Constructing Confidence Intervals In Exercises 17 and 18, you are given the sample mean and the sample standard deviation. Assume the random variable is normally distributed and use a t-distribution to find the margin of error and construct a 95\% confidence interval for the population mean. Interpret the results. If convenient, use technology to construct the confidence interval. 17. Commute Time to Work In a random sample of eight people, the mean commute time to work was 35.5 minutes and the sample standard deviation was 7.2 minutes. 18. Driving Distance to Work In a random sample of five people, the mean driving distance to work was 22.2 miles and the sample standard deviation was 5.8 miles.

To calculate the margin of error for the first example, use the formula: Margin of Error = t* × (s/√n), where s is the sample standard deviation, n is the sample size, and t* is the t-value for 95% confidence (based on degrees of freedom, which is n-1). For the 35.5-minute commute time (n=8, s=7.2), the t-value is about 2.365. This results in a margin of error of approximately 5.75. The 95% confidence interval is (35.5 - 5.75, 35.5 + 5.75) or (29.75, 41.25). Meanwhile, for the driving distance (n=5, s=5.8), using a t-value of about 4.604 gives a margin of error around 5.29, resulting in an interval of (22.2 - 5.29, 22.2 + 5.29) or (16.91, 27.49). Interpreting the results: For the commute time, we can be 95% confident that the true mean commute time for the population lies between 29.75 and 41.25 minutes. For the driving distance, we're 95% confident that the average distance to work is between 16.91 and 27.49 miles. Confidence intervals not only give us a range of plausible values for the population mean but also reflect the precision of our estimate. The smaller the interval, the more confident we are, but this confidence also depends on sample size and variability.