What has a greater effect, if any, on the power of the Wilcoxon rank sum test, skewness

6.21 Refer to Exercises 6.19 and 6.20. a. For what type of population distributions would you recommend using the t test? Justify your answer.

b. For what type of population distributions would you recommend using the Wilcoxon

rank sum test? Justify your answer. 6.4 Inferences about M 1 ⴚ M 2 : Paired Data 6.22 Set up the rejection regions for testing the following:

a. H

: m d ⫽ 0 versus H a : m d ⫽ 0, with n 1 ⫽ 11, n 2 ⫽ 14, and a ⫽ .05

b. H

: m d ⱕ 0 versus H a : m d ⬎ 0, with n 1 ⫽ n 2 ⫽ 17, and a ⫽ .01

c. H

: m d ⱖ 0 versus H a : m d ⬍ 0, with n 1 ⫽ 8, n 2 ⫽ 12, and a ⫽ .025

6.23 Consider the data given here. Pair

1 2 3 4 5 6 y 1 48.2 44.6 49.7 40.5 54.6 47.1 y 1 41.5 40.1 44.0 41.2 49.8 41.7

a. Conduct a paired t test of H

: m d ⱕ 0 versus H a : m d ⬎ 0 with d ⫽ y 1 ⫺ y 2 . Use a ⫽ .05

b. Using a testing procedure related to the binomial distribution, test the

hypotheses in a. Does your conclusion agree with the conclusion reached in part a? c. When might the two approaches used in parts a and b not agree? 6.24 Refer to the data of Exercise 6.23.

a. Give the level of significance for your test. b. Place a 95 confidence interval on m

d . 6.25 Refer to the data of Exercise 6.11. A potential criticism of analyzing these data as if they were two independent samples is that the measurements taken in 1996 were taken at the same site as the measurements taken in 1982. Thus, there is the possibility that there will be a strong positive correlation between the pair of observations at each site.

a. Plot the pairs of observations in a scatterplot with the 1982 values on the horizontal

axis and the 1996 values on the vertical axis. Does there appear to be a positive correlation between the pairs of measurements? Estimate the correlation between the pair of observations?

b. Compute the correlation coefficient between the pair of observations.

Does this value confirm your observations from the scatterplot? Explain your answer.

c. Answer the questions posed in Exercise 6.11 parts a and b using a paired data

analysis. Are your conclusions different from the conclusions you reached treating the data as two independent samples? Engin. 6.26 Researchers are studying two existing coatings used to prevent corrosion in pipes that transport natural gas. The study involves examining sections of pipe that had been in the ground at least 5 years. The effectiveness of the coating depends on the pH of the soil, so the researchers recorded the pH of the soil at all 20 sites at which the pipe was buried prior to measuring the amount of corrosion on the pipes. The pH readings are given here. Describe how the researchers could conduct the study to reduce the effect of the differences in the pH readings on the evalua- tion of the difference in the two coatings’ corrosion protection. pH Readings at Twenty Research Sites Coating A 3.2 4.9 5.1 6.3 7.1 3.8 8.1 7.3 5.9 8.9 Coating B 3.7 8.2 7.4 5.8 8.8 3.4 4.7 5.3 6.8 7.2 Med. 6.27 Suppose you are a participant in a project to study the effectiveness of a new treatment for high cholesterol. The new treatment will be compared to a current treatment by recording the change in cholesterol readings over a 10-week treatment period. The effectiveness of the treatment may depend on the participant’s age, body fat percentage, diet, and general health. The study will involve at most 30 participants because of cost considerations. a. Describe how you would conduct the study using independent samples. b. Describe how you would conduct the study using paired samples.

c. How would you decide which method, paired or independent samples, would be more

efficient in evaluating the change in cholesterol readings? Med. 6.28 The paper “Effect of long-term blood pressure control on salt sensitivity” [Journal of Medicine 1997 28:147–156] describes a study evaluating salt sensitivity SENS after a period of antihypertensive treatment. Ten hypertensive patients diastolic blood pressure between 90 and 115 mmHg were studied after at least 18 months on antihypertensive treatment. SENS readings, which were obtained before and after the patients were placed on an antihypertensive treatment, are given here: Patient 1 2 3 4 5 6 7 8 9 10 Before treatment 22.86 7.74 15.49 9.97 1.44 9.39 11.40 1.86 ⫺ 6.71 6.42 After treatment 6.11 ⫺ 4.02 8.04 3.29 ⫺ 0.77 6.99 10.19 2.09 11.40 10.70

a. Is there significant evidence that the mean SENS value decreased after the patient

received antihypertensive treatment? b. Estimate the size of the change in the mean SENS value. c. Do the conditions required for using the t procedures appear to be valid for these data? Justify your answer. Edu. 6.29 A study was designed to measure the effect of home environment on academic achieve- ment of 12-year-old students. Because genetic differences may also contribute to academic achievement, the researcher wanted to control for this factor. Thirty sets of identical twins were identified who had been adopted prior to their first birthday, with one twin placed in a home in which academics were emphasized Academic and the other twin placed in a home in which academics were not emphasized Nonacademic. The final grades based on 100 points for the 60 students are given here: Set of Set of Twins Academic Nonacademic Twins Academic Nonacademic 1 78 71 16 90 88 2 75 70 17 89 80 3 68 66 18 73 65 4 92 85 19 61 60 5 55 60 20 76 74 6 74 72 21 81 76 7 65 57 22 89 78 8 80 75 23 82 78 9 98 92 24 70 62 10 52 56 25 68 73 11 67 63 26 74 73 12 55 52 27 85 75 13 49 48 28 97 88 14 66 67 29 95 94 15 75 70 30 78 75

a. Use the following computer output to evaluate whether there is a difference in the

mean final grade between the students in an academically oriented home environment and those in a nonacademically oriented home environment.

b. Estimate the size of the difference in the mean final grades of the students in academic

and nonacademic home environments.