Constructing Confidence Intervals In Exercises 13–24, assume the sample is from a normally distributed population and construct the indicated confidence intervals for (b) the population standard deviation σ. Interpret the results.
Drive-Thru Times The times (in seconds) spent by a random sample of 28 customers in the drive-thru of a fast-food restaurant have a sample standard deviation of 56.1. Use a 98% level of confidence.

Verified step by step guidance

1. Identify the given information: The sample size (n) is 28, the sample standard deviation (s) is 56.1, and the confidence level is 98%. The goal is to construct a confidence interval for the population standard deviation (σ).
2. Recognize that the confidence interval for the population standard deviation is based on the chi-square distribution. The formula for the confidence interval is: \( \left( \sqrt{\frac{(n-1)s^2}{\chi^2_{\text{upper}}}}, \sqrt{\frac{(n-1)s^2}{\chi^2_{\text{lower}}}} \right) \), where \( \chi^2_{\text{upper}} \) and \( \chi^2_{\text{lower}} \) are the critical values of the chi-square distribution.
3. Determine the degrees of freedom (df), which is \( n-1 \). In this case, \( df = 28 - 1 = 27 \).
4. Find the critical values \( \chi^2_{\text{upper}} \) and \( \chi^2_{\text{lower}} \) for a 98% confidence level and 27 degrees of freedom. Use a chi-square table or statistical software to find these values. The confidence level of 98% corresponds to a significance level of \( \alpha = 0.02 \), so divide \( \alpha \) into two tails: \( \alpha/2 = 0.01 \).
5. Substitute the values into the formula: \( n-1 = 27 \), \( s = 56.1 \), and the critical values \( \chi^2_{\text{upper}} \) and \( \chi^2_{\text{lower}} \). Compute the lower and upper bounds of the confidence interval for \( \sigma \) by solving the formula. Interpret the results by stating that you are 98% confident the true population standard deviation lies within the calculated interval.

Key Concepts

Confidence Interval

A confidence interval is a range of values, derived from a sample, that is likely to contain the population parameter with a specified level of confidence. For example, a 98% confidence interval suggests that if we were to take many samples and construct intervals in the same way, approximately 98% of those intervals would contain the true population parameter.

Population Standard Deviation (σ)

The population standard deviation (σ) is a measure of the dispersion or spread of a set of values in a population. It quantifies how much the individual data points deviate from the population mean. In the context of confidence intervals, estimating σ is crucial for understanding the variability of the population from which the sample is drawn.

Sample Standard Deviation

The sample standard deviation is an estimate of the population standard deviation based on a sample. It reflects the variability of the sample data points around the sample mean. In this case, the sample standard deviation of 56.1 seconds is used to construct the confidence interval for the population standard deviation, providing insight into the drive-thru times' variability.