Various plots of the data are given here. Based on these plots, which of the test proce- Computer output is provided for two versions of the t test and a Wilcoxon rank sum Place a 95 confidence interval on the differences between the daily mileages for th

Mann-Whitney Test and CI: Treatment, Control Point estimate for ETA1-ETA2 is –5.20 95.2 Percent CI for ETA1-ETA2 is –12.89, 0.81 W = 278.5 Test of ETA1 = ETA2 vs ETA1 ETA2 is significant at 0.0438 The test is significant at 0.0438 adjusted for ties N Median Treatment Control 18 18 88 .25 93.75 5 1 50 100 Control Normal probability plot for control group Normal 150 200

10 20

Percent 30 40 50 60 70 80 90 95 99 P -value .010 RJ .826 N 18 StDev 33.20 Mean 102.1 5 1 60 70 80 90 Treatment Normal probability plot for treatment group Normal 100 110

10 20

Percent 30 40 50 60 70 80 90 95 99 P -value .046 RJ .944 N 18 StDev 9.059 Mean 88.33 Med. 6.18 The paper “Serum beta-2-microglobulin SB2M in patients with multiple myeloma treated with alpha interferon” [Journal of Medicine 1997 28:311–318] reports on the influence of alpha interferon administration in the treatment of patients with multiple myeloma MM. Twenty newly diagnosed patients with MM were entered into the study. The researchers ran- domly assigned the 20 patients into the two groups. Ten patients were treated with both intermit- tent melphalan and sumiferon treatment group, whereas the remaining ten patients were treated only with intermittent melphalan. The SB2M levels were measured before and at days 3, 8, 15, and months 1, 3, and 6 from the start of therapy. The measurement of SB2M was performed using a radioimmunoassay method. The measurements before treatment are given here: Treatment Group 2.9 2.7 3.9 2.7 2.1 2.6 2.2 4.2 5.0 0.7 Control Group 3.5 2.5 3.8 8.1 3.6 2.2 5.0 2.9 2.3 2.9 a. Plot the sample data for both groups using boxplots or normal probability plots. b. Based on your findings in part a, which procedure appears more appropriate for comparing the distributions of SB2M?

c. Is there significant evidence that there is a difference in the distribution of SB2M for

the two groups?

d. Discuss the implications of your findings in part c on the evaluation of the influence

of alpha interferon. 6.19 The simulation study described in Section 6.3 evaluated the effect of heavy-tailed and skewed distributions on the level of significance and power of the t test and Wilcoxon rank sum test. Examine the results displayed in Table 6.13 and then answer the following questions.

a. What has a greater effect, if any, on the level of significance of the t test, skewness or

heavy-tailness?

b. What has a greater effect, if any, on the level of significance of the Wilcoxon rank sum

test, skewness or heavy-tailness? 6.20 Refer to Exercise 6.19. a. What has a greater effect, if any, on the power of the t test, skewness or heavy-tailness?

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

or heavy-tailness? Two-Sample T-Test and CI: Treatment, Control Two-sample T for Treatment vs Control Difference = mu Treatment – mu Control Estimate for difference: –13.7889 95 CI for difference: –30.2727, 2.6950 T-Test of difference = 0 vs not =: T-Value = –1.70 P-Value = 0.098 DF = 34 Both use Pooled StDev = 24.3335 N Mean StDev SE Mean Treatment Control 18 18 88 .33 102.1 9.06 33.2 2.1 7.8 Two-Sample T-Test and CI: Treatment, Control Two-sample T for Treatment vs Control Difference = mu Treatment – mu Control Estimate for difference: –13.7889 95 CI for difference: –30.7657, 3.1880 T-Test of difference = 0 vs not =: T-Value = –1.70 P-Value = 0.105 DF = 19 N Mean StDev SE Mean Treatment Control 18 18 88 .33 102.1 9.06 33.2 2.1 7.8 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