Question

Exercises 3.96 to 3.101 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. How Close Do You Feel to Others? Use the closeness ratings before the activity (CloseBefore) to estimate the mean closeness rating one person would assign to others in a group.

Short answer

The quantity we are estimating, denoted by \( µ_{CB} \), is the mean closeness rating a person would assign to others in a group before the activity. The value of the sample statistic (sample mean) and the standard error can be calculated using the statistical software. A 95% confidence interval for this quantity is calculated using the sample mean and standard error. The interpretation of the confidence interval is that we can be 95% confident that the true mean closeness rating falls within this interval.

Step-by-step solution

Identifying the Quantity and Parameters

We are trying to estimate the mean closeness rating that a person would assign to others in a group before doing synchronized movement or any activity. Thus, we denote the quantity we are estimating as \( µ_{CB} \), where 'CB' represents 'CloseBefore' referring to the closeness measure before the activity. 'µ' represents the mean value we are trying to estimate.

Finding the Value of Sample Statistic

Using the given dataset in the statistical software (like StatKey), we need to find the mean value of the 'CloseBefore' ratings. It gives us the sample mean which is represented as \( \bar{x}_{CB} \). This is our sample statistic.

Calculating Standard Error for the Estimate

The standard error for the estimate can be calculated using the software too. It can be done by finding the standard deviation of the 'CloseBefore' ratings and dividing it by the square root of the number of observations. This value represents the standard error (SE) and it measures the accuracy of our estimate.

Constructing a Confidence Interval

A 95% confidence interval for the quantity we are estimating can be calculated using the formula: \[ \bar{x}_{CB} \pm 1.96 * SE \] where \( \bar{x}_{CB} \) is the sample mean, SE is the standard error, 1.96 is the z-score for a 95% confidence interval. This interval gives us a range in which we can be 95% confident that the true population mean falls in.

Interpretation of Confidence Interval

The interpretation of the confidence interval is in the context of the problem. It means that we can be 95% confident that the true mean closeness rating that one person would assign to others in a group before activity falls within the calculated interval.