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. Does Synchronization Boost Pain Tolerance? Use the pain tolerance ratings ( PainTolerance) after the activity to estimate the difference in mean pain tolerance between those who just completed a synchronized activity and those who did a nonsynchronized activity.

Short answer

The steps required are firstly defining the parameters and quantities we are looking for. Then, using a statistical tool to calculate the mean of the two groups and the standard error of these means. This gives us the estimated difference in mean pain tolerance of the two groups and the standard error of this estimate. With this, the 95% confidence interval for this difference can be constructed. Lastly, the confidence interval is interpreted in terms of the context.

Step-by-step solution

(a) Define parameters and notation

We are estimating the difference in mean pain tolerance between those who just completed a synchronized and nonsynchronized activity. Let \( \mu_{1} \) denote the mean pain tolerance of those who just completed a synchronized activity, let \( \mu_{2} \) denote the mean pain tolerance of those who just completed a nonsynchronized activity. The quantity we are estimating is the difference in means, denoted as \( \mu_{1} - \mu_{2} \).

(b) Find the value of the sample statistic

Using statistical software like StatKey or any other, input the set of pain tolerance ratings of the two groups. Calculate the mean of each group. Let's denote the sample means as \( \bar{x}_{1} \) and \( \bar{x}_{2} \), for synchronized and nonsynchronized groups, respectively. The sample statistic or the estimated difference in means is \( \bar{x}_{1} - \bar{x}_{2} \).

(c) Find the standard error for the estimate

Standard error (SE) is a measure of how spread out the values of the sample mean is likely to be from the population mean. Using the statistical software, calculate the standard deviation of each group's pain tolerance, denoted \( s_{1} \) and \( s_{2} \). The standard error for the estimated difference in means can be calculated using the formula: \( SE = \sqrt{\frac{s_{1}^{2}}{n_{1}} + \frac{s_{2}^{2}}{n_{2}}} \), where \( n_{1} \) and \( n_{2} \) are the sizes of the synchronized and nonsynchronized groups, respectively.

(d) 95% confidence interval

The 95% confidence interval for the difference in means is given by the sample difference plus or minus a margin of error. The margin of error is given by \( 1.96*SE \). Thus, the 95% confidence interval for the difference in means is \( (\bar{x}_{1} - \bar{x}_{2}) \pm 1.96*SE \).

(e) Interpretation

The result can be interpreted as follows: We are 95% confident that the true difference in mean pain tolerance between the population who completed a synchronized activity and those who completed a nonsynchronized activity is within the range given by the 95% confidence interval.

Key concepts

Pain Tolerance Measurement

Understanding pain tolerance is an important aspect of medical research and psychological studies. Pain tolerance refers to the maximum level of pain that a person can handle before it becomes too uncomfortable. In the context of the given exercise, pain tolerance was measured using the pressure from a blood pressure cuff. Participants indicated when the pressure became too uncomfortable, with higher numbers suggesting higher pain tolerance.

This method of pain tolerance measurement is not only relevant in the context of synchronized movements but also has broader implications in understanding how individuals cope with physical stress and pain. When assessing pain tolerance, it is critical to consider the subjectivity of pain perception, as pain tolerance thresholds can vary widely among individuals.

Synchronized Movements Study

The study of synchronized movements involves analyzing the effects of activities that are performed in unison, like marching or dancing. In synchronized movements, individuals coordinate their actions with one another, which could potentially impact their psychological state and physical abilities.

In the exercise, high school students either engaged in synchronized or nonsynchronized movements, combined with varying levels of physical exertion. The study aimed to explore how these variables affected pain tolerance and feelings of closeness among the participants. The implication is that synchronized movements might not only improve physical coordination but could also influence social cohesion and tolerance to discomfort.

Confidence Interval Estimation

Confidence interval estimation is a fundamental concept in statistics used to infer about a population parameter based on sample data. In simple terms, a confidence interval provides a range of values within which we can expect the true population parameter (such as a mean or a difference in means) to lie, with a certain level of certainty.

For the exercise in question, a 95% confidence interval was calculated to estimate the difference in mean pain tolerance between groups engaging in synchronized versus nonsynchronized activities. This interval is constructed from the sample data and indicates that we can be 95% confident that the true difference in the population means falls within this range. Notably, the confidence interval does not guarantee that the true difference is within the interval, but it provides a plausible range based on the data obtained.