals in a population have an unknown population distribution.

In a sample of size 245, the mean serum cholesterol level was found to be 198.96.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

If the population standard deviation is known to be 40.89, give a \(91\%\) symmetric confidence interval for the mean.

Lower Bound: Upper Bound:

If the sample standard deviation is found to be 40.89, give a \(91\%\) symmetric confidence interval for the mean.

Lower Bound: Upper Bound:

If the population standard deviation is known to be 40.89, give a \(91\%\) upper bound for the mean.
If the sample standard deviation is found to be 40.89, give a \(91\%\) upper bound for the mean.

(Question ID: 0686596984)

Suppose that systolic blood pressures of individuals in a population are approximately normally distributed.

In a sample of size 13, the mean blood pressure was found to be 119.76.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

If the population standard deviation is known to be 10.58, give an \(82\%\) symmetric confidence interval for the mean.

Lower Bound: Upper Bound:

If the sample standard deviation is found to be 10.58, give an \(82\%\) symmetric confidence interval for the mean.

Lower Bound: Upper Bound:

If the population standard deviation is known to be 10.58, give an \(82\%\) lower bound for the mean.
If the sample standard deviation is found to be 10.58, give an \(82\%\) lower bound for the mean.

(Question ID: 0219628637)

Suppose that the serum cholesterol levels of individuals in a population are approximately normally distributed.

In a sample of size 14, the mean serum cholesterol level was found to be 205.87.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

If the population standard deviation is known to be 57.44, give a \(93\%\) symmetric confidence interval for the mean.

Lower Bound: Upper Bound:

If the sample standard deviation is found to be 57.44, give a \(93\%\) symmetric confidence interval for the mean.

Lower Bound: Upper Bound:

If the population standard deviation is known to be 57.44, give a \(93\%\) lower bound for the mean.
If the sample standard deviation is found to be 57.44, give a \(93\%\) lower bound for the mean.

(Question ID: 0779402287)

Suppose that the heights of individuals in a population have an unknown population distribution.

In a sample of size 61, the mean height was found to be 166.98.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

If the population standard deviation is known to be 7.41, give a \(96\%\) symmetric confidence interval for the mean.

Lower Bound: Upper Bound:

If the sample standard deviation is found to be 7.41, give a \(96\%\) symmetric confidence interval for the mean.

Lower Bound: Upper Bound:

If the population standard deviation is known to be 7.41, give a \(96\%\) lower bound for the mean.
If the sample standard deviation is found to be 7.41, give a \(96\%\) lower bound for the mean.

(Question ID: 0717640944)

Suppose that the serum cholesterol levels of individuals in a population have an unknown population distribution.

In a sample of size 247, the mean serum cholesterol level was found to be 202.13.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

If the population standard deviation is known to be 43.01, give a \(94\%\) symmetric confidence interval for the mean.

Lower Bound: Upper Bound:

If the sample standard deviation is found to be 43.01, give a \(94\%\) symmetric confidence interval for the mean.

Lower Bound: Upper Bound:

If the population standard deviation is known to be 43.01, give a \(94\%\) lower bound for the mean.
If the sample standard deviation is found to be 43.01, give a \(94\%\) lower bound for the mean.

(Question ID: 0427995167)

Suppose that the serum cholesterol levels of individuals in a population are approximately normally distributed.

In a sample of size 13, the mean serum cholesterol level was found to be 197.5.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

If the population standard deviation is known to be 37.27, give an \(89\%\) symmetric confidence interval for the mean.

Lower Bound: Upper Bound:

If the sample standard deviation is found to be 37.27, give an \(89\%\) symmetric confidence interval for the mean.

Lower Bound: Upper Bound:

If the population standard deviation is known to be 37.27, give an \(89\%\) lower bound for the mean.
If the sample standard deviation is found to be 37.27, give an \(89\%\) lower bound for the mean.

(Question ID: 0104509627)

Suppose that systolic blood pressures of individuals in a population have an unknown population distribution.

In a sample of size 205, the mean blood pressure was found to be 120.13.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

If the population standard deviation is known to be 10.6, give a \(91\%\) symmetric confidence interval for the mean.

Lower Bound: Upper Bound:

If the sample standard deviation is found to be 10.6, give a \(91\%\) symmetric confidence interval for the mean.

Lower Bound: Upper Bound:

If the population standard deviation is known to be 10.6, give a \(91\%\) upper bound for the mean.
If the sample standard deviation is found to be 10.6, give a \(91\%\) upper bound for the mean.

(Question ID: 0016719047)

Suppose that diastolic blood pressures of individuals in a population have an unknown population distribution.

In a sample of size 215, the mean blood pressure was found to be 89.13.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

If the population standard deviation is known to be 7.18, give a \(98\%\) symmetric confidence interval for the mean.

Lower Bound: Upper Bound:

If the sample standard deviation is found to be 7.18, give a \(98\%\) symmetric confidence interval for the mean.

Lower Bound: Upper Bound:

If the population standard deviation is known to be 7.18, give a \(98\%\) upper bound for the mean.
If the sample standard deviation is found to be 7.18, give a \(98\%\) upper bound for the mean.

(Question ID: 0893310266)

Suppose that body temperatures of individuals in a population have an unknown population distribution.

In a sample of size 346, the mean temperature was found to be 36.48.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

If the population standard deviation is known to be 0.49, give a \(96\%\) symmetric confidence interval for the mean.

Lower Bound: Upper Bound:

If the sample standard deviation is found to be 0.49, give a \(96\%\) symmetric confidence interval for the mean.

Lower Bound: Upper Bound:

If the population standard deviation is known to be 0.49, give a \(96\%\) lower bound for the mean.
If the sample standard deviation is found to be 0.49, give a \(96\%\) lower bound for the mean.

(Question ID: 0434129521)

Suppose that the reaction times of individuals in a population have an unknown population distribution.

In a sample of size 428, the mean reaction time was found to be 245.63.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

If the population standard deviation is known to be 47.91, give a \(96\%\) symmetric confidence interval for the mean.

Lower Bound: Upper Bound:

If the sample standard deviation is found to be 47.91, give a \(96\%\) symmetric confidence interval for the mean.

Lower Bound: Upper Bound:

If the population standard deviation is known to be 47.91, give a \(96\%\) upper bound for the mean.
If the sample standard deviation is found to be 47.91, give a \(96\%\) upper bound for the mean.

(Question ID: 0565753664)

Suppose that systolic blood pressures of individuals in a population are approximately normally distributed.

In a sample of size 16, the mean blood pressure was found to be 120.85.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

If the population standard deviation is known to be 10.65, give an \(82\%\) symmetric confidence interval for the mean.

Lower Bound: Upper Bound:

If the sample standard deviation is found to be 10.65, give an \(82\%\) symmetric confidence interval for the mean.

Lower Bound: Upper Bound:

If the population standard deviation is known to be 10.65, give an \(82\%\) lower bound for the mean.
If the sample standard deviation is found to be 10.65, give an \(82\%\) lower bound for the mean.

(Question ID: 0690957478)

Suppose that the heights of individuals in a population have an unknown population distribution.

In a sample of size 158, the mean height was found to be 168.33.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

If the population standard deviation is known to be 6.81, give a \(96\%\) symmetric confidence interval for the mean.

Lower Bound: Upper Bound:

If the sample standard deviation is found to be 6.81, give a \(96\%\) symmetric confidence interval for the mean.

Lower Bound: Upper Bound:

If the population standard deviation is known to be 6.81, give a \(96\%\) upper bound for the mean.
If the sample standard deviation is found to be 6.81, give a \(96\%\) upper bound for the mean.

(Question ID: 0282405184)

Suppose that weights of newborns in a population are approximately normally distributed.

In a sample of size 40, the mean weight was found to be 3570.18.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

If the population standard deviation is known to be 529.7, give a \(97\%\) symmetric confidence interval for the mean.

Lower Bound: Upper Bound:

If the sample standard deviation is found to be 529.7, give a \(97\%\) symmetric confidence interval for the mean.

Lower Bound: Upper Bound:

If the population standard deviation is known to be 529.7, give a \(97\%\) lower bound for the mean.
If the sample standard deviation is found to be 529.7, give a \(97\%\) lower bound for the mean.

(Question ID: 0518239900)

Suppose that the red blood cell counts of individuals in a population have an unknown population distribution.

In a sample of size 492, the mean red blood cell count was found to be 5.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

If the population standard deviation is known to be 0.48, give a \(95\%\) symmetric confidence interval for the mean.

Lower Bound: Upper Bound:

If the sample standard deviation is found to be 0.48, give a \(95\%\) symmetric confidence interval for the mean.

Lower Bound: Upper Bound:

If the population standard deviation is known to be 0.48, give a \(95\%\) upper bound for the mean.
If the sample standard deviation is found to be 0.48, give a \(95\%\) upper bound for the mean.

(Question ID: 0124188277)

Suppose that diastolic blood pressures of individuals in a population are approximately normally distributed.

In a sample of size 23, the mean blood pressure was found to be 90.14.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

If the population standard deviation is known to be 7.85, give a \(92\%\) symmetric confidence interval for the mean.

Lower Bound: Upper Bound:

If the sample standard deviation is found to be 7.85, give a \(92\%\) symmetric confidence interval for the mean.

Lower Bound: Upper Bound:

If the population standard deviation is known to be 7.85, give a \(92\%\) upper bound for the mean.
If the sample standard deviation is found to be 7.85, give a \(92\%\) upper bound for the mean.

(Question ID: 0015480111)

Suppose that diastolic blood pressures of individuals in a population have an unknown population distribution.

In a sample of size 117, the mean blood pressure was found to be 89.27.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

If the population standard deviation is known to be 8.21, give a \(90\%\) symmetric confidence interval for the mean.

Lower Bound: Upper Bound:

If the sample standard deviation is found to be 8.21, give a \(90\%\) symmetric confidence interval for the mean.

Lower Bound: Upper Bound:

If the population standard deviation is known to be 8.21, give a \(90\%\) lower bound for the mean.
If the sample standard deviation is found to be 8.21, give a \(90\%\) lower bound for the mean.

(Question ID: 0823315388)

Suppose that the red blood cell counts of individuals in a population have an unknown population distribution.

In a sample of size 204, the mean red blood cell count was found to be 5.01.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

If the population standard deviation is known to be 0.5, give a \(92\%\) symmetric confidence interval for the mean.

Lower Bound: Upper Bound:

If the sample standard deviation is found to be 0.5, give a \(92\%\) symmetric confidence interval for the mean.

Lower Bound: Upper Bound:

If the population standard deviation is known to be 0.5, give a \(92\%\) upper bound for the mean.
If the sample standard deviation is found to be 0.5, give a \(92\%\) upper bound for the mean.

(Question ID: 0297248030)

Suppose that diastolic blood pressures of individuals in a population are approximately normally distributed.

In a sample of size 20, the mean blood pressure was found to be 89.25.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

If the population standard deviation is known to be 7.35, give a \(97\%\) symmetric confidence interval for the mean.

Lower Bound: Upper Bound:

If the sample standard deviation is found to be 7.35, give a \(97\%\) symmetric confidence interval for the mean.

Lower Bound: Upper Bound:

If the population standard deviation is known to be 7.35, give a \(97\%\) upper bound for the mean.
If the sample standard deviation is found to be 7.35, give a \(97\%\) upper bound for the mean.

(Question ID: 0314925577)

Suppose that weights of newborns in a population are approximately normally distributed.

In a sample of size 38, the mean weight was found to be 3595.09.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

If the population standard deviation is known to be 434.59, give a \(93\%\) symmetric confidence interval for the mean.

Lower Bound: Upper Bound:

If the sample standard deviation is found to be 434.59, give a \(93\%\) symmetric confidence interval for the mean.

Lower Bound: Upper Bound:

If the population standard deviation is known to be 434.59, give a \(93\%\) lower bound for the mean.
If the sample standard deviation is found to be 434.59, give a \(93\%\) lower bound for the mean.

(Question ID: 0403612547)

Suppose that the reaction times of individuals in a population have an unknown population distribution.

In a sample of size 248, the mean reaction time was found to be 249.4.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

If the population standard deviation is known to be 46.57, give an \(85\%\) symmetric confidence interval for the mean.

Lower Bound: Upper Bound:

If the sample standard deviation is found to be 46.57, give an \(85\%\) symmetric confidence interval for the mean.

Lower Bound: Upper Bound:

If the population standard deviation is known to be 46.57, give an \(85\%\) upper bound for the mean.
If the sample standard deviation is found to be 46.57, give an \(85\%\) upper bound for the mean.

(Question ID: 0574136187)

Suppose that the red blood cell counts of individuals in a population have an unknown population distribution.

In a sample of size 474, the mean red blood cell count was found to be 4.99.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

If the population standard deviation is known to be 0.47, give an \(80\%\) symmetric confidence interval for the mean.

Lower Bound: Upper Bound:

If the sample standard deviation is found to be 0.47, give an \(80\%\) symmetric confidence interval for the mean.

Lower Bound: Upper Bound:

If the population standard deviation is known to be 0.47, give an \(80\%\) upper bound for the mean.
If the sample standard deviation is found to be 0.47, give an \(80\%\) upper bound for the mean.

(Question ID: 0063605694)

Suppose that the heights of individuals in a population have an unknown population distribution.

In a sample of size 182, the mean height was found to be 167.59.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

If the population standard deviation is known to be 7.06, give a \(97\%\) symmetric confidence interval for the mean.

Lower Bound: Upper Bound:

If the sample standard deviation is found to be 7.06, give a \(97\%\) symmetric confidence interval for the mean.

Lower Bound: Upper Bound:

If the population standard deviation is known to be 7.06, give a \(97\%\) lower bound for the mean.
If the sample standard deviation is found to be 7.06, give a \(97\%\) lower bound for the mean.

(Question ID: 0054901845)

Suppose that the red blood cell counts of individuals in a population are approximately normally distributed.

In a sample of size 29, the mean red blood cell count was found to be 4.92.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

If the population standard deviation is known to be 0.5, give an \(83\%\) symmetric confidence interval for the mean.

Lower Bound: Upper Bound:

If the sample standard deviation is found to be 0.5, give an \(83\%\) symmetric confidence interval for the mean.

Lower Bound: Upper Bound:

If the population standard deviation is known to be 0.5, give an \(83\%\) upper bound for the mean.
If the sample standard deviation is found to be 0.5, give an \(83\%\) upper bound for the mean.

