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www.downloadslide.com Exercises 649 12.11 When Y 1i , for i = 1, 2,...,n, and Y 2i , for i = 1, 2,...,n, represent independent samples from two populations **with** means µ 1 and µ 2 and variances σ1 2 and σ 2 2 , respectively, we determined that σ 2 = (1/n)(σ 2 (Y 1 −Y 2 ) 1 + σ 2 2 ). If the samples were paired and we computed the differences, D i , for i = 1, 2,...,n, we determined that σ(2/D) = (1/n)(σ1 2 + σ 2 2 − 2ρσ 1σ 2 ). a When is σ 2 greater than σ(2/D)? (Y 1 −Y 2 ) b When is σ 2 equal to σ(2/D)? (Y 1 −Y 2 ) c When is σ 2 less than σ(2/D)? (Y 1 −Y 2 ) d Based on the discussion in the text and your answers to parts (a)–(c), when would it be better to implement the matched-pairs experiment and when would it be better to implement the independent samples experiment? 12.12 Refer to Exercise 12.11. Assume that σ 2 1 = σ 2 2 = σ 2 . The table values used to implement a test of hypothesis or construct a confidence interval depend, for small samples, on the number of degrees of freedom associated **with** the estimates for σ 2 or σ 2 D . a Assuming two independent samples, each of size n, and that σ1 2 = σ 2 2 = σ 2 , how many degrees of freedom are associated **with** the estimator for the common variance σ 2 ? b Assuming a matched-pairs experiment consisting of n pairs of observations, how many degrees of freedom are associated **with** the estimator of σD 2 ? c Assume that all of the assumptions necessary to implement the independent samples t procedures are satisfied and that we want to find a 95% confidence interval for the difference in means. What are the values of t .025 used to construct confidence intervals for the difference in means based on the independent samples and matched-pairs experiments if n = 5? If n = 10? If n = 30? d If all of the assumptions necessary