Question

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.

Short answer

The relevant parameter we are trying to estimate is the population proportion \(p\) of people who would allow the blood pressure cuff to reach its maximum level. The sample proportion, \(\hat{p}\), is calculated by dividing the number of people who allowed max pressure (75) by the total number of participants (264). The standard error is calculated using the formula \( SE = \sqrt{\frac{\hat{p}(1- \hat{p})}{n}} \), and the \(95\%\) confidence interval is given by \((\hat{p} - 1.96 \times SE, \hat{p} + 1.96 \times SE)\). The interpretation of the confidence interval would be based on the actual calculated interval.

Step-by-step solution

Identifying the Relevant Parameters

We need to find the proportion of people who allowed the blood pressure cuff to reach the maximum level of 300 mmHg. The quantity we are estimating is thus denoted as \(p\), the proportion of people who would allow the pressure to reach the maximum level.

Calculate the Sample Statistic

From the problem, it's stated that 75 out of 264 people in the study allowed the pressure to reach the maximum level. Therefore, the sample proportion \(\hat{p}\) is calculated as \(\hat{p} = \frac{75}{264}\).

Find the Standard Error

The standard error (SE) for the estimate can be calculated using the formula for the standard error of a proportion which is \( SE = \sqrt{\frac{\hat{p}(1- \hat{p})}{n}} \), where \(\hat{p}\) is the sample proportion and \(n\) is the sample size.

Compute the Confidence Interval

The \(95\%\) confidence interval (CI) for the quantity we are estimating can be computed using the formula for the confidence interval of a proportion which is \((\hat{p} - 1.96 \times SE, \hat{p} + 1.96 \times SE)\) where 1.96 is the z-score for a \(95\%\) confidence interval and SE is the standard error.

Interpret the Confidence Interval

The \(95\%\) confidence interval represents the range of values within which we can be \(95\%\) confident that the true population proportion (the proportion of all people who would allow the pressure to reach its maximum level) lies. The interpretation relies on the confidence interval calculated in step 4.