(Question ID: 0292009676)

Suppose that systolic blood pressures of individuals in a population are approximately normally distributed.

In a sample of size 16, the mean blood pressure was found to be 118.25.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

If the population standard deviation is known to be 12.74, give a \(97\%\) symmetric confidence interval for the mean.

Lower Bound: Upper Bound:

If the sample standard deviation is found to be 12.74, give a \(97\%\) symmetric confidence interval for the mean.

Lower Bound: Upper Bound:

If the population standard deviation is known to be 12.74, give a \(97\%\) lower bound for the mean.
If the sample standard deviation is found to be 12.74, give a \(97\%\) lower bound for the mean.

(Question ID: 0405490350)

Suppose that body temperatures of individuals in a population are approximately normally distributed.

In a sample of size 40, the mean temperature was found to be 36.62.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

If the population standard deviation is known to be 0.48, give a \(97\%\) symmetric confidence interval for the mean.

Lower Bound: Upper Bound:

If the sample standard deviation is found to be 0.48, give a \(97\%\) symmetric confidence interval for the mean.

Lower Bound: Upper Bound:

If the population standard deviation is known to be 0.48, give a \(97\%\) upper bound for the mean.
If the sample standard deviation is found to be 0.48, give a \(97\%\) upper bound for the mean.

(Question ID: 0099412802)

Suppose that diastolic blood pressures of individuals in a population have an unknown population distribution.

In a sample of size 113, the mean blood pressure was found to be 90.48.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

If the population standard deviation is known to be 8.15, give an \(87\%\) symmetric confidence interval for the mean.

Lower Bound: Upper Bound:

If the sample standard deviation is found to be 8.15, give an \(87\%\) symmetric confidence interval for the mean.

Lower Bound: Upper Bound:

If the population standard deviation is known to be 8.15, give an \(87\%\) lower bound for the mean.
If the sample standard deviation is found to be 8.15, give an \(87\%\) lower bound for the mean.

(Question ID: 0096527109)

Suppose that diastolic blood pressures of individuals in a population have an unknown population distribution.

In a sample of size 366, the mean blood pressure was found to be 89.69.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

If the population standard deviation is known to be 7.99, give a \(91\%\) symmetric confidence interval for the mean.

Lower Bound: Upper Bound:

If the sample standard deviation is found to be 7.99, give a \(91\%\) symmetric confidence interval for the mean.

Lower Bound: Upper Bound:

If the population standard deviation is known to be 7.99, give a \(91\%\) upper bound for the mean.
If the sample standard deviation is found to be 7.99, give a \(91\%\) upper bound for the mean.

(Question ID: 0450671406)

Suppose that the reaction times of individuals in a population are approximately normally distributed.

In a sample of size 34, the mean reaction time was found to be 260.88.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

If the population standard deviation is known to be 48.17, give an \(85\%\) symmetric confidence interval for the mean.

Lower Bound: Upper Bound:

If the sample standard deviation is found to be 48.17, give an \(85\%\) symmetric confidence interval for the mean.

Lower Bound: Upper Bound:

If the population standard deviation is known to be 48.17, give an \(85\%\) lower bound for the mean.
If the sample standard deviation is found to be 48.17, give an \(85\%\) lower bound for the mean.

(Question ID: 0349259052)

Suppose that diastolic blood pressures of individuals in a population have an unknown population distribution.

In a sample of size 319, the mean blood pressure was found to be 90.18.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

If the population standard deviation is known to be 7.79, give a \(94\%\) symmetric confidence interval for the mean.

Lower Bound: Upper Bound:

If the sample standard deviation is found to be 7.79, give a \(94\%\) symmetric confidence interval for the mean.

Lower Bound: Upper Bound:

If the population standard deviation is known to be 7.79, give a \(94\%\) upper bound for the mean.
If the sample standard deviation is found to be 7.79, give a \(94\%\) upper bound for the mean.

(Question ID: 0411105162)

Suppose that the heights of individuals in a population have an unknown population distribution.

In a sample of size 132, the mean height was found to be 168.99.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

If the population standard deviation is known to be 6.75, give an \(84\%\) symmetric confidence interval for the mean.

Lower Bound: Upper Bound:

If the sample standard deviation is found to be 6.75, give an \(84\%\) symmetric confidence interval for the mean.

Lower Bound: Upper Bound:

If the population standard deviation is known to be 6.75, give an \(84\%\) lower bound for the mean.
If the sample standard deviation is found to be 6.75, give an \(84\%\) lower bound for the mean.

(Question ID: 0910883603)

Suppose that the reaction times of individuals in a population have an unknown population distribution.

In a sample of size 352, the mean reaction time was found to be 251.27.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

If the population standard deviation is known to be 52.95, give a \(97\%\) symmetric confidence interval for the mean.

Lower Bound: Upper Bound:

If the sample standard deviation is found to be 52.95, give a \(97\%\) symmetric confidence interval for the mean.

Lower Bound: Upper Bound:

If the population standard deviation is known to be 52.95, give a \(97\%\) lower bound for the mean.
If the sample standard deviation is found to be 52.95, give a \(97\%\) lower bound for the mean.

(Question ID: 0340791156)

Suppose that the heights of individuals in a population have an unknown population distribution.

In a sample of size 307, the mean height was found to be 168.16.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

If the population standard deviation is known to be 6.33, give an \(88\%\) symmetric confidence interval for the mean.

Lower Bound: Upper Bound:

If the sample standard deviation is found to be 6.33, give an \(88\%\) symmetric confidence interval for the mean.

Lower Bound: Upper Bound:

If the population standard deviation is known to be 6.33, give an \(88\%\) lower bound for the mean.
If the sample standard deviation is found to be 6.33, give an \(88\%\) lower bound for the mean.

(Question ID: 0782222392)

Suppose that body temperatures of individuals in a population have an unknown population distribution.

In a sample of size 435, the mean temperature was found to be 36.45.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

If the population standard deviation is known to be 0.48, give a \(91\%\) symmetric confidence interval for the mean.

Lower Bound: Upper Bound:

If the sample standard deviation is found to be 0.48, give a \(91\%\) symmetric confidence interval for the mean.

Lower Bound: Upper Bound:

If the population standard deviation is known to be 0.48, give a \(91\%\) lower bound for the mean.
If the sample standard deviation is found to be 0.48, give a \(91\%\) lower bound for the mean.

(Question ID: 0939716652)

Suppose that systolic blood pressures of individuals in a population are approximately normally distributed.

In a sample of size 28, the mean blood pressure was found to be 123.13.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

If the population standard deviation is known to be 11.21, give an \(84\%\) symmetric confidence interval for the mean.

Lower Bound: Upper Bound:

If the sample standard deviation is found to be 11.21, give an \(84\%\) symmetric confidence interval for the mean.

Lower Bound: Upper Bound:

If the population standard deviation is known to be 11.21, give an \(84\%\) lower bound for the mean.
If the sample standard deviation is found to be 11.21, give an \(84\%\) lower bound for the mean.

(Question ID: 0750019605)

Suppose that the red blood cell counts of individuals in a population have an unknown population distribution.

In a sample of size 468, the mean red blood cell count was found to be 5.01.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

If the population standard deviation is known to be 0.5, give an \(80\%\) symmetric confidence interval for the mean.

Lower Bound: Upper Bound:

If the sample standard deviation is found to be 0.5, give an \(80\%\) symmetric confidence interval for the mean.

Lower Bound: Upper Bound:

If the population standard deviation is known to be 0.5, give an \(80\%\) lower bound for the mean.
If the sample standard deviation is found to be 0.5, give an \(80\%\) lower bound for the mean.

(Question ID: 0508645374)

Suppose that the heights of individuals in a population have an unknown population distribution.

In a sample of size 355, the mean height was found to be 168.61.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

If the population standard deviation is known to be 6.6, give a \(98\%\) symmetric confidence interval for the mean.

Lower Bound: Upper Bound:

If the sample standard deviation is found to be 6.6, give a \(98\%\) symmetric confidence interval for the mean.

Lower Bound: Upper Bound:

If the population standard deviation is known to be 6.6, give a \(98\%\) upper bound for the mean.
If the sample standard deviation is found to be 6.6, give a \(98\%\) upper bound for the mean.

(Question ID: 0383144309)

Suppose that the heights of individuals in a population have an unknown population distribution.

In a sample of size 73, the mean height was found to be 167.69.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

If the population standard deviation is known to be 6.98, give a \(98\%\) symmetric confidence interval for the mean.

Lower Bound: Upper Bound:

If the sample standard deviation is found to be 6.98, give a \(98\%\) symmetric confidence interval for the mean.

Lower Bound: Upper Bound:

If the population standard deviation is known to be 6.98, give a \(98\%\) upper bound for the mean.
If the sample standard deviation is found to be 6.98, give a \(98\%\) upper bound for the mean.

(Question ID: 0857330695)

Suppose that diastolic blood pressures of individuals in a population have an unknown population distribution.

In a sample of size 71, the mean blood pressure was found to be 90.92.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

If the population standard deviation is known to be 9.44, give a \(92\%\) symmetric confidence interval for the mean.

Lower Bound: Upper Bound:

If the sample standard deviation is found to be 9.44, give a \(92\%\) symmetric confidence interval for the mean.

Lower Bound: Upper Bound:

If the population standard deviation is known to be 9.44, give a \(92\%\) lower bound for the mean.
If the sample standard deviation is found to be 9.44, give a \(92\%\) lower bound for the mean.

(Question ID: 0539253278)

Suppose that weights of newborns in a population have an unknown population distribution.

In a sample of size 120, the mean weight was found to be 3535.33.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

If the population standard deviation is known to be 520.24, give an \(81\%\) symmetric confidence interval for the mean.

Lower Bound: Upper Bound:

If the sample standard deviation is found to be 520.24, give an \(81\%\) symmetric confidence interval for the mean.

Lower Bound: Upper Bound:

If the population standard deviation is known to be 520.24, give an \(81\%\) lower bound for the mean.
If the sample standard deviation is found to be 520.24, give an \(81\%\) lower bound for the mean.

(Question ID: 0534700349)

Suppose that the red blood cell counts of individuals in a population are approximately normally distributed.

In a sample of size 46, the mean red blood cell count was found to be 4.84.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

If the population standard deviation is known to be 0.59, give a \(99\%\) symmetric confidence interval for the mean.

Lower Bound: Upper Bound:

If the sample standard deviation is found to be 0.59, give a \(99\%\) symmetric confidence interval for the mean.

Lower Bound: Upper Bound:

If the population standard deviation is known to be 0.59, give a \(99\%\) lower bound for the mean.
If the sample standard deviation is found to be 0.59, give a \(99\%\) lower bound for the mean.

(Question ID: 0041366616)

Suppose that diastolic blood pressures of individuals in a population are approximately normally distributed.

In a sample of size 40, the mean blood pressure was found to be 90.22.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

If the population standard deviation is known to be 8.18, give a \(98\%\) symmetric confidence interval for the mean.

Lower Bound: Upper Bound:

If the sample standard deviation is found to be 8.18, give a \(98\%\) symmetric confidence interval for the mean.

Lower Bound: Upper Bound:

If the population standard deviation is known to be 8.18, give a \(98\%\) lower bound for the mean.
If the sample standard deviation is found to be 8.18, give a \(98\%\) lower bound for the mean.

(Question ID: 0385258174)

Suppose that the serum cholesterol levels of individuals in a population have an unknown population distribution.

In a sample of size 415, the mean serum cholesterol level was found to be 201.88.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

If the population standard deviation is known to be 40.76, give an \(81\%\) symmetric confidence interval for the mean.

Lower Bound: Upper Bound:

If the sample standard deviation is found to be 40.76, give an \(81\%\) symmetric confidence interval for the mean.

Lower Bound: Upper Bound:

If the population standard deviation is known to be 40.76, give an \(81\%\) lower bound for the mean.
If the sample standard deviation is found to be 40.76, give an \(81\%\) lower bound for the mean.

(Question ID: 0938605271)

Suppose that the heights of individuals in a population are approximately normally distributed.

In a sample of size 31, the mean height was found to be 169.75.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

If the population standard deviation is known to be 5.51, give a \(99\%\) symmetric confidence interval for the mean.

Lower Bound: Upper Bound:

If the sample standard deviation is found to be 5.51, give a \(99\%\) symmetric confidence interval for the mean.

Lower Bound: Upper Bound:

If the population standard deviation is known to be 5.51, give a \(99\%\) lower bound for the mean.
If the sample standard deviation is found to be 5.51, give a \(99\%\) lower bound for the mean.

(Question ID: 0622850899)

Suppose that the serum cholesterol levels of individuals in a population have an unknown population distribution.

In a sample of size 446, the mean serum cholesterol level was found to be 201.54.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

If the population standard deviation is known to be 39.09, give an \(82\%\) symmetric confidence interval for the mean.

Lower Bound: Upper Bound:

If the sample standard deviation is found to be 39.09, give an \(82\%\) symmetric confidence interval for the mean.

Lower Bound: Upper Bound:

If the population standard deviation is known to be 39.09, give an \(82\%\) upper bound for the mean.
If the sample standard deviation is found to be 39.09, give an \(82\%\) upper bound for the mean.

(Question ID: 0675157327)

Suppose that weights of newborns in a population have an unknown population distribution.

In a sample of size 270, the mean weight was found to be 3547.56.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

If the population standard deviation is known to be 514.18, give a \(96\%\) symmetric confidence interval for the mean.

Lower Bound: Upper Bound:

If the sample standard deviation is found to be 514.18, give a \(96\%\) symmetric confidence interval for the mean.

Lower Bound: Upper Bound:

If the population standard deviation is known to be 514.18, give a \(96\%\) upper bound for the mean.
If the sample standard deviation is found to be 514.18, give a \(96\%\) upper bound for the mean.

(Question ID: 0427660423)

Suppose that the reaction times of individuals in a population are approximately normally distributed.

In a sample of size 49, the mean reaction time was found to be 235.87.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

If the population standard deviation is known to be 54.2, give an \(87\%\) symmetric confidence interval for the mean.

Lower Bound: Upper Bound:

If the sample standard deviation is found to be 54.2, give an \(87\%\) symmetric confidence interval for the mean.

