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Chapter 3: Problem 81

Exercises 3.81 to 3.84 give information about the proportion of a sample that agrees with a certain statement. Use StatKey or other technology to estimate the standard error from a bootstrap distribution generated from the sample. Then use the standard error to give a \(95 \%\) confidence interval for the proportion of the population to agree with the statement. StatKey tip: Use "CI for Single Proportion" and then "Edit Data" to enter the sample information. In a random sample of 100 people, 35 agree.

Short Answer

Expert verified
The detailed solution is obtained by using statistical software and following the procedure of bootstrapping and confidence interval calculation outlined in the step-by-step instructions. The exact values will depend on the estimated standard error obtained from the bootstrap procedure, and thus cannot be provided without executing this procedure.

Step by step solution

01

Understand the sample data

There is a sample of 100 people, 35 of whom agree with a statement. This means our sample proportion is \(35 / 100 = 0.35\). This is the statistic we want to construct a confidence interval around.
02

Estimate the standard error using bootstrapping

Using software like StatKey, you can generate a bootstrap distribution from the sample and estimate the standard error. This involves simulating the drawing of multiple samples from the population, calculating the sample proportion for each, and using the standard deviation of these bootstrap sample proportions as an estimate of the standard error.
03

Calculate the confidence interval

A 95% confidence interval is calculated as \(p \pm Z_{0.025} * SE\), where \(p\) is the sample proportion, \(Z_{0.025}\) is the Z-score that corresponds to a 95% confidence interval (approximately 1.96), and \(SE\) is the standard error. Use your estimated standard error from step 2 to calculate the width of the confidence interval and then add and subtract that from the sample proportion to get the confidence interval's lower and upper boundaries, respectively.

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Most popular questions from this chapter

Investigating the Width of a Confidence Interval Comparing Exercise 3.120 to Exercise \(3.121,\) you should have found that the confidence interval when utilizing the paired structure of the data was narrower than the confidence interval ignoring this structure (this will generally be the case, and is the primary reason for pairing). How else could we change the width of the confidence interval? More specifically, for each of the following changes, would the width of the confidence interval likely increase, decrease, or remain the same? (a) Increase the sample size. (b) Simulate more bootstrap samples. (c) Decrease the confidence level from \(99 \%\) to \(95 \%\).

Use data from a study designed to examine the effect of doing synchronized movements (such as marching in step or doing synchronized dance steps) and the effect of exertion on many different variables, such as pain tolerance and attitudes toward others. In the study, 264 high school students in Brazil were randomly assigned to one of four groups reflecting whether or not movements were synchronized (Synch= yes or no) and level of activity (Exertion= high or low). \(^{49}\) Participants rated how close they felt to others in their group both before (CloseBefore) and after (CloseAfter) the activity, using a 7-point scale (1=least close to \(7=\) most close ). Participants also had their pain tolerance measured using pressure from a blood pressure cuff, by indicating when the pressure became too uncomfortable (up to a maximum pressure of \(300 \mathrm{mmHg}\) ). Higher numbers for this Pain Tolerance measure indicate higher pain tolerance. The full dataset is available in SynchronizedMovement. For each of the following problems: (a) Give notation for the quantity we are estimating, and define any relevant parameters. (b) Use StatKey or other technology to find the value of the sample statistic. Give the correct notation with your answer. (c) Use StatKey or other technology to find the standard error for the estimate. (d) Use the standard error to give a \(95 \%\) confidence interval for the quantity we are estimating. (e) Interpret the confidence interval in context. What Proportion Go to Maximum Pressure? We see that 75 of the 264 people in the study allowed the pressure to reach its maximum level of \(300 \mathrm{mmHg}\), without ever saying that the pain was too much (MaxPressure=yes). Use this information to estimate the proportion of people who would allow the pressure to reach its maximum level.

Adolescent Brains Are Different Researchers continue to find evidence that brains of adolescents behave quite differently than either brains of adults or brains of children. In particular, adolescents seem to hold on more strongly to fear associations than either children or adults, suggesting that frightening connections made during the teen years are particularly hard to unlearn. In one study, \({ }^{25}\) participants first learned to associate fear with a particular sound. In the second part of the study, participants heard the sound without the fear-causing mechanism, and their ability to "unlearn" the connection was measured. A physiological measure of fear was used, and larger numbers indicate less fear. We are estimating the difference in mean response between adults and teenagers. The mean response for adults in the study was 0.225 and the mean response for teenagers in the study was \(0.059 .\) We are told that the standard error of the estimate is 0.091 . (a) Give notation for the quantity being estimated. (b) Give notation for the quantity that gives the best estimate, and give its value. (c) Give a \(95 \%\) confidence interval for the quantity being estimated. (d) Is this an experiment or an observational study?

Small Sample Size and Outliers As we have seen, bootstrap distributions are generally symmetric and bell-shaped and centered at the value of the original sample statistic. However, strange things can happen when the sample size is small and there is an outlier present. Use StatKey or other technology to create a bootstrap distribution for the standard deviation based on the following data: \(8 \quad 10\) 72 \(13 \quad 8\) \(\begin{array}{ll}10 & 50\end{array}\) Describe the shape of the distribution. Is it appropriate to construct a confidence interval from this distribution? Explain why the distribution might have the shape it does.

Use data from a study designed to examine the effect of doing synchronized movements (such as marching in step or doing synchronized dance steps) and the effect of exertion on many different variables, such as pain tolerance and attitudes toward others. In the study, 264 high school students in Brazil were randomly assigned to one of four groups reflecting whether or not movements were synchronized (Synch= yes or no) and level of activity (Exertion= high or low). \(^{49}\) Participants rated how close they felt to others in their group both before (CloseBefore) and after (CloseAfter) the activity, using a 7-point scale (1=least close to \(7=\) most close ). Participants also had their pain tolerance measured using pressure from a blood pressure cuff, by indicating when the pressure became too uncomfortable (up to a maximum pressure of \(300 \mathrm{mmHg}\) ). Higher numbers for this Pain Tolerance measure indicate higher pain tolerance. The full dataset is available in SynchronizedMovement. For each of the following problems: (a) Give notation for the quantity we are estimating, and define any relevant parameters. (b) Use StatKey or other technology to find the value of the sample statistic. Give the correct notation with your answer. (c) Use StatKey or other technology to find the standard error for the estimate. (d) Use the standard error to give a \(95 \%\) confidence interval for the quantity we are estimating. (e) Interpret the confidence interval in context. Does Exertion Boost Pain Tolerance? Use the pain tolerance ratings after the activity to estimate the difference in mean pain tolerance between those who just completed a high exertion activity and those who completed a low exertion activity.
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