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.
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.

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Step 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 population is assumed to be normally distributed.

Step 2: Recall the formula for the confidence interval for the population variance (σ²). 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 \( \chi^2_{\text{right}} \) and \( \chi^2_{\text{left}} \) are the critical values of the chi-square distribution for the given confidence level.

Step 3: Calculate the degrees of freedom (df). The degrees of freedom is \( n-1 \), so \( df = 28 - 1 = 27 \).

Step 4: Determine the critical values \( \chi^2_{\text{right}} \) and \( \chi^2_{\text{left}} \) from the chi-square distribution table for \( df = 27 \) and a 98% confidence level. The critical values correspond to the upper and lower tails of the distribution, with \( \alpha/2 = 0.01 \) in each tail.

Step 5: Substitute the values into the confidence interval formula. Use \( s^2 = (56.1)^2 \), \( n-1 = 27 \), and the critical values \( \chi^2_{\text{right}} \) and \( \chi^2_{\text{left}} \) to compute the lower and upper bounds of the confidence interval for the population variance. Finally, interpret the results by stating that you are 98% confident the true population variance lies within the calculated interval.

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주요 개념

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Confidence Interval

A confidence interval is a range of values, derived from a sample statistic, 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 Variance

Population variance (σ²) measures the dispersion of a set of values in a population. It is calculated as the average of the squared differences from the mean. Understanding population variance is crucial for constructing confidence intervals, as it helps quantify the uncertainty around the sample estimate.

Sample Standard Deviation

Sample standard deviation (s) is a statistic that measures the amount of variation or dispersion in a sample data set. It is calculated as the square root of the sample variance. In the context of constructing confidence intervals, the sample standard deviation is used to estimate the population standard deviation, which is essential for determining the width of the confidence interval.