Lower Bound: Upper Bound:

If the population standard deviation is known to be 54.2, give an \(87\%\) upper bound for the mean.
If the sample standard deviation is found to be 54.2, give an \(87\%\) upper bound for the mean.

(Question ID: 0253033177)

Suppose that weights of newborns in a population are approximately normally distributed.

In a sample of size 50, the mean weight was found to be 3496.23.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

If the population standard deviation is known to be 489.72, give a \(97\%\) symmetric confidence interval for the mean.

Lower Bound: Upper Bound:

If the sample standard deviation is found to be 489.72, give a \(97\%\) symmetric confidence interval for the mean.

Lower Bound: Upper Bound:

If the population standard deviation is known to be 489.72, give a \(97\%\) lower bound for the mean.
If the sample standard deviation is found to be 489.72, give a \(97\%\) lower bound for the mean.

(Question ID: 0086317969)

Self Assessment 14.6

Suppose that researchers are interested in estimating the proportion of individuals who smoke cigarettes.

In a sample of size 93, the total number of individuals who smoke was found to be 40.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

Give a \(96\%\) symmetric confidence interval for the population proportion.

Lower Bound: Upper Bound:

Give a conservative \(96\%\) symmetric confidence interval for the population proportion.

Lower Bound: Upper Bound:

Give a \(96\%\) upper bound for the population proportion.
Give a conservative \(96\%\) upper bound for the population proportion.

(Question ID: 0262955202)

Suppose that researchers are interested in estimating the proportion of individuals in a particular region living in poverty.

In a sample of size 329, the total number of people living in poverty was found to be 28.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

Give an \(80\%\) symmetric confidence interval for the population proportion.

Lower Bound: Upper Bound:

Give a conservative \(80\%\) symmetric confidence interval for the population proportion.

Lower Bound: Upper Bound:

Give an \(80\%\) upper bound for the population proportion.
Give a conservative \(80\%\) upper bound for the population proportion.

(Question ID: 0454798508)

Suppose that researchers are interested in estimating the proportion of individuals who are vegan.

In a sample of size 257, the total number of people who are vegan was found to be 8.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

Give an \(87\%\) symmetric confidence interval for the population proportion.

Lower Bound: Upper Bound:

Give a conservative \(87\%\) symmetric confidence interval for the population proportion.

Lower Bound: Upper Bound:

Give an \(87\%\) lower bound for the population proportion.
Give a conservative \(87\%\) lower bound for the population proportion.

(Question ID: 0614428620)

Suppose that researchers are interested in estimating the proportion of individuals who are satisfied with their jobs.

In a sample of size 67, the total number of individuals who report that they are satisfied with their jobs was found to be 40.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

Give a \(93\%\) symmetric confidence interval for the population proportion.

Lower Bound: Upper Bound:

Give a conservative \(93\%\) symmetric confidence interval for the population proportion.

Lower Bound: Upper Bound:

Give a \(93\%\) upper bound for the population proportion.
Give a conservative \(93\%\) upper bound for the population proportion.

(Question ID: 0454529055)

Suppose that researchers are interested in estimating the proportion of individuals in a particular region living in poverty.

In a sample of size 325, the total number of people living in poverty was found to be 23.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

Give a \(95\%\) symmetric confidence interval for the population proportion.

Lower Bound: Upper Bound:

Give a conservative \(95\%\) symmetric confidence interval for the population proportion.

Lower Bound: Upper Bound:

Give a \(95\%\) upper bound for the population proportion.
Give a conservative \(95\%\) upper bound for the population proportion.

(Question ID: 0755170226)

Suppose that researchers are interested in estimating the proportion of individuals who are satisfied with their jobs.

In a sample of size 57, the total number of individuals who report that they are satisfied with their jobs was found to be 37.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

Give a \(93\%\) symmetric confidence interval for the population proportion.

Lower Bound: Upper Bound:

Give a conservative \(93\%\) symmetric confidence interval for the population proportion.

Lower Bound: Upper Bound:

Give a \(93\%\) upper bound for the population proportion.
Give a conservative \(93\%\) upper bound for the population proportion.

(Question ID: 0788949158)

Suppose that researchers are interested in estimating the proportion of individuals who are left-handed.

In a sample of size 184, the total number of left-handed individuals was found to be 23.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

Give an \(85\%\) symmetric confidence interval for the population proportion.

Lower Bound: Upper Bound:

Give a conservative \(85\%\) symmetric confidence interval for the population proportion.

Lower Bound: Upper Bound:

Give an \(85\%\) lower bound for the population proportion.
Give a conservative \(85\%\) lower bound for the population proportion.

(Question ID: 0624913333)

Suppose that researchers are interested in estimating the proportion of high schoolers in a school board who own a smartphone.

In a sample of size 74, the total number of high schoolers with a smartphone was found to be 49.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

Give an \(87\%\) symmetric confidence interval for the population proportion.

Lower Bound: Upper Bound:

Give a conservative \(87\%\) symmetric confidence interval for the population proportion.

Lower Bound: Upper Bound:

Give an \(87\%\) upper bound for the population proportion.
Give a conservative \(87\%\) upper bound for the population proportion.

(Question ID: 0318143325)

Suppose that researchers are interested in estimating the proportion of individuals who are vegan.

In a sample of size 304, the total number of people who are vegan was found to be 30.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

Give a \(98\%\) symmetric confidence interval for the population proportion.

Lower Bound: Upper Bound:

Give a conservative \(98\%\) symmetric confidence interval for the population proportion.

Lower Bound: Upper Bound:

Give a \(98\%\) lower bound for the population proportion.
Give a conservative \(98\%\) lower bound for the population proportion.

(Question ID: 0793206641)

Suppose that researchers are interested in estimating the proportion of individuals who are left-handed.

In a sample of size 324, the total number of left-handed individuals was found to be 28.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

Give a \(98\%\) symmetric confidence interval for the population proportion.

Lower Bound: Upper Bound:

Give a conservative \(98\%\) symmetric confidence interval for the population proportion.

Lower Bound: Upper Bound:

Give a \(98\%\) lower bound for the population proportion.
Give a conservative \(98\%\) lower bound for the population proportion.

(Question ID: 0192819384)

Suppose that researchers are interested in estimating the proportion of individuals who smoke cigarettes.

In a sample of size 80, the total number of individuals who smoke was found to be 16.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

Give a \(98\%\) symmetric confidence interval for the population proportion.

Lower Bound: Upper Bound:

Give a conservative \(98\%\) symmetric confidence interval for the population proportion.

Lower Bound: Upper Bound:

Give a \(98\%\) upper bound for the population proportion.
Give a conservative \(98\%\) upper bound for the population proportion.

(Question ID: 0323446172)

Suppose that researchers are interested in estimating the proportion of individuals who smoke cigarettes.

In a sample of size 57, the total number of individuals who smoke was found to be 14.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

Give an \(81\%\) symmetric confidence interval for the population proportion.

Lower Bound: Upper Bound:

Give a conservative \(81\%\) symmetric confidence interval for the population proportion.

Lower Bound: Upper Bound:

Give an \(81\%\) lower bound for the population proportion.
Give a conservative \(81\%\) lower bound for the population proportion.

(Question ID: 0059591358)

Suppose that researchers are interested in estimating the proportion of individuals who are left-handed.

In a sample of size 183, the total number of left-handed individuals was found to be 13.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

Give a \(91\%\) symmetric confidence interval for the population proportion.

Lower Bound: Upper Bound:

Give a conservative \(91\%\) symmetric confidence interval for the population proportion.

Lower Bound: Upper Bound:

Give a \(91\%\) upper bound for the population proportion.
Give a conservative \(91\%\) upper bound for the population proportion.

(Question ID: 0841261656)

Suppose that researchers are interested in estimating the proportion of individuals in a particular region who recycle.

In a sample of size 72, the total number of individuals who recycle was found to be 29.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

Give a \(96\%\) symmetric confidence interval for the population proportion.

Lower Bound: Upper Bound:

Give a conservative \(96\%\) symmetric confidence interval for the population proportion.

Lower Bound: Upper Bound:

Give a \(96\%\) upper bound for the population proportion.
Give a conservative \(96\%\) upper bound for the population proportion.

(Question ID: 0131113001)

Suppose that researchers are interested in estimating the proportion of individuals in a particular region who recycle.

In a sample of size 103, the total number of individuals who recycle was found to be 74.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

Give a \(99\%\) symmetric confidence interval for the population proportion.

Lower Bound: Upper Bound:

Give a conservative \(99\%\) symmetric confidence interval for the population proportion.

Lower Bound: Upper Bound:

Give a \(99\%\) lower bound for the population proportion.
Give a conservative \(99\%\) lower bound for the population proportion.

(Question ID: 0748790105)

Suppose that researchers are interested in estimating the proportion of individuals who are left-handed.

In a sample of size 383, the total number of left-handed individuals was found to be 20.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

Give an \(88\%\) symmetric confidence interval for the population proportion.

Lower Bound: Upper Bound:

Give a conservative \(88\%\) symmetric confidence interval for the population proportion.

Lower Bound: Upper Bound:

Give an \(88\%\) lower bound for the population proportion.
Give a conservative \(88\%\) lower bound for the population proportion.

(Question ID: 0233191679)

Suppose that researchers are interested in estimating the proportion of individuals in a particular region who recycle.

In a sample of size 65, the total number of individuals who recycle was found to be 36.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

Give an \(82\%\) symmetric confidence interval for the population proportion.

Lower Bound: Upper Bound:

Give a conservative \(82\%\) symmetric confidence interval for the population proportion.

Lower Bound: Upper Bound:

Give an \(82\%\) upper bound for the population proportion.
Give a conservative \(82\%\) upper bound for the population proportion.

(Question ID: 0731106175)

Suppose that researchers are interested in estimating the proportion of individuals who are vegan.

In a sample of size 172, the total number of people who are vegan was found to be 23.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

Give a \(92\%\) symmetric confidence interval for the population proportion.

Lower Bound: Upper Bound:

Give a conservative \(92\%\) symmetric confidence interval for the population proportion.

Lower Bound: Upper Bound:

Give a \(92\%\) upper bound for the population proportion.
Give a conservative \(92\%\) upper bound for the population proportion.

(Question ID: 0140313504)

Suppose that researchers are interested in estimating the proportion of individuals who are satisfied with their jobs.

In a sample of size 57, the total number of individuals who report that they are satisfied with their jobs was found to be 30.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

Give an \(88\%\) symmetric confidence interval for the population proportion.

Lower Bound: Upper Bound:

Give a conservative \(88\%\) symmetric confidence interval for the population proportion.

Lower Bound: Upper Bound:

Give an \(88\%\) upper bound for the population proportion.
Give a conservative \(88\%\) upper bound for the population proportion.

(Question ID: 0686111276)

Suppose that researchers are interested in estimating the proportion of individuals in a particular region who recycle.

In a sample of size 75, the total number of individuals who recycle was found to be 17.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

Give a \(91\%\) symmetric confidence interval for the population proportion.

Lower Bound: Upper Bound:

Give a conservative \(91\%\) symmetric confidence interval for the population proportion.

Lower Bound: Upper Bound:

Give a \(91\%\) lower bound for the population proportion.
Give a conservative \(91\%\) lower bound for the population proportion.

(Question ID: 0995762332)

Suppose that researchers are interested in estimating the proportion of high schoolers in a school board who own a smartphone.

In a sample of size 69, the total number of high schoolers with a smartphone was found to be 53.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

Give an \(86\%\) symmetric confidence interval for the population proportion.

Lower Bound: Upper Bound:

Give a conservative \(86\%\) symmetric confidence interval for the population proportion.

Lower Bound: Upper Bound:

Give an \(86\%\) lower bound for the population proportion.
Give a conservative \(86\%\) lower bound for the population proportion.

(Question ID: 0198542307)

Suppose that researchers are interested in estimating the proportion of individuals who are vegan.

In a sample of size 1419, the total number of people who are vegan was found to be 32.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

Give an \(83\%\) symmetric confidence interval for the population proportion.

Lower Bound: Upper Bound:

Give a conservative \(83\%\) symmetric confidence interval for the population proportion.

Lower Bound: Upper Bound:

Give an \(83\%\) upper bound for the population proportion.
Give a conservative \(83\%\) upper bound for the population proportion.

(Question ID: 0356966679)

Suppose that researchers are interested in estimating the proportion of individuals in a particular region living in poverty.

In a sample of size 269, the total number of people living in poverty was found to be 36.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

Give a \(94\%\) symmetric confidence interval for the population proportion.

Lower Bound: Upper Bound:

Give a conservative \(94\%\) symmetric confidence interval for the population proportion.

Lower Bound: Upper Bound:

Give a \(94\%\) upper bound for the population proportion.
Give a conservative \(94\%\) upper bound for the population proportion.

(Question ID: 0602954323)

Suppose that researchers are interested in estimating the proportion of individuals who are satisfied with their jobs.

In a sample of size 44, the total number of individuals who report that they are satisfied with their jobs was found to be 28.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

Give a \(95\%\) symmetric confidence interval for the population proportion.

Lower Bound: Upper Bound:

Give a conservative \(95\%\) symmetric confidence interval for the population proportion.

Lower Bound: Upper Bound:

Give a \(95\%\) upper bound for the population proportion.
Give a conservative \(95\%\) upper bound for the population proportion.

(Question ID: 0357431815)

Suppose that researchers are interested in estimating the proportion of high schoolers in a school board who own a smartphone.

In a sample of size 125, the total number of high schoolers with a smartphone was found to be 91.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

Give a \(97\%\) symmetric confidence interval for the population proportion.

Lower Bound: Upper Bound:

Give a conservative \(97\%\) symmetric confidence interval for the population proportion.

Lower Bound: Upper Bound:

Give a \(97\%\) upper bound for the population proportion.
Give a conservative \(97\%\) upper bound for the population proportion.

(Question ID: 0271893710)

Suppose that researchers are interested in estimating the proportion of individuals who are satisfied with their jobs.

In a sample of size 61, the total number of individuals who report that they are satisfied with their jobs was found to be 40.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

Give an \(81\%\) symmetric confidence interval for the population proportion.

Lower Bound: Upper Bound:

Give a conservative \(81\%\) symmetric confidence interval for the population proportion.

Lower Bound: Upper Bound:

Give an \(81\%\) upper bound for the population proportion.
Give a conservative \(81\%\) upper bound for the population proportion.

