SECTION 8.2 Testing the Difference Between Means (Independent Samples, s1 and s2 Unknown) 435 Extending Concepts Constructing Confidence Intervals for M1 − M2 When the sampling distribution for x1 - x2 is approximated by a t-distribution and the population variances are not equal, you can construct a confidence interval for m1 - m2, as shown below. 1x1 - x22 - tcBs2 1 n1 + s2 2 n2 6 m1 - m2 6 1x1 - x22 + tcBs2 1 n1 + s2 2 n2 where d.f. is the smaller of n1 - 1 and n2 - 1 In Exercises 23 and 24, construct the indicated confidence interval for m1 - m2. Assume the populations are approximately normal with unequal variances. 23. 10K Race To compare the mean finishing times of male and female participants in a 10K race, you randomly select several finishing times from both sexes. The results are shown at the left. Construct an 80% confidence interval for the difference in mean finishing times of male and female participants in the race. (Adapted from Great Race) 24. Golf To compare the mean driving distances for two golfers, you randomly select several drives from each golfer. The results are shown at the left. Construct a 90% confidence interval for the difference in mean driving distances for the two golfers. Constructing Confidence Intervals for M1 − M2 When the sampling distribution for x1 - x2 is approximated by a t-distribution and the populations have equal variances, you can construct a confidence interval for m1 - m2, as shown below. 1x1 - x22 - tc ns# A1 n1 + 1 n2 6 m1 - m2 6 1x1 - x22 + tc ns# A1 n1 + 1 n2 where ns = B1n1 - 12s1 2 + 1n 2 - 12s 2 2 n1 + n2 - 2 and d.f. = n1 + n2 - 2 In Exercises 25 and 26, construct the indicated confidence interval for m1 - m2. Assume the populations are approximately normal with equal variances. 25. Family Doctor To compare the mean number of days spent waiting to see a family doctor for two large cities, you randomly select several people in each city who have had an appointment with a family doctor. The results are shown at the left. Construct a 90% confidence interval for the difference in mean number of days spent waiting to see a family doctor for the two cities. (Adapted from Merritt Hawkins) 26. 10K Race To compare the mean ages of male and female participants in a 10K race, you randomly select several ages from both sexes. The results are shown below. Construct a 95% confidence interval for the difference in mean ages of male and female participants in the race. (Adapted from Great Race) Males Females x1 = 40 years s1 = 12.3 years n1 = 20 x2 = 39 years s2 = 14.5 years n2 = 18 Sample Statistics for Ages of 10K Race Participants Miami Seattle x1 = 28 days s1 = 39.7 days n1 = 20 x2 = 26 days s2 = 42.4 days n2 = 17 TABLE FOR EXERCISE 25 Sample Statistics for Number of Days Waiting for an Appointment with a Family Doctor Golfer 1 Golfer 2 x1 = 267 yd s1 = 6 yd n1 = 9 x2 = 244 yd s2 = 12 yd n2 = 5 TABLE FOR EXERCISE 24 Sample Statistics for Driving Distances Males Females x1 = 65.8 min s1 = 34.1 min n1 = 20 x2 = 65.3 min s2 = 17.7 min n2 = 18 TABLE FOR EXERCISE 23 Sample Statistics for Finishing Times of 10K Race Participants
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