91Ó°ÊÓ

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.

Step by step solution

01

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.

02

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.

03

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.

04

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.

05

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.

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Key Concepts

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

Understanding Standard Error

Standard error (SE) is a crucial concept in statistics that helps in understanding the reliability and accuracy of a sample mean as an estimate of the true population mean. It essentially measures the extent to which the sample mean might vary from the actual population mean.
To calculate the standard error, one needs to determine the standard deviation of the sample and then divide it by the square root of the sample size. The formula is given by:

• \[SE = \frac{s}{\sqrt{n}}\]

The smaller the standard error, the more precise the estimate of the population mean. This is because smaller SE values indicate less variation among the sample means that would be obtained if we took multiple samples from the population.
When you encounter a standard error value, it helps quantify the uncertainty associated with the sample mean. A high SE value suggests more variability which might lead to a wider confidence interval. Conversely, a low SE value typically results in a narrow confidence interval.

Sample Statistic: Mean of 'CloseBefore' Ratings

A sample statistic is a single measure, derived from sample data, which is used to estimate a population parameter. In this study, the sample statistic of interest is the mean of 'CloseBefore' ratings, denoted as \( \bar{x}_{CB} \).
This mean provides an average rating that high school students used to quantify how close they felt to their peers before engaging in synchronized movements.
To obtain this sample mean, you would compile all the 'CloseBefore' ratings from the collected data and calculate their average. This simple arithmetic mean serves as the sample statistic that estimates the true population mean (\( µ_{CB} \)).
Understanding the sample statistic is essential because it offers a base from which other calculations, like the standard error and confidence interval, are derived. It reflects the central tendency of your data and provides a summary measure that is easy to interpret.

Estimating the Mean with Confidence

Mean estimation is a statistical process for using sample data to make inferences about the population mean. In this problem, the goal is to estimate the mean closeness rating \( µ_{CB} \), representing how close students feel to each other before an activity.
The main tool used in this estimation is the confidence interval, which offers a range of values that likely encompass the true population mean. For a 95% confidence level, the general formula is:

• \[ \bar{x}_{CB} \pm 1.96 \times SE \]

This formula uses the sample mean (\( \bar{x}_{CB} \)) and the calculated standard error (SE). The constant 1.96 is a z-score value corresponding to a 95% confidence level, signifying that if the study were repeated many times, 95% of the calculated intervals would contain the true population mean.
Estimating the mean through a confidence interval not only provides a specific range but also expresses the degree of uncertainty associated with the estimate. It's a foundational concept in inferential statistics that helps researchers understand the precision of their sample estimates.