(Question ID: 0347275765)

Suppose that researchers are interested in estimating the proportion of individuals who smoke cigarettes.

In a sample of size 280, the total number of individuals who smoke was found to be 27.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

Give an \(87\%\) symmetric confidence interval for the population proportion.

Lower Bound: Upper Bound:

Give a conservative \(87\%\) symmetric confidence interval for the population proportion.

Lower Bound: Upper Bound:

Give an \(87\%\) upper bound for the population proportion.
Give a conservative \(87\%\) upper bound for the population proportion.

(Question ID: 0318743559)

Suppose that researchers are interested in estimating the proportion of individuals who are satisfied with their jobs.

In a sample of size 39, the total number of individuals who report that they are satisfied with their jobs was found to be 19.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

Give a \(90\%\) symmetric confidence interval for the population proportion.

Lower Bound: Upper Bound:

Give a conservative \(90\%\) symmetric confidence interval for the population proportion.

Lower Bound: Upper Bound:

Give a \(90\%\) lower bound for the population proportion.
Give a conservative \(90\%\) lower bound for the population proportion.

(Question ID: 0808526686)

Suppose that researchers are interested in estimating the proportion of high schoolers in a school board who own a smartphone.

In a sample of size 74, the total number of high schoolers with a smartphone was found to be 49.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

Give a \(90\%\) symmetric confidence interval for the population proportion.

Lower Bound: Upper Bound:

Give a conservative \(90\%\) symmetric confidence interval for the population proportion.

Lower Bound: Upper Bound:

Give a \(90\%\) upper bound for the population proportion.
Give a conservative \(90\%\) upper bound for the population proportion.

(Question ID: 0893900127)

Suppose that researchers are interested in estimating the proportion of individuals who are vegan.

In a sample of size 165, the total number of people who are vegan was found to be 24.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

Give a \(93\%\) symmetric confidence interval for the population proportion.

Lower Bound: Upper Bound:

Give a conservative \(93\%\) symmetric confidence interval for the population proportion.

Lower Bound: Upper Bound:

Give a \(93\%\) upper bound for the population proportion.
Give a conservative \(93\%\) upper bound for the population proportion.

(Question ID: 0975506692)

Suppose that researchers are interested in estimating the proportion of high schoolers in a school board who own a smartphone.

In a sample of size 66, the total number of high schoolers with a smartphone was found to be 55.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

Give a \(95\%\) symmetric confidence interval for the population proportion.

Lower Bound: Upper Bound:

Give a conservative \(95\%\) symmetric confidence interval for the population proportion.

Lower Bound: Upper Bound:

Give a \(95\%\) lower bound for the population proportion.
Give a conservative \(95\%\) lower bound for the population proportion.

(Question ID: 0203706468)

Suppose that researchers are interested in estimating the proportion of individuals who use social media daily.

In a sample of size 150, the total number of individuals who use social media daily was found to be 124.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

Give an \(88\%\) symmetric confidence interval for the population proportion.

Lower Bound: Upper Bound:

Give a conservative \(88\%\) symmetric confidence interval for the population proportion.

Lower Bound: Upper Bound:

Give an \(88\%\) lower bound for the population proportion.
Give a conservative \(88\%\) lower bound for the population proportion.

(Question ID: 0017068784)

Suppose that researchers are interested in estimating the proportion of individuals who are satisfied with their jobs.

In a sample of size 69, the total number of individuals who report that they are satisfied with their jobs was found to be 30.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

Give an \(88\%\) symmetric confidence interval for the population proportion.

Lower Bound: Upper Bound:

Give a conservative \(88\%\) symmetric confidence interval for the population proportion.

Lower Bound: Upper Bound:

Give an \(88\%\) upper bound for the population proportion.
Give a conservative \(88\%\) upper bound for the population proportion.

(Question ID: 0210547183)

Suppose that researchers are interested in estimating the proportion of individuals who use social media daily.

In a sample of size 99, the total number of individuals who use social media daily was found to be 65.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

Give a \(98\%\) symmetric confidence interval for the population proportion.

Lower Bound: Upper Bound:

Give a conservative \(98\%\) symmetric confidence interval for the population proportion.

Lower Bound: Upper Bound:

Give a \(98\%\) lower bound for the population proportion.
Give a conservative \(98\%\) lower bound for the population proportion.

(Question ID: 0839842284)

Suppose that researchers are interested in estimating the proportion of individuals who are satisfied with their jobs.

In a sample of size 80, the total number of individuals who report that they are satisfied with their jobs was found to be 23.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

Give an \(84\%\) symmetric confidence interval for the population proportion.

Lower Bound: Upper Bound:

Give a conservative \(84\%\) symmetric confidence interval for the population proportion.

Lower Bound: Upper Bound:

Give an \(84\%\) upper bound for the population proportion.
Give a conservative \(84\%\) upper bound for the population proportion.

(Question ID: 0541844012)

Suppose that researchers are interested in estimating the proportion of individuals who are satisfied with their jobs.

In a sample of size 58, the total number of individuals who report that they are satisfied with their jobs was found to be 34.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

Give a \(90\%\) symmetric confidence interval for the population proportion.

Lower Bound: Upper Bound:

Give a conservative \(90\%\) symmetric confidence interval for the population proportion.

Lower Bound: Upper Bound:

Give a \(90\%\) upper bound for the population proportion.
Give a conservative \(90\%\) upper bound for the population proportion.

(Question ID: 0637267733)

Suppose that researchers are interested in estimating the proportion of individuals who are satisfied with their jobs.

In a sample of size 58, the total number of individuals who report that they are satisfied with their jobs was found to be 35.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

Give a \(92\%\) symmetric confidence interval for the population proportion.

Lower Bound: Upper Bound:

Give a conservative \(92\%\) symmetric confidence interval for the population proportion.

Lower Bound: Upper Bound:

Give a \(92\%\) upper bound for the population proportion.
Give a conservative \(92\%\) upper bound for the population proportion.

(Question ID: 0244707147)

Suppose that researchers are interested in estimating the proportion of individuals in a particular region who recycle.

In a sample of size 54, the total number of individuals who recycle was found to be 14.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

Give a \(93\%\) symmetric confidence interval for the population proportion.

Lower Bound: Upper Bound:

Give a conservative \(93\%\) symmetric confidence interval for the population proportion.

Lower Bound: Upper Bound:

Give a \(93\%\) lower bound for the population proportion.
Give a conservative \(93\%\) lower bound for the population proportion.

(Question ID: 0088740015)

Suppose that researchers are interested in estimating the proportion of individuals who are left-handed.

In a sample of size 307, the total number of left-handed individuals was found to be 15.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

Give a \(94\%\) symmetric confidence interval for the population proportion.

Lower Bound: Upper Bound:

Give a conservative \(94\%\) symmetric confidence interval for the population proportion.

Lower Bound: Upper Bound:

Give a \(94\%\) lower bound for the population proportion.
Give a conservative \(94\%\) lower bound for the population proportion.

(Question ID: 0255891216)

Suppose that researchers are interested in estimating the proportion of individuals who smoke cigarettes.

In a sample of size 52, the total number of individuals who smoke was found to be 21.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

Give a \(90\%\) symmetric confidence interval for the population proportion.

Lower Bound: Upper Bound:

Give a conservative \(90\%\) symmetric confidence interval for the population proportion.

Lower Bound: Upper Bound:

Give a \(90\%\) lower bound for the population proportion.
Give a conservative \(90\%\) lower bound for the population proportion.

(Question ID: 0632406341)

Suppose that researchers are interested in estimating the proportion of high schoolers in a school board who own a smartphone.

In a sample of size 149, the total number of high schoolers with a smartphone was found to be 125.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

Give a \(92\%\) symmetric confidence interval for the population proportion.

Lower Bound: Upper Bound:

Give a conservative \(92\%\) symmetric confidence interval for the population proportion.

Lower Bound: Upper Bound:

Give a \(92\%\) upper bound for the population proportion.
Give a conservative \(92\%\) upper bound for the population proportion.

(Question ID: 0174119924)

Suppose that researchers are interested in estimating the proportion of individuals who are vegan.

In a sample of size 219, the total number of people who are vegan was found to be 23.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

Give a \(93\%\) symmetric confidence interval for the population proportion.

Lower Bound: Upper Bound:

Give a conservative \(93\%\) symmetric confidence interval for the population proportion.

Lower Bound: Upper Bound:

Give a \(93\%\) upper bound for the population proportion.
Give a conservative \(93\%\) upper bound for the population proportion.

(Question ID: 0008090319)

Suppose that researchers are interested in estimating the proportion of individuals in a particular region living in poverty.

In a sample of size 187, the total number of people living in poverty was found to be 15.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

Give a \(97\%\) symmetric confidence interval for the population proportion.

Lower Bound: Upper Bound:

Give a conservative \(97\%\) symmetric confidence interval for the population proportion.

Lower Bound: Upper Bound:

Give a \(97\%\) upper bound for the population proportion.
Give a conservative \(97\%\) upper bound for the population proportion.

(Question ID: 0451578885)

Suppose that researchers are interested in estimating the proportion of individuals who are satisfied with their jobs.

In a sample of size 62, the total number of individuals who report that they are satisfied with their jobs was found to be 30.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

Give an \(80\%\) symmetric confidence interval for the population proportion.

Lower Bound: Upper Bound:

Give a conservative \(80\%\) symmetric confidence interval for the population proportion.

Lower Bound: Upper Bound:

Give an \(80\%\) upper bound for the population proportion.
Give a conservative \(80\%\) upper bound for the population proportion.

(Question ID: 0809080280)

Suppose that researchers are interested in estimating the proportion of individuals who are left-handed.

In a sample of size 283, the total number of left-handed individuals was found to be 22.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

Give a \(99\%\) symmetric confidence interval for the population proportion.

Lower Bound: Upper Bound:

Give a conservative \(99\%\) symmetric confidence interval for the population proportion.

Lower Bound: Upper Bound:

Give a \(99\%\) upper bound for the population proportion.
Give a conservative \(99\%\) upper bound for the population proportion.

(Question ID: 0836537250)

Suppose that researchers are interested in estimating the proportion of high schoolers in a school board who own a smartphone.

In a sample of size 53, the total number of high schoolers with a smartphone was found to be 39.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

Give an \(89\%\) symmetric confidence interval for the population proportion.

Lower Bound: Upper Bound:

Give a conservative \(89\%\) symmetric confidence interval for the population proportion.

Lower Bound: Upper Bound:

Give an \(89\%\) lower bound for the population proportion.
Give a conservative \(89\%\) lower bound for the population proportion.

(Question ID: 0158292919)

Suppose that researchers are interested in estimating the proportion of individuals who use social media daily.

In a sample of size 58, the total number of individuals who use social media daily was found to be 33.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

Give an \(82\%\) symmetric confidence interval for the population proportion.

Lower Bound: Upper Bound:

Give a conservative \(82\%\) symmetric confidence interval for the population proportion.

Lower Bound: Upper Bound:

Give an \(82\%\) lower bound for the population proportion.
Give a conservative \(82\%\) lower bound for the population proportion.

(Question ID: 0573664495)

Suppose that researchers are interested in estimating the proportion of individuals who are vegan.

In a sample of size 231, the total number of people who are vegan was found to be 29.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

Give an \(82\%\) symmetric confidence interval for the population proportion.

Lower Bound: Upper Bound:

Give a conservative \(82\%\) symmetric confidence interval for the population proportion.

Lower Bound: Upper Bound:

Give an \(82\%\) upper bound for the population proportion.
Give a conservative \(82\%\) upper bound for the population proportion.

(Question ID: 0973034501)

Suppose that researchers are interested in estimating the proportion of individuals in a particular region living in poverty.

In a sample of size 265, the total number of people living in poverty was found to be 25.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

Give an \(82\%\) symmetric confidence interval for the population proportion.

Lower Bound: Upper Bound:

Give a conservative \(82\%\) symmetric confidence interval for the population proportion.

Lower Bound: Upper Bound:

Give an \(82\%\) lower bound for the population proportion.
Give a conservative \(82\%\) lower bound for the population proportion.

(Question ID: 0173703952)

Suppose that researchers are interested in estimating the proportion of high schoolers in a school board who own a smartphone.

In a sample of size 41, the total number of high schoolers with a smartphone was found to be 26.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

Give a \(91\%\) symmetric confidence interval for the population proportion.

Lower Bound: Upper Bound:

Give a conservative \(91\%\) symmetric confidence interval for the population proportion.

Lower Bound: Upper Bound:

Give a \(91\%\) lower bound for the population proportion.
Give a conservative \(91\%\) lower bound for the population proportion.

(Question ID: 0244120234)

Self Assessment 14.7

Suppose that the red blood cell counts of individuals in a population are approximately normally distributed.

In a sample of size 41, the mean red blood cell count was found to be 5.07, with a standard deviation of 0.46.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

Give a \(97\%\) confidence interval for the variance.

Lower Bound: Upper Bound:

Give an approximate \(97\%\) confidence interval for the standard deviation.

Lower Bound: Upper Bound:

(Question ID: 0742930226)

Suppose that weights of newborns in a population are approximately normally distributed.

In a sample of size 37, the mean weight was found to be 3501.38, with a standard deviation of 461.05.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

Give a \(99\%\) confidence interval for the variance.

Lower Bound: Upper Bound:

Give an approximate \(99\%\) confidence interval for the standard deviation.

Lower Bound: Upper Bound:

(Question ID: 0953866743)

Suppose that the reaction times of individuals in a population are approximately normally distributed.

In a sample of size 36, the mean reaction time was found to be 238.05, with a standard deviation of 45.03.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

Give a \(90\%\) confidence interval for the variance.

Lower Bound: Upper Bound:

Give an approximate \(90\%\) confidence interval for the standard deviation.

Lower Bound: Upper Bound:

(Question ID: 0582250788)

Suppose that diastolic blood pressures of individuals in a population are approximately normally distributed.

In a sample of size 28, the mean blood pressure was found to be 89.03, with a standard deviation of 7.86.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

Give an \(82\%\) confidence interval for the variance.

Lower Bound: Upper Bound:

Give an approximate \(82\%\) confidence interval for the standard deviation.

Lower Bound: Upper Bound:

(Question ID: 0553472622)

Suppose that the serum cholesterol levels of individuals in a population are approximately normally distributed.

