A man who moves to a new city sees that there are two routes he could take to work. A neighbor who has lived there a long time tells him Route A will average 5 minutes faster than Route B. The man decides to experiment. Each day, he flips a coin to determine which way to go, driving each route 20 days. He finds that Route A takes an average of 40 minutes, with standard deviation 3 minutes, and Route B takes an average of 43 minutes, with standard deviation 2 minutes. Histograms of travel times for the routes are roughly symmetric and show no outliers.
a) Find a 95% confidence interval for the difference in average commuting time for the two routes. (From technology, df = 33.1.)
b) Should the man believe the old-timer's claim that he can save an average of 5 minutes a day by always driving Route A? Explain.