91Ó°ÊÓ

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. Are Males or Females More Likely to Go to Maximum Pressure? The study recorded whether participants were female or male \((\operatorname{Sex}=\mathrm{F}\) or \(\mathrm{M})\), and we see that 33 of the 165 females and 42 of the 99 males allowed the pressure to reach its maximum level of \(300 \mathrm{mmHg}\) after treatment, without ever saying that the pain was too much. Use this information to estimate the difference in proportion of people who would allow the pressure to reach its maximum level after treatment, between females and males.

Step by step solution

01

Define Parameters

For this problem, the parameters are the actual proportions of males and females who can withstand maximum pressure. Let's denote \( p_M \) for males and \( p_F \) for females.

02

Calculate Sample Statistic

According to the information, 33 out of 165 females and 42 out of 99 males can tolerate maximum pressure. We can calculate the sample proportions for each group using the formula: \( p = \frac{x}{n} \), where x is the number of successes and n is the total sample size. We find, \( \hat{p}_F = \frac{33}{165} = 0.20 \) and \( \hat{p}_M = \frac{42}{99} = 0.42 \). Thus, the point estimate of the difference between the two proportions is \( \hat{p}_M - \hat{p}_F = 0.42 - 0.20 = 0.22 \)

03

Calculate Standard Error

The standard error for the difference between two proportions is calculated with the formula: SE = \( \sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2}} \). Substituting the values we have: SE = \( \sqrt{\frac{0.42(1-0.42)}{99} + \frac{0.20(1-0.20)}{165}} = 0.07 \)

04

Construct Confidence Interval

A 95% confidence interval can be constructed using the formula: \( \hat{p}_1 - \hat{p}_2 \pm Z \times SE \) where Z value for 95% confidence is approximately 1.96. So, the confidence interval is \( 0.22 \pm 1.96 \times 0.07 = [0.09, 0.35] \)

05

Interpret Confidence Interval

The interval [0.09, 0.35] can be interpreted as: We are 95% confident that the true difference in proportion of people who would allow the pressure to reach its maximum level after treatment, between males and females, is between 9% and 35%.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sample Proportion

When we talk about sample proportion, we are referring to a statistic that represents a portion of a population observed in a sample. In this study, you saw how sample proportions were calculated for determining how many males and females tolerated the maximum pressure applied. For both groups, the sample proportion is computed using the formula:

• \( p = \frac{x}{n} \)

Here, \( x \) represents the number of successes, and \( n \) is the total sample size. For example, for females, 33 of 165 could withstand the pressure, giving a sample proportion \( \hat{p}_F = \frac{33}{165} = 0.20 \).
Similarly, for males, \( \hat{p}_M = \frac{42}{99} = 0.42 \).
These proportions help in understanding the characteristics of the sample allowing us to make inferences about the entire population.

Standard Error

The standard error is a measure that describes how much we expect the sample statistic to fluctuate if we were to take multiple samples. Think of it as a gauge for reliability; the smaller the standard error, the more reliable your sample statistic is.
The formula to calculate the standard error of the difference between two sample proportions is given by:

• \[ SE = \sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2}} \]

Here, \( \hat{p}_1 \) and \( \hat{p}_2 \) are the sample proportions, and \( n_1 \) and \( n_2 \) are the corresponding sample sizes.
In our case, substituting the values gives us a standard error of 0.07. This number tells us about the expected range of variability in the difference in pressure tolerance between males and females.

Difference of Proportions

When comparing two groups, like males and females in our study, the difference in their sample proportions shows how those groups are distinct based on the attribute in question. In this exercise, we focused on how likely each gender was to endure a specified pressure.
The difference in sample proportions between males and females is calculated as:

• \( \hat{p}_M - \hat{p}_F = 0.42 - 0.20 = 0.22 \)

This difference of 0.22 or 22% suggests that a significantly greater proportion of males reach the maximum pressure limit than females.
We use this difference to create a confidence interval that provides a range where the true difference in population proportions is likely to be. By following the given calculations, we find a 95% confidence interval between 9% and 35%.