In a sample of size 42, the mean serum cholesterol level was found to be 201.77, with a standard deviation of 45.45.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

Give an \(86\%\) confidence interval for the variance.

Lower Bound: Upper Bound:

Give an approximate \(86\%\) confidence interval for the standard deviation.

Lower Bound: Upper Bound:

(Question ID: 0646315891)

Suppose that the reaction times of individuals in a population are approximately normally distributed.

In a sample of size 32, the mean reaction time was found to be 258.45, with a standard deviation of 45.62.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

Give a \(98\%\) confidence interval for the variance.

Lower Bound: Upper Bound:

Give an approximate \(98\%\) confidence interval for the standard deviation.

Lower Bound: Upper Bound:

(Question ID: 0921667867)

Suppose that the red blood cell counts of individuals in a population are approximately normally distributed.

In a sample of size 31, the mean red blood cell count was found to be 5, with a standard deviation of 0.58.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

Give an \(87\%\) confidence interval for the variance.

Lower Bound: Upper Bound:

Give an approximate \(87\%\) confidence interval for the standard deviation.

Lower Bound: Upper Bound:

(Question ID: 0500018612)

Suppose that the reaction times of individuals in a population are approximately normally distributed.

In a sample of size 48, the mean reaction time was found to be 243.11, with a standard deviation of 44.05.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

Give a \(95\%\) confidence interval for the variance.

Lower Bound: Upper Bound:

Give an approximate \(95\%\) confidence interval for the standard deviation.

Lower Bound: Upper Bound:

(Question ID: 0039758655)

Suppose that systolic blood pressures of individuals in a population are approximately normally distributed.

In a sample of size 13, the mean blood pressure was found to be 120.17, with a standard deviation of 11.25.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

Give a \(90\%\) confidence interval for the variance.

Lower Bound: Upper Bound:

Give an approximate \(90\%\) confidence interval for the standard deviation.

Lower Bound: Upper Bound:

(Question ID: 0558165203)

Suppose that the serum cholesterol levels of individuals in a population are approximately normally distributed.

In a sample of size 47, the mean serum cholesterol level was found to be 200.77, with a standard deviation of 42.45.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

Give an \(87\%\) confidence interval for the variance.

Lower Bound: Upper Bound:

Give an approximate \(87\%\) confidence interval for the standard deviation.

Lower Bound: Upper Bound:

(Question ID: 0158699444)

Suppose that diastolic blood pressures of individuals in a population are approximately normally distributed.

In a sample of size 17, the mean blood pressure was found to be 88.91, with a standard deviation of 8.12.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

Give a \(99\%\) confidence interval for the variance.

Lower Bound: Upper Bound:

Give an approximate \(99\%\) confidence interval for the standard deviation.

Lower Bound: Upper Bound:

(Question ID: 0155232583)

Suppose that the red blood cell counts of individuals in a population are approximately normally distributed.

In a sample of size 36, the mean red blood cell count was found to be 5.06, with a standard deviation of 0.49.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

Give a \(94\%\) confidence interval for the variance.

Lower Bound: Upper Bound:

Give an approximate \(94\%\) confidence interval for the standard deviation.

Lower Bound: Upper Bound:

(Question ID: 0544846579)

Suppose that weights of newborns in a population are approximately normally distributed.

In a sample of size 10, the mean weight was found to be 3443.56, with a standard deviation of 446.86.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

Give an \(81\%\) confidence interval for the variance.

Lower Bound: Upper Bound:

Give an approximate \(81\%\) confidence interval for the standard deviation.

Lower Bound: Upper Bound:

(Question ID: 0259512259)

Suppose that the serum cholesterol levels of individuals in a population are approximately normally distributed.

In a sample of size 46, the mean serum cholesterol level was found to be 204.31, with a standard deviation of 37.3.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

Give a \(94\%\) confidence interval for the variance.

Lower Bound: Upper Bound:

Give an approximate \(94\%\) confidence interval for the standard deviation.

Lower Bound: Upper Bound:

(Question ID: 0770243050)

Suppose that weights of newborns in a population are approximately normally distributed.

In a sample of size 36, the mean weight was found to be 3595.47, with a standard deviation of 463.45.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

Give a \(99\%\) confidence interval for the variance.

Lower Bound: Upper Bound:

Give an approximate \(99\%\) confidence interval for the standard deviation.

Lower Bound: Upper Bound:

(Question ID: 0759895220)

Suppose that the serum cholesterol levels of individuals in a population are approximately normally distributed.

In a sample of size 31, the mean serum cholesterol level was found to be 194.68, with a standard deviation of 39.77.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

Give an \(81\%\) confidence interval for the variance.

Lower Bound: Upper Bound:

Give an approximate \(81\%\) confidence interval for the standard deviation.

Lower Bound: Upper Bound:

(Question ID: 0674018190)

Suppose that the reaction times of individuals in a population are approximately normally distributed.

In a sample of size 49, the mean reaction time was found to be 245.01, with a standard deviation of 49.45.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

Give an \(85\%\) confidence interval for the variance.

Lower Bound: Upper Bound:

Give an approximate \(85\%\) confidence interval for the standard deviation.

Lower Bound: Upper Bound:

(Question ID: 0732436450)

Suppose that body temperatures of individuals in a population are approximately normally distributed.

In a sample of size 48, the mean temperature was found to be 36.55, with a standard deviation of 0.58.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

Give an \(89\%\) confidence interval for the variance.

Lower Bound: Upper Bound:

Give an approximate \(89\%\) confidence interval for the standard deviation.

Lower Bound: Upper Bound:

(Question ID: 0643984675)

Suppose that the red blood cell counts of individuals in a population are approximately normally distributed.

In a sample of size 32, the mean red blood cell count was found to be 4.94, with a standard deviation of 0.51.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

Give an \(80\%\) confidence interval for the variance.

Lower Bound: Upper Bound:

Give an approximate \(80\%\) confidence interval for the standard deviation.

Lower Bound: Upper Bound:

(Question ID: 0088227468)

Suppose that body temperatures of individuals in a population are approximately normally distributed.

In a sample of size 33, the mean temperature was found to be 36.66, with a standard deviation of 0.5.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

Give an \(82\%\) confidence interval for the variance.

Lower Bound: Upper Bound:

Give an approximate \(82\%\) confidence interval for the standard deviation.

Lower Bound: Upper Bound:

(Question ID: 0459458424)

Suppose that systolic blood pressures of individuals in a population are approximately normally distributed.

In a sample of size 48, the mean blood pressure was found to be 118.99, with a standard deviation of 9.74.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

Give a \(93\%\) confidence interval for the variance.

Lower Bound: Upper Bound:

Give an approximate \(93\%\) confidence interval for the standard deviation.

Lower Bound: Upper Bound:

(Question ID: 0998754688)

Suppose that diastolic blood pressures of individuals in a population are approximately normally distributed.

In a sample of size 10, the mean blood pressure was found to be 83.05, with a standard deviation of 9.09.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

Give an \(89\%\) confidence interval for the variance.

Lower Bound: Upper Bound:

Give an approximate \(89\%\) confidence interval for the standard deviation.

Lower Bound: Upper Bound:

(Question ID: 0471188723)

Suppose that body temperatures of individuals in a population are approximately normally distributed.

In a sample of size 11, the mean temperature was found to be 36.62, with a standard deviation of 0.68.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

Give a \(90\%\) confidence interval for the variance.

Lower Bound: Upper Bound:

Give an approximate \(90\%\) confidence interval for the standard deviation.

Lower Bound: Upper Bound:

(Question ID: 0837640566)

Suppose that systolic blood pressures of individuals in a population are approximately normally distributed.

In a sample of size 43, the mean blood pressure was found to be 119.91, with a standard deviation of 10.57.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

Give an \(85\%\) confidence interval for the variance.

Lower Bound: Upper Bound:

Give an approximate \(85\%\) confidence interval for the standard deviation.

Lower Bound: Upper Bound:

(Question ID: 0769838755)

Suppose that the red blood cell counts of individuals in a population are approximately normally distributed.

In a sample of size 13, the mean red blood cell count was found to be 4.66, with a standard deviation of 0.52.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

Give an \(85\%\) confidence interval for the variance.

Lower Bound: Upper Bound:

Give an approximate \(85\%\) confidence interval for the standard deviation.

Lower Bound: Upper Bound:

(Question ID: 0937734499)

Suppose that systolic blood pressures of individuals in a population are approximately normally distributed.

In a sample of size 36, the mean blood pressure was found to be 123.5, with a standard deviation of 7.97.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

Give a \(90\%\) confidence interval for the variance.

Lower Bound: Upper Bound:

Give an approximate \(90\%\) confidence interval for the standard deviation.

Lower Bound: Upper Bound:

(Question ID: 0292258245)

Suppose that diastolic blood pressures of individuals in a population are approximately normally distributed.

In a sample of size 22, the mean blood pressure was found to be 89.4, with a standard deviation of 9.16.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

Give a \(90\%\) confidence interval for the variance.

Lower Bound: Upper Bound:

Give an approximate \(90\%\) confidence interval for the standard deviation.

Lower Bound: Upper Bound:

(Question ID: 0983017302)

Suppose that the heights of individuals in a population are approximately normally distributed.

In a sample of size 26, the mean height was found to be 166.32, with a standard deviation of 6.66.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

Give a \(95\%\) confidence interval for the variance.

Lower Bound: Upper Bound:

Give an approximate \(95\%\) confidence interval for the standard deviation.

Lower Bound: Upper Bound:

(Question ID: 0282712131)

Suppose that body temperatures of individuals in a population are approximately normally distributed.

In a sample of size 15, the mean temperature was found to be 36.75, with a standard deviation of 0.28.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

Give an \(84\%\) confidence interval for the variance.

Lower Bound: Upper Bound:

Give an approximate \(84\%\) confidence interval for the standard deviation.

Lower Bound: Upper Bound:

(Question ID: 0010944452)

Suppose that systolic blood pressures of individuals in a population are approximately normally distributed.

In a sample of size 13, the mean blood pressure was found to be 123.07, with a standard deviation of 11.6.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

Give a \(92\%\) confidence interval for the variance.

Lower Bound: Upper Bound:

Give an approximate \(92\%\) confidence interval for the standard deviation.

Lower Bound: Upper Bound:

(Question ID: 0343483626)

Suppose that the serum cholesterol levels of individuals in a population are approximately normally distributed.

In a sample of size 36, the mean serum cholesterol level was found to be 200.25, with a standard deviation of 41.41.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

Give an \(83\%\) confidence interval for the variance.

Lower Bound: Upper Bound:

Give an approximate \(83\%\) confidence interval for the standard deviation.

Lower Bound: Upper Bound:

(Question ID: 0336339999)

Suppose that the reaction times of individuals in a population are approximately normally distributed.

In a sample of size 50, the mean reaction time was found to be 261.19, with a standard deviation of 40.02.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

Give an \(87\%\) confidence interval for the variance.

Lower Bound: Upper Bound:

Give an approximate \(87\%\) confidence interval for the standard deviation.

Lower Bound: Upper Bound:

(Question ID: 0811172852)

Suppose that the reaction times of individuals in a population are approximately normally distributed.

In a sample of size 17, the mean reaction time was found to be 263.03, with a standard deviation of 53.11.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

Give an \(83\%\) confidence interval for the variance.

Lower Bound: Upper Bound:

Give an approximate \(83\%\) confidence interval for the standard deviation.

Lower Bound: Upper Bound:

(Question ID: 0487676733)

Suppose that the reaction times of individuals in a population are approximately normally distributed.

In a sample of size 20, the mean reaction time was found to be 250.61, with a standard deviation of 62.72.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

Give an \(85\%\) confidence interval for the variance.

Lower Bound: Upper Bound:

Give an approximate \(85\%\) confidence interval for the standard deviation.

Lower Bound: Upper Bound:

(Question ID: 0515473133)

Suppose that the serum cholesterol levels of individuals in a population are approximately normally distributed.

In a sample of size 12, the mean serum cholesterol level was found to be 214.67, with a standard deviation of 24.07.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

Give an \(81\%\) confidence interval for the variance.

Lower Bound: Upper Bound:

Give an approximate \(81\%\) confidence interval for the standard deviation.

Lower Bound: Upper Bound:

(Question ID: 0522977671)

Suppose that weights of newborns in a population are approximately normally distributed.

In a sample of size 33, the mean weight was found to be 3412.45, with a standard deviation of 547.97.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

Give a \(93\%\) confidence interval for the variance.

Lower Bound: Upper Bound:

Give an approximate \(93\%\) confidence interval for the standard deviation.

Lower Bound: Upper Bound:

(Question ID: 0278260483)

Suppose that the heights of individuals in a population are approximately normally distributed.

In a sample of size 19, the mean height was found to be 168.39, with a standard deviation of 6.24.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

Give a \(96\%\) confidence interval for the variance.

Lower Bound: Upper Bound:

Give an approximate \(96\%\) confidence interval for the standard deviation.

Lower Bound: Upper Bound:

(Question ID: 0796670397)

Suppose that the heights of individuals in a population are approximately normally distributed.

In a sample of size 10, the mean height was found to be 166.38, with a standard deviation of 9.19.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

Give an \(85\%\) confidence interval for the variance.

Lower Bound: Upper Bound:

Give an approximate \(85\%\) confidence interval for the standard deviation.

Lower Bound: Upper Bound:

(Question ID: 0313492853)

Suppose that the serum cholesterol levels of individuals in a population are approximately normally distributed.

In a sample of size 39, the mean serum cholesterol level was found to be 201.22, with a standard deviation of 36.54.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

Give a \(95\%\) confidence interval for the variance.

Lower Bound: Upper Bound:

Give an approximate \(95\%\) confidence interval for the standard deviation.

Lower Bound: Upper Bound:

(Question ID: 0476681710)

Suppose that body temperatures of individuals in a population are approximately normally distributed.

In a sample of size 18, the mean temperature was found to be 36.51, with a standard deviation of 0.37.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

Give a \(96\%\) confidence interval for the variance.

Lower Bound: Upper Bound:

Give an approximate \(96\%\) confidence interval for the standard deviation.

Lower Bound: Upper Bound:

(Question ID: 0047044667)

Suppose that weights of newborns in a population are approximately normally distributed.

In a sample of size 41, the mean weight was found to be 3464.15, with a standard deviation of 433.07.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

Give an \(83\%\) confidence interval for the variance.

