Question: Several retired bicycle racers are coaching a large group of young prospects. They randomly select seven of their riders to take part in a test of the effectiveness of a new dietary supplement that is supposed to increase strength and stamina. Each of the seven riders does a time trial on the same course. Then they all take the dietary supplement for 4 weeks. All other aspects of their training program remain as they were prior to the time trial. At the end of the 4 weeks, these riders do another time trial on the same course. The times (in minutes) recorded by each rider for these trials before and after the 4-week period are shown in the following table.
Before 103 97 111 95 102 96 108
After 100 95 104 101 96 91 101
a. Construct a 99% confidence interval for the mean d of the population paired differences, where a paired difference is equal to the time taken before the dietary supplement minus the time taken after the dietary supplement.
b. Test at the 2.5% significance level whether taking this dietary supplement results in faster times in the time trials. Assume that the population of paired differences is (approximately) normally distributed.