Constructing Confidence Intervals In Exercises 13–24, assume the sample is from a normally distributed population and construct the indicated confidence intervals for (a) the population variance σ^2. Interpret the results.
Volleyball The numbers of service aces scored by 15 teams randomly selected from the top 50 NCAA Division I Women’s Volleyball teams for the 2021 season have a sample standard deviation of 26.1. Use an 80% level of confidence. (Source: National Collegiate Athletic Association)

Verified step by step guidance

1. Step 1: Understand the problem. We are tasked with constructing an 80% confidence interval for the population variance (σ²) based on a sample standard deviation (s = 26.1) and a sample size (n = 15). The population is assumed to be normally distributed.
2. Step 2: Recall the formula for the confidence interval of the population variance. The confidence interval is given by: \( \left( \frac{(n-1)s^2}{\chi^2_{\text{right}}}, \frac{(n-1)s^2}{\chi^2_{\text{left}}} \right) \), where \( n \) is the sample size, \( s^2 \) is the sample variance, and \( \chi^2_{\text{right}} \) and \( \chi^2_{\text{left}} \) are the critical values of the chi-square distribution for the given confidence level.
3. Step 3: Calculate the sample variance \( s^2 \). The sample variance is the square of the sample standard deviation: \( s^2 = (26.1)^2 \).
4. Step 4: Determine the degrees of freedom and the critical chi-square values. The degrees of freedom (df) are \( n-1 = 15-1 = 14 \). For an 80% confidence level, find the critical values \( \chi^2_{\text{right}} \) and \( \chi^2_{\text{left}} \) from the chi-square distribution table for df = 14. These correspond to the upper and lower tails of the distribution, with 10% in each tail (since 80% is the middle area).
5. Step 5: Plug the values into the confidence interval formula. Substitute \( n-1 = 14 \), \( s^2 \), and the critical chi-square values into the formula \( \left( \frac{(n-1)s^2}{\chi^2_{\text{right}}}, \frac{(n-1)s^2}{\chi^2_{\text{left}}} \right) \). Simplify the expressions to find the confidence interval for the population variance. Finally, interpret the interval in the context of the problem.

Key Concepts

Confidence Interval

A confidence interval is a range of values, derived from a sample, that is likely to contain the true population parameter with a specified level of confidence. For example, an 80% confidence interval suggests that if we were to take many samples and construct intervals in the same way, approximately 80% of those intervals would contain the true population parameter.

Population Variance

Population variance (σ²) measures the spread of a set of values in a population. It is calculated as the average of the squared differences from the mean. In the context of the question, constructing a confidence interval for the population variance allows us to estimate the range within which the true variance of service aces scored by all teams lies.

Sample Standard Deviation

Sample standard deviation (s) quantifies the amount of variation or dispersion in a sample data set. It is calculated as the square root of the sample variance. In this exercise, the sample standard deviation of 26.1 is crucial for estimating the population variance and constructing the confidence interval, as it reflects the variability of service aces among the selected teams.