Lower Bound: Upper Bound:

Give an approximate \(83\%\) confidence interval for the standard deviation.

Lower Bound: Upper Bound:

(Question ID: 0933194930)

Suppose that weights of newborns in a population are approximately normally distributed.

In a sample of size 25, the mean weight was found to be 3432.49, with a standard deviation of 466.95.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

Give a \(92\%\) confidence interval for the variance.

Lower Bound: Upper Bound:

Give an approximate \(92\%\) confidence interval for the standard deviation.

Lower Bound: Upper Bound:

(Question ID: 0699191989)

Suppose that the heights of individuals in a population are approximately normally distributed.

In a sample of size 36, the mean height was found to be 169.01, with a standard deviation of 7.49.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

Give a \(97\%\) confidence interval for the variance.

Lower Bound: Upper Bound:

Give an approximate \(97\%\) confidence interval for the standard deviation.

Lower Bound: Upper Bound:

(Question ID: 0698582702)

Suppose that the serum cholesterol levels of individuals in a population are approximately normally distributed.

In a sample of size 24, the mean serum cholesterol level was found to be 199.65, with a standard deviation of 38.53.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

Give an \(82\%\) confidence interval for the variance.

Lower Bound: Upper Bound:

Give an approximate \(82\%\) confidence interval for the standard deviation.

Lower Bound: Upper Bound:

(Question ID: 0658788762)

Suppose that diastolic blood pressures of individuals in a population are approximately normally distributed.

In a sample of size 39, the mean blood pressure was found to be 88.51, with a standard deviation of 8.24.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

Give a \(95\%\) confidence interval for the variance.

Lower Bound: Upper Bound:

Give an approximate \(95\%\) confidence interval for the standard deviation.

Lower Bound: Upper Bound:

(Question ID: 0104607563)

Suppose that the reaction times of individuals in a population are approximately normally distributed.

In a sample of size 42, the mean reaction time was found to be 238.46, with a standard deviation of 51.38.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

Give a \(99\%\) confidence interval for the variance.

Lower Bound: Upper Bound:

Give an approximate \(99\%\) confidence interval for the standard deviation.

Lower Bound: Upper Bound:

(Question ID: 0974296254)

Suppose that the reaction times of individuals in a population are approximately normally distributed.

In a sample of size 43, the mean reaction time was found to be 254.51, with a standard deviation of 43.54.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

Give a \(91\%\) confidence interval for the variance.

Lower Bound: Upper Bound:

Give an approximate \(91\%\) confidence interval for the standard deviation.

Lower Bound: Upper Bound:

(Question ID: 0431894619)

Suppose that weights of newborns in a population are approximately normally distributed.

In a sample of size 50, the mean weight was found to be 3490.94, with a standard deviation of 509.6.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

Give a \(92\%\) confidence interval for the variance.

Lower Bound: Upper Bound:

Give an approximate \(92\%\) confidence interval for the standard deviation.

Lower Bound: Upper Bound:

(Question ID: 0460519243)

Suppose that weights of newborns in a population are approximately normally distributed.

In a sample of size 41, the mean weight was found to be 3301.41, with a standard deviation of 445.51.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

Give a \(90\%\) confidence interval for the variance.

Lower Bound: Upper Bound:

Give an approximate \(90\%\) confidence interval for the standard deviation.

Lower Bound: Upper Bound:

(Question ID: 0688169090)

Suppose that the heights of individuals in a population are approximately normally distributed.

In a sample of size 21, the mean height was found to be 168.43, with a standard deviation of 5.43.

(Answer -1 if you do not have sufficient information to answer any part of the question.)

Give a \(92\%\) confidence interval for the variance.

Lower Bound: Upper Bound:

Give an approximate \(92\%\) confidence interval for the standard deviation.

Lower Bound: Upper Bound:

(Question ID: 0927368499)

Self Assessment 14.8

Suppose that the serum cholesterol levels of individuals in a population are approximately normally distributed, with a known population standard deviation of 39.47.

How large of a sample would be required in order for a \(92\%\) confidence interval to be within 2.71 of the truth?
How large of a sample would be required in order for a \(92\%\) confidence interval to no more than 7.3 wide?

(Question ID: 0177679825)

Suppose that weights of newborns in a population are approximately normally distributed, with a known population standard deviation of 555.49.

How large of a sample would be required in order for an \(80\%\) confidence interval to be within 11.68 of the truth?
How large of a sample would be required in order for an \(80\%\) confidence interval to no more than 54.4 wide?

(Question ID: 0610028132)

Suppose that systolic blood pressures of individuals in a population are approximately normally distributed, with a known population standard deviation of 8.66.

How large of a sample would be required in order for a \(92\%\) confidence interval to be within 1.8 of the truth?
How large of a sample would be required in order for a \(92\%\) confidence interval to no more than 0.76 wide?

(Question ID: 0359604317)

Suppose that diastolic blood pressures of individuals in a population are approximately normally distributed, with a known population standard deviation of 7.98.

How large of a sample would be required in order for an \(85\%\) confidence interval to be within 0.19 of the truth?
How large of a sample would be required in order for an \(85\%\) confidence interval to no more than 0.36 wide?

(Question ID: 0610779991)

Suppose that weights of newborns in a population are approximately normally distributed, with a known population standard deviation of 473.55.

How large of a sample would be required in order for an \(80\%\) confidence interval to be within 13.53 of the truth?
How large of a sample would be required in order for an \(80\%\) confidence interval to no more than 19.34 wide?

(Question ID: 0967155284)

Suppose that the red blood cell counts of individuals in a population are approximately normally distributed, with a known population standard deviation of 0.55.

How large of a sample would be required in order for a \(95\%\) confidence interval to be within 0.02 of the truth?
How large of a sample would be required in order for a \(95\%\) confidence interval to no more than 0.08 wide?

(Question ID: 0144830397)

Suppose that body temperatures of individuals in a population are approximately normally distributed, with a known population standard deviation of 0.41.

How large of a sample would be required in order for an \(80\%\) confidence interval to be within 0.01 of the truth?
How large of a sample would be required in order for an \(80\%\) confidence interval to no more than 0.02 wide?

(Question ID: 0387022971)

Suppose that the reaction times of individuals in a population are approximately normally distributed, with a known population standard deviation of 47.7.

How large of a sample would be required in order for an \(81\%\) confidence interval to be within 1.07 of the truth?
How large of a sample would be required in order for an \(81\%\) confidence interval to no more than 1.84 wide?

(Question ID: 0680451453)

Suppose that body temperatures of individuals in a population are approximately normally distributed, with a known population standard deviation of 0.59.

How large of a sample would be required in order for an \(83\%\) confidence interval to be within 0.02 of the truth?
How large of a sample would be required in order for an \(83\%\) confidence interval to no more than 0.04 wide?

(Question ID: 0572170443)

Suppose that body temperatures of individuals in a population are approximately normally distributed, with a known population standard deviation of 0.47.

How large of a sample would be required in order for a \(96\%\) confidence interval to be within 0.02 of the truth?
How large of a sample would be required in order for a \(96\%\) confidence interval to no more than 0.04 wide?

(Question ID: 0377229763)

Suppose that the heights of individuals in a population are approximately normally distributed, with a known population standard deviation of 6.91.

How large of a sample would be required in order for a \(90\%\) confidence interval to be within 0.18 of the truth?
How large of a sample would be required in order for a \(90\%\) confidence interval to no more than 0.74 wide?

(Question ID: 0451143320)

Suppose that body temperatures of individuals in a population are approximately normally distributed, with a known population standard deviation of 0.5.

How large of a sample would be required in order for an \(87\%\) confidence interval to be within 0.02 of the truth?
How large of a sample would be required in order for an \(87\%\) confidence interval to no more than 0.06 wide?

(Question ID: 0660199024)

Suppose that the reaction times of individuals in a population are approximately normally distributed, with a known population standard deviation of 52.96.

How large of a sample would be required in order for an \(88\%\) confidence interval to be within 1.81 of the truth?
How large of a sample would be required in order for an \(88\%\) confidence interval to no more than 4.24 wide?

(Question ID: 0835634611)

Suppose that the serum cholesterol levels of individuals in a population are approximately normally distributed, with a known population standard deviation of 34.01.

How large of a sample would be required in order for an \(86\%\) confidence interval to be within 1.03 of the truth?
How large of a sample would be required in order for an \(86\%\) confidence interval to no more than 1.86 wide?

(Question ID: 0273303900)

Suppose that the serum cholesterol levels of individuals in a population are approximately normally distributed, with a known population standard deviation of 39.

How large of a sample would be required in order for an \(88\%\) confidence interval to be within 1.09 of the truth?
How large of a sample would be required in order for an \(88\%\) confidence interval to no more than 2.64 wide?

(Question ID: 0084666522)

Suppose that the serum cholesterol levels of individuals in a population are approximately normally distributed, with a known population standard deviation of 40.27.

How large of a sample would be required in order for a \(95\%\) confidence interval to be within 1.12 of the truth?
How large of a sample would be required in order for a \(95\%\) confidence interval to no more than 2.76 wide?

(Question ID: 0283731731)

Suppose that weights of newborns in a population are approximately normally distributed, with a known population standard deviation of 500.6.

How large of a sample would be required in order for a \(96\%\) confidence interval to be within 47.07 of the truth?
How large of a sample would be required in order for a \(96\%\) confidence interval to no more than 33.36 wide?

(Question ID: 0468460502)

Suppose that the reaction times of individuals in a population are approximately normally distributed, with a known population standard deviation of 53.74.

How large of a sample would be required in order for a \(95\%\) confidence interval to be within 3.24 of the truth?
How large of a sample would be required in order for a \(95\%\) confidence interval to no more than 3.18 wide?

(Question ID: 0730441115)

Suppose that body temperatures of individuals in a population are approximately normally distributed, with a known population standard deviation of 0.45.

How large of a sample would be required in order for an \(87\%\) confidence interval to be within 0.03 of the truth?
How large of a sample would be required in order for an \(87\%\) confidence interval to no more than 0.02 wide?

(Question ID: 0184610971)

Suppose that body temperatures of individuals in a population are approximately normally distributed, with a known population standard deviation of 0.46.

How large of a sample would be required in order for a \(99\%\) confidence interval to be within 0.02 of the truth?
How large of a sample would be required in order for a \(99\%\) confidence interval to no more than 0.04 wide?

(Question ID: 0537874205)

Suppose that the heights of individuals in a population are approximately normally distributed, with a known population standard deviation of 6.9.

How large of a sample would be required in order for a \(95\%\) confidence interval to be within 0.25 of the truth?
How large of a sample would be required in order for a \(95\%\) confidence interval to no more than 0.66 wide?

(Question ID: 0931621308)

Suppose that weights of newborns in a population are approximately normally distributed, with a known population standard deviation of 507.28.

How large of a sample would be required in order for a \(99\%\) confidence interval to be within 51.57 of the truth?
How large of a sample would be required in order for a \(99\%\) confidence interval to no more than 42.46 wide?

(Question ID: 0380670458)

Suppose that body temperatures of individuals in a population are approximately normally distributed, with a known population standard deviation of 0.51.

How large of a sample would be required in order for a \(96\%\) confidence interval to be within 0.02 of the truth?
How large of a sample would be required in order for a \(96\%\) confidence interval to no more than 0.04 wide?

(Question ID: 0519051230)

Suppose that diastolic blood pressures of individuals in a population are approximately normally distributed, with a known population standard deviation of 7.66.

How large of a sample would be required in order for a \(96\%\) confidence interval to be within 0.23 of the truth?
How large of a sample would be required in order for a \(96\%\) confidence interval to no more than 0.74 wide?

(Question ID: 0656880834)

Suppose that the red blood cell counts of individuals in a population are approximately normally distributed, with a known population standard deviation of 0.51.

How large of a sample would be required in order for an \(88\%\) confidence interval to be within 0.03 of the truth?
How large of a sample would be required in order for an \(88\%\) confidence interval to no more than 0.04 wide?

(Question ID: 0945255071)

Suppose that weights of newborns in a population are approximately normally distributed, with a known population standard deviation of 505.57.

How large of a sample would be required in order for an \(85\%\) confidence interval to be within 13.69 of the truth?
How large of a sample would be required in order for an \(85\%\) confidence interval to no more than 22 wide?

(Question ID: 0987452053)

Suppose that the reaction times of individuals in a population are approximately normally distributed, with a known population standard deviation of 41.04.

How large of a sample would be required in order for a \(93\%\) confidence interval to be within 1.9 of the truth?
How large of a sample would be required in order for a \(93\%\) confidence interval to no more than 2.38 wide?

(Question ID: 0089433390)

Suppose that weights of newborns in a population are approximately normally distributed, with a known population standard deviation of 768.2.

How large of a sample would be required in order for a \(91\%\) confidence interval to be within 35.47 of the truth?
How large of a sample would be required in order for a \(91\%\) confidence interval to no more than 49.68 wide?

(Question ID: 0544612521)

Suppose that the red blood cell counts of individuals in a population are approximately normally distributed, with a known population standard deviation of 0.53.

How large of a sample would be required in order for an \(88\%\) confidence interval to be within 0.01 of the truth?
How large of a sample would be required in order for an \(88\%\) confidence interval to no more than 0.04 wide?

(Question ID: 0787763919)

Suppose that the serum cholesterol levels of individuals in a population are approximately normally distributed, with a known population standard deviation of 41.41.

How large of a sample would be required in order for a \(93\%\) confidence interval to be within 1.21 of the truth?
How large of a sample would be required in order for a \(93\%\) confidence interval to no more than 3.22 wide?

(Question ID: 0094233779)

Suppose that the serum cholesterol levels of individuals in a population are approximately normally distributed, with a known population standard deviation of 38.25.

How large of a sample would be required in order for an \(82\%\) confidence interval to be within 0.83 of the truth?
How large of a sample would be required in order for an \(82\%\) confidence interval to no more than 1.68 wide?

(Question ID: 0253434166)

Suppose that the serum cholesterol levels of individuals in a population are approximately normally distributed, with a known population standard deviation of 46.37.

How large of a sample would be required in order for an \(87\%\) confidence interval to be within 1.74 of the truth?
How large of a sample would be required in order for an \(87\%\) confidence interval to no more than 2.14 wide?

(Question ID: 0196765207)

Suppose that weights of newborns in a population are approximately normally distributed, with a known population standard deviation of 500.32.

How large of a sample would be required in order for a \(93\%\) confidence interval to be within 17.7 of the truth?
How large of a sample would be required in order for a \(93\%\) confidence interval to no more than 41.9 wide?

(Question ID: 0254894267)

Suppose that weights of newborns in a population are approximately normally distributed, with a known population standard deviation of 457.98.

How large of a sample would be required in order for an \(81\%\) confidence interval to be within 11.33 of the truth?
How large of a sample would be required in order for an \(81\%\) confidence interval to no more than 24.6 wide?

(Question ID: 0784282455)

Suppose that body temperatures of individuals in a population are approximately normally distributed, with a known population standard deviation of 0.49.

How large of a sample would be required in order for a \(98\%\) confidence interval to be within 0.02 of the truth?
How large of a sample would be required in order for a \(98\%\) confidence interval to no more than 0.04 wide?

(Question ID: 0314509604)

Suppose that the heights of individuals in a population are approximately normally distributed, with a known population standard deviation of 5.71.

How large of a sample would be required in order for an \(83\%\) confidence interval to be within 0.45 of the truth?
How large of a sample would be required in order for an \(83\%\) confidence interval to no more than 0.3 wide?

(Question ID: 0110336311)

Suppose that the red blood cell counts of individuals in a population are approximately normally distributed, with a known population standard deviation of 0.42.

How large of a sample would be required in order for an \(85\%\) confidence interval to be within 0.02 of the truth?
How large of a sample would be required in order for an \(85\%\) confidence interval to no more than 0.02 wide?

(Question ID: 0367065191)

Suppose that the red blood cell counts of individuals in a population are approximately normally distributed, with a known population standard deviation of 0.54.

How large of a sample would be required in order for a \(99\%\) confidence interval to be within 0.04 of the truth?
How large of a sample would be required in order for a \(99\%\) confidence interval to no more than 0.04 wide?

(Question ID: 0104170757)

Suppose that the red blood cell counts of individuals in a population are approximately normally distributed, with a known population standard deviation of 0.5.

How large of a sample would be required in order for a \(96\%\) confidence interval to be within 0.02 of the truth?
How large of a sample would be required in order for a \(96\%\) confidence interval to no more than 0.06 wide?

(Question ID: 0524745991)

Suppose that the serum cholesterol levels of individuals in a population are approximately normally distributed, with a known population standard deviation of 39.73.

How large of a sample would be required in order for an \(81\%\) confidence interval to be within 1.85 of the truth?
How large of a sample would be required in order for an \(81\%\) confidence interval to no more than 3.46 wide?

(Question ID: 0217376896)

Suppose that the red blood cell counts of individuals in a population are approximately normally distributed, with a known population standard deviation of 0.49.

How large of a sample would be required in order for an \(80\%\) confidence interval to be within 0.01 of the truth?
How large of a sample would be required in order for an \(80\%\) confidence interval to no more than 0.02 wide?

(Question ID: 0004843199)

Suppose that the reaction times of individuals in a population are approximately normally distributed, with a known population standard deviation of 46.81.

How large of a sample would be required in order for an \(86\%\) confidence interval to be within 1.29 of the truth?
How large of a sample would be required in order for an \(86\%\) confidence interval to no more than 2.1 wide?

(Question ID: 0914613887)

Suppose that the red blood cell counts of individuals in a population are approximately normally distributed, with a known population standard deviation of 0.4.

How large of a sample would be required in order for a \(90\%\) confidence interval to be within 0.01 of the truth?
How large of a sample would be required in order for a \(90\%\) confidence interval to no more than 0.02 wide?

(Question ID: 0400318919)

Suppose that the red blood cell counts of individuals in a population are approximately normally distributed, with a known population standard deviation of 0.49.

How large of a sample would be required in order for a \(98\%\) confidence interval to be within 0.02 of the truth?
How large of a sample would be required in order for a \(98\%\) confidence interval to no more than 0.04 wide?

(Question ID: 0932262328)

Suppose that diastolic blood pressures of individuals in a population are approximately normally distributed, with a known population standard deviation of 8.45.

How large of a sample would be required in order for an \(80\%\) confidence interval to be within 0.23 of the truth?
How large of a sample would be required in order for an \(80\%\) confidence interval to no more than 0.42 wide?

(Question ID: 0245493505)

Suppose that body temperatures of individuals in a population are approximately normally distributed, with a known population standard deviation of 0.58.

How large of a sample would be required in order for a \(92\%\) confidence interval to be within 0.01 of the truth?
How large of a sample would be required in order for a \(92\%\) confidence interval to no more than 0.1 wide?

(Question ID: 0983068650)

Suppose that body temperatures of individuals in a population are approximately normally distributed, with a known population standard deviation of 0.51.

How large of a sample would be required in order for a \(92\%\) confidence interval to be within 0.01 of the truth?
How large of a sample would be required in order for a \(92\%\) confidence interval to no more than 0.04 wide?

(Question ID: 0801458312)

Suppose that the serum cholesterol levels of individuals in a population are approximately normally distributed, with a known population standard deviation of 45.65.

How large of a sample would be required in order for a \(93\%\) confidence interval to be within 2.34 of the truth?
How large of a sample would be required in order for a \(93\%\) confidence interval to no more than 2.42 wide?

(Question ID: 0722501885)

Suppose that diastolic blood pressures of individuals in a population are approximately normally distributed, with a known population standard deviation of 8.06.

How large of a sample would be required in order for an \(87\%\) confidence interval to be within 0.38 of the truth?
How large of a sample would be required in order for an \(87\%\) confidence interval to no more than 0.62 wide?

(Question ID: 0841808787)

Suppose that systolic blood pressures of individuals in a population are approximately normally distributed, with a known population standard deviation of 11.59.

How large of a sample would be required in order for an \(81\%\) confidence interval to be within 0.22 of the truth?
How large of a sample would be required in order for an \(81\%\) confidence interval to no more than 2.64 wide?

(Question ID: 0819650829)

Self Assessment 14.9

Suppose that researchers are interested in estimating the proportion of high schoolers in a school board who own a smartphone.

In a sample of size 52 the total number of high schoolers with a smartphone was found to be 27.

(If you have insufficient information to answer the question, enter -1.)

How large of a sample would be required in order for a \(94\%\) confidence interval to be within 0.01 of the truth?
How large of a sample would be required in order for a \(94\%\) confidence interval to no more than 0.02 wide?

(Question ID: 0993794093)

Suppose that researchers are interested in estimating the proportion of high schoolers in a school board who own a smartphone.

(If you have insufficient information to answer the question, enter -1.)

How large of a sample would be required in order for an \(85\%\) confidence interval to be within 0.01 of the truth?
How large of a sample would be required in order for an \(85\%\) confidence interval to no more than 0.2 wide?

(Question ID: 0638937106)

Suppose that researchers are interested in estimating the proportion of individuals who are satisfied with their jobs.

In a sample of size 54 the total number of individuals who report that they are satisfied with their jobs was found to be 35.

(If you have insufficient information to answer the question, enter -1.)

How large of a sample would be required in order for an \(80\%\) confidence interval to be within 0.02 of the truth?
How large of a sample would be required in order for an \(80\%\) confidence interval to no more than 0.04 wide?

(Question ID: 0648793784)

Suppose that researchers are interested in estimating the proportion of individuals who use social media daily.

(If you have insufficient information to answer the question, enter -1.)

How large of a sample would be required in order for an \(88\%\) confidence interval to be within 0.01 of the truth?
How large of a sample would be required in order for an \(88\%\) confidence interval to no more than 0.02 wide?

(Question ID: 0636001230)

Suppose that researchers are interested in estimating the proportion of individuals in a particular region living in poverty.

In a sample of size 128 the total number of people living in poverty was found to be 18.

(If you have insufficient information to answer the question, enter -1.)

How large of a sample would be required in order for a \(99\%\) confidence interval to be within 0.02 of the truth?
How large of a sample would be required in order for a \(99\%\) confidence interval to no more than 0.02 wide?

(Question ID: 0406300143)

Suppose that researchers are interested in estimating the proportion of individuals who use social media daily.

(If you have insufficient information to answer the question, enter -1.)

How large of a sample would be required in order for a \(95\%\) confidence interval to be within 0.01 of the truth?
How large of a sample would be required in order for a \(95\%\) confidence interval to no more than 0.02 wide?

(Question ID: 0313974601)

Suppose that researchers are interested in estimating the proportion of individuals who smoke cigarettes.

In a sample of size 55 the total number of individuals who smoke was found to be 23.

(If you have insufficient information to answer the question, enter -1.)

How large of a sample would be required in order for a \(99\%\) confidence interval to be within 0.01 of the truth?
How large of a sample would be required in order for a \(99\%\) confidence interval to no more than 0.04 wide?

(Question ID: 0783405142)

Suppose that researchers are interested in estimating the proportion of individuals who are satisfied with their jobs.

In a sample of size 60 the total number of individuals who report that they are satisfied with their jobs was found to be 29.

(If you have insufficient information to answer the question, enter -1.)

How large of a sample would be required in order for a \(97\%\) confidence interval to be within 0.01 of the truth?
How large of a sample would be required in order for a \(97\%\) confidence interval to no more than 0.06 wide?

(Question ID: 0579445751)

Suppose that researchers are interested in estimating the proportion of individuals in a particular region living in poverty.

In a sample of size 224 the total number of people living in poverty was found to be 25.

(If you have insufficient information to answer the question, enter -1.)

How large of a sample would be required in order for an \(88\%\) confidence interval to be within 0.01 of the truth?
How large of a sample would be required in order for an \(88\%\) confidence interval to no more than 0.02 wide?

(Question ID: 0173075268)

Suppose that researchers are interested in estimating the proportion of individuals who use social media daily.

(If you have insufficient information to answer the question, enter -1.)

How large of a sample would be required in order for a \(96\%\) confidence interval to be within 0.01 of the truth?
How large of a sample would be required in order for a \(96\%\) confidence interval to no more than 0.04 wide?

(Question ID: 0138479375)

Suppose that researchers are interested in estimating the proportion of individuals in a particular region living in poverty.

In a sample of size 287 the total number of people living in poverty was found to be 33.

(If you have insufficient information to answer the question, enter -1.)

How large of a sample would be required in order for a \(90\%\) confidence interval to be within 0.02 of the truth?
How large of a sample would be required in order for a \(90\%\) confidence interval to no more than 0.02 wide?

(Question ID: 0454058457)

Suppose that researchers are interested in estimating the proportion of individuals in a particular region living in poverty.

(If you have insufficient information to answer the question, enter -1.)

How large of a sample would be required in order for an \(82\%\) confidence interval to be within 0.01 of the truth?
How large of a sample would be required in order for an \(82\%\) confidence interval to no more than 0.02 wide?

(Question ID: 0964913323)

Suppose that researchers are interested in estimating the proportion of high schoolers in a school board who own a smartphone.

In a sample of size 56 the total number of high schoolers with a smartphone was found to be 36.

(If you have insufficient information to answer the question, enter -1.)

How large of a sample would be required in order for a \(94\%\) confidence interval to be within 0.01 of the truth?
How large of a sample would be required in order for a \(94\%\) confidence interval to no more than 0.14 wide?

(Question ID: 0576908855)

Suppose that researchers are interested in estimating the proportion of individuals who are vegan.

In a sample of size 462 the total number of people who are vegan was found to be 19.

(If you have insufficient information to answer the question, enter -1.)

How large of a sample would be required in order for a \(93\%\) confidence interval to be within 0.01 of the truth?
How large of a sample would be required in order for a \(93\%\) confidence interval to no more than 0.02 wide?

(Question ID: 0955974449)

Suppose that researchers are interested in estimating the proportion of individuals who are vegan.

In a sample of size 673 the total number of people who are vegan was found to be 23.

(If you have insufficient information to answer the question, enter -1.)

How large of a sample would be required in order for an \(80\%\) confidence interval to be within 0.01 of the truth?
How large of a sample would be required in order for an \(80\%\) confidence interval to no more than 0.02 wide?

(Question ID: 0665108430)

Suppose that researchers are interested in estimating the proportion of individuals in a particular region living in poverty.

(If you have insufficient information to answer the question, enter -1.)

How large of a sample would be required in order for an \(86\%\) confidence interval to be within 0.01 of the truth?
How large of a sample would be required in order for an \(86\%\) confidence interval to no more than 0.04 wide?

(Question ID: 0824194285)

Suppose that researchers are interested in estimating the proportion of individuals who are satisfied with their jobs.

In a sample of size 55 the total number of individuals who report that they are satisfied with their jobs was found to be 31.

(If you have insufficient information to answer the question, enter -1.)

How large of a sample would be required in order for an \(86\%\) confidence interval to be within 0.02 of the truth?
How large of a sample would be required in order for an \(86\%\) confidence interval to no more than 0.04 wide?

(Question ID: 0610404089)

Suppose that researchers are interested in estimating the proportion of individuals in a particular region who recycle.

(If you have insufficient information to answer the question, enter -1.)

How large of a sample would be required in order for a \(96\%\) confidence interval to be within 0.01 of the truth?
How large of a sample would be required in order for a \(96\%\) confidence interval to no more than 0.02 wide?

(Question ID: 0716840541)

Suppose that researchers are interested in estimating the proportion of individuals who are left-handed.

In a sample of size 240 the total number of left-handed individuals was found to be 20.

(If you have insufficient information to answer the question, enter -1.)

How large of a sample would be required in order for an \(88\%\) confidence interval to be within 0.04 of the truth?
How large of a sample would be required in order for an \(88\%\) confidence interval to no more than 0.04 wide?

(Question ID: 0805283066)

Suppose that researchers are interested in estimating the proportion of individuals who are satisfied with their jobs.

(If you have insufficient information to answer the question, enter -1.)

How large of a sample would be required in order for a \(98\%\) confidence interval to be within 0.03 of the truth?
How large of a sample would be required in order for a \(98\%\) confidence interval to no more than 0.04 wide?

(Question ID: 0461964064)

Suppose that researchers are interested in estimating the proportion of individuals who smoke cigarettes.

In a sample of size 126 the total number of individuals who smoke was found to be 18.

(If you have insufficient information to answer the question, enter -1.)

How large of a sample would be required in order for an \(80\%\) confidence interval to be within 0.01 of the truth?
How large of a sample would be required in order for an \(80\%\) confidence interval to no more than 0.02 wide?

(Question ID: 0492351366)

Suppose that researchers are interested in estimating the proportion of individuals who use social media daily.

In a sample of size 52 the total number of individuals who use social media daily was found to be 29.

(If you have insufficient information to answer the question, enter -1.)

How large of a sample would be required in order for a \(92\%\) confidence interval to be within 0.01 of the truth?
How large of a sample would be required in order for a \(92\%\) confidence interval to no more than 0.06 wide?

(Question ID: 0534610851)

Suppose that researchers are interested in estimating the proportion of high schoolers in a school board who own a smartphone.

In a sample of size 64 the total number of high schoolers with a smartphone was found to be 37.

(If you have insufficient information to answer the question, enter -1.)

How large of a sample would be required in order for a \(95\%\) confidence interval to be within 0.01 of the truth?
How large of a sample would be required in order for a \(95\%\) confidence interval to no more than 0.02 wide?

(Question ID: 0498980676)

Suppose that researchers are interested in estimating the proportion of individuals who are left-handed.

(If you have insufficient information to answer the question, enter -1.)

How large of a sample would be required in order for a \(94\%\) confidence interval to be within 0.01 of the truth?
How large of a sample would be required in order for a \(94\%\) confidence interval to no more than 0.02 wide?

(Question ID: 0575992724)

Suppose that researchers are interested in estimating the proportion of high schoolers in a school board who own a smartphone.

In a sample of size 43 the total number of high schoolers with a smartphone was found to be 27.

(If you have insufficient information to answer the question, enter -1.)

How large of a sample would be required in order for a \(93\%\) confidence interval to be within 0.04 of the truth?
How large of a sample would be required in order for a \(93\%\) confidence interval to no more than 0.02 wide?

(Question ID: 0219642383)

Suppose that researchers are interested in estimating the proportion of individuals in a particular region living in poverty.

In a sample of size 217 the total number of people living in poverty was found to be 22.

(If you have insufficient information to answer the question, enter -1.)

How large of a sample would be required in order for an \(87\%\) confidence interval to be within 0.02 of the truth?
How large of a sample would be required in order for an \(87\%\) confidence interval to no more than 0.02 wide?

(Question ID: 0479786784)

Suppose that researchers are interested in estimating the proportion of high schoolers in a school board who own a smartphone.

(If you have insufficient information to answer the question, enter -1.)

How large of a sample would be required in order for a \(96\%\) confidence interval to be within 0.01 of the truth?
How large of a sample would be required in order for a \(96\%\) confidence interval to no more than 0.02 wide?

(Question ID: 0911559045)

Suppose that researchers are interested in estimating the proportion of individuals who smoke cigarettes.

In a sample of size 61 the total number of individuals who smoke was found to be 19.

(If you have insufficient information to answer the question, enter -1.)

How large of a sample would be required in order for an \(87\%\) confidence interval to be within 0.02 of the truth?
How large of a sample would be required in order for an \(87\%\) confidence interval to no more than 0.02 wide?

(Question ID: 0054275599)

Suppose that researchers are interested in estimating the proportion of high schoolers in a school board who own a smartphone.

(If you have insufficient information to answer the question, enter -1.)

How large of a sample would be required in order for an \(82\%\) confidence interval to be within 0.02 of the truth?
How large of a sample would be required in order for an \(82\%\) confidence interval to no more than 0.04 wide?

(Question ID: 0787718939)

Suppose that researchers are interested in estimating the proportion of high schoolers in a school board who own a smartphone.

(If you have insufficient information to answer the question, enter -1.)

How large of a sample would be required in order for an \(85\%\) confidence interval to be within 0.01 of the truth?
How large of a sample would be required in order for an \(85\%\) confidence interval to no more than 0.04 wide?

(Question ID: 0987567695)

Suppose that researchers are interested in estimating the proportion of individuals in a particular region who recycle.

(If you have insufficient information to answer the question, enter -1.)

How large of a sample would be required in order for a \(95\%\) confidence interval to be within 0.04 of the truth?
How large of a sample would be required in order for a \(95\%\) confidence interval to no more than 0.02 wide?

(Question ID: 0027991232)

Suppose that researchers are interested in estimating the proportion of individuals in a particular region who recycle.

In a sample of size 38 the total number of individuals who recycle was found to be 17.

(If you have insufficient information to answer the question, enter -1.)

How large of a sample would be required in order for a \(93\%\) confidence interval to be within 0.01 of the truth?
How large of a sample would be required in order for a \(93\%\) confidence interval to no more than 0.02 wide?

(Question ID: 0876973352)

Suppose that researchers are interested in estimating the proportion of individuals in a particular region who recycle.

In a sample of size 46 the total number of individuals who recycle was found to be 16.

(If you have insufficient information to answer the question, enter -1.)

How large of a sample would be required in order for a \(96\%\) confidence interval to be within 0.01 of the truth?
How large of a sample would be required in order for a \(96\%\) confidence interval to no more than 0.02 wide?

(Question ID: 0814998166)

Suppose that researchers are interested in estimating the proportion of individuals who use social media daily.

(If you have insufficient information to answer the question, enter -1.)

How large of a sample would be required in order for an \(80\%\) confidence interval to be within 0.01 of the truth?
How large of a sample would be required in order for an \(80\%\) confidence interval to no more than 0.04 wide?

(Question ID: 0351671088)

Suppose that researchers are interested in estimating the proportion of individuals who use social media daily.

(If you have insufficient information to answer the question, enter -1.)

How large of a sample would be required in order for an \(80\%\) confidence interval to be within 0.02 of the truth?
How large of a sample would be required in order for an \(80\%\) confidence interval to no more than 0.02 wide?

(Question ID: 0364723638)

Suppose that researchers are interested in estimating the proportion of high schoolers in a school board who own a smartphone.

(If you have insufficient information to answer the question, enter -1.)

How large of a sample would be required in order for an \(88\%\) confidence interval to be within 0.01 of the truth?
How large of a sample would be required in order for an \(88\%\) confidence interval to no more than 0.04 wide?

(Question ID: 0469390520)

Suppose that researchers are interested in estimating the proportion of individuals in a particular region who recycle.

(If you have insufficient information to answer the question, enter -1.)

How large of a sample would be required in order for a \(92\%\) confidence interval to be within 0.02 of the truth?
How large of a sample would be required in order for a \(92\%\) confidence interval to no more than 0.02 wide?

(Question ID: 0399438572)

Suppose that researchers are interested in estimating the proportion of high schoolers in a school board who own a smartphone.

(If you have insufficient information to answer the question, enter -1.)

How large of a sample would be required in order for a \(97\%\) confidence interval to be within 0.01 of the truth?
How large of a sample would be required in order for a \(97\%\) confidence interval to no more than 0.06 wide?

(Question ID: 0996495493)

Suppose that researchers are interested in estimating the proportion of individuals in a particular region who recycle.

(If you have insufficient information to answer the question, enter -1.)

How large of a sample would be required in order for a \(94\%\) confidence interval to be within 0.01 of the truth?
How large of a sample would be required in order for a \(94\%\) confidence interval to no more than 0.02 wide?

(Question ID: 0532878796)

Suppose that researchers are interested in estimating the proportion of individuals in a particular region who recycle.

(If you have insufficient information to answer the question, enter -1.)

How large of a sample would be required in order for an \(81\%\) confidence interval to be within 0.02 of the truth?
How large of a sample would be required in order for an \(81\%\) confidence interval to no more than 0.02 wide?

(Question ID: 0080643917)

Suppose that researchers are interested in estimating the proportion of individuals who are satisfied with their jobs.

In a sample of size 53 the total number of individuals who report that they are satisfied with their jobs was found to be 12.

(If you have insufficient information to answer the question, enter -1.)

How large of a sample would be required in order for an \(80\%\) confidence interval to be within 0.01 of the truth?
How large of a sample would be required in order for an \(80\%\) confidence interval to no more than 0.02 wide?

(Question ID: 0991100737)

Suppose that researchers are interested in estimating the proportion of individuals who are satisfied with their jobs.

In a sample of size 49 the total number of individuals who report that they are satisfied with their jobs was found to be 19.

(If you have insufficient information to answer the question, enter -1.)

How large of a sample would be required in order for an \(83\%\) confidence interval to be within 0.03 of the truth?
How large of a sample would be required in order for an \(83\%\) confidence interval to no more than 0.02 wide?

(Question ID: 0227584316)

Suppose that researchers are interested in estimating the proportion of individuals who smoke cigarettes.

In a sample of size 81 the total number of individuals who smoke was found to be 13.

(If you have insufficient information to answer the question, enter -1.)

How large of a sample would be required in order for an \(89\%\) confidence interval to be within 0.01 of the truth?
How large of a sample would be required in order for an \(89\%\) confidence interval to no more than 0.02 wide?

(Question ID: 0632269501)

Suppose that researchers are interested in estimating the proportion of individuals who use social media daily.

In a sample of size 90 the total number of individuals who use social media daily was found to be 58.

(If you have insufficient information to answer the question, enter -1.)

How large of a sample would be required in order for a \(90\%\) confidence interval to be within 0.03 of the truth?
How large of a sample would be required in order for a \(90\%\) confidence interval to no more than 0.02 wide?

(Question ID: 0741018954)

Suppose that researchers are interested in estimating the proportion of individuals in a particular region who recycle.

In a sample of size 72 the total number of individuals who recycle was found to be 48.

(If you have insufficient information to answer the question, enter -1.)

How large of a sample would be required in order for an \(81\%\) confidence interval to be within 0.01 of the truth?
How large of a sample would be required in order for an \(81\%\) confidence interval to no more than 0.02 wide?

(Question ID: 0322446062)

Suppose that researchers are interested in estimating the proportion of individuals who are vegan.

In a sample of size 518 the total number of people who are vegan was found to be 17.

(If you have insufficient information to answer the question, enter -1.)

How large of a sample would be required in order for a \(91\%\) confidence interval to be within 0.01 of the truth?
How large of a sample would be required in order for a \(91\%\) confidence interval to no more than 0.02 wide?

(Question ID: 0621285014)

Suppose that researchers are interested in estimating the proportion of high schoolers in a school board who own a smartphone.

In a sample of size 40 the total number of high schoolers with a smartphone was found to be 23.

(If you have insufficient information to answer the question, enter -1.)

How large of a sample would be required in order for an \(85\%\) confidence interval to be within 0.01 of the truth?
How large of a sample would be required in order for an \(85\%\) confidence interval to no more than 0.04 wide?

(Question ID: 0828710176)

Suppose that researchers are interested in estimating the proportion of individuals who are left-handed.

(If you have insufficient information to answer the question, enter -1.)

How large of a sample would be required in order for a \(95\%\) confidence interval to be within 0.01 of the truth?
How large of a sample would be required in order for a \(95\%\) confidence interval to no more than 0.04 wide?

(Question ID: 0885329812)

Suppose that researchers are interested in estimating the proportion of individuals in a particular region living in poverty.

In a sample of size 191 the total number of people living in poverty was found to be 16.

(If you have insufficient information to answer the question, enter -1.)

How large of a sample would be required in order for an \(86\%\) confidence interval to be within 0.02 of the truth?
How large of a sample would be required in order for an \(86\%\) confidence interval to no more than 0.04 wide?

(Question ID: 0958335517)

Suppose that researchers are interested in estimating the proportion of individuals in a particular region living in poverty.

(If you have insufficient information to answer the question, enter -1.)

How large of a sample would be required in order for a \(94\%\) confidence interval to be within 0.01 of the truth?
How large of a sample would be required in order for a \(94\%\) confidence interval to no more than 0.02 wide?

(Question ID: 0871160282)

Credible intervals are similar to confidence intervals, but approached through the Bayesian paradigm. Likelihood intervals are related to maximum likelihood estimation, and provide a rather interpretable framework for interval estimation. Prediction intervals, as the name suggests, are related to predictive statistics and allow the uncertainty of predictions to be quantified.↩︎
In this case, it is zero since \(2\) is not in the interval \((4, 6)\).↩︎
Recall that an inverse function is the function, \(f^{-1}\) such that \(f^{-1}(f(x)) = x\). So if \(f(x) = \log(x)\), then \(f^{-1}(x) = \exp(x)\). If \(f(x) = x\) then \(f^{-1} = \frac{1}{x}\) and so forth.↩︎
Note that very often we will find that \(a^{-1}(\widehat{\theta}) \geq b^{-1}(\widehat{\theta})\), and so the endpoints switch. This is not always the case, but frequently occurs.↩︎
With its randomness endowed thanks to the random sampling↩︎
Note that it is not uncommon to see that an interval is interpretable as “We are \(100p\%\) confident that the parameter is in the interval.” This confidence is, importantly, not a probability, and this is taken as shorthand to mean that, if we were to repeatedly find confidence intervals following this practice, in \(100p\%\) of them we would find the truth. I think that this statement tends to lead to more confusion, and as such, I would advise against this interpretation until you are very confident in your understanding.↩︎
For instance, we will see how to calculate confidence intervals for both the mean and the variance of a random sample. These will be different and are disconnected from the sample values.↩︎
Note, in this case, the true value would be \(-2\), which is fairly close.↩︎
Note that, for large \(n\) these two sets of values are essentially equivalent. This means that if you had used the \(t\) values, it would not result in meaningfully different results, however, the \(Z\) values are justified in this setting.↩︎
By nicely behaved, we mean monotonic. That is, if \(a \leq b\), then \(g(a) \leq g(b)\) or \(g(a) \geq g(b)\) should hold. This means that taking \(g(x) = \sqrt{x}\) is nicely behaved, but taking \(g(x) = x^2\) is not. The concern with a function like \(g(x) = x^2\) comes from what would happen if, for instance, our initial interval were \((-2, 2)\). Then transforming this interval gives \((4, 4)\) which is clearly incorrect. There are ways of working around this, however, considerations of more complex transformations are beyond the scope of these notes.↩︎
This should make intuitive sense. If you want to have a \(80\%\) chance of containing the truth you can afford to give a narrower interval than if you wanted to have a \(99\%\) chance of containing the truth.↩︎
This should also make intuitive sense. If we are trying to understand the center of a distribution, the more spread out our data are, the less certain we will be of our results. The less spread out, the more certain we can be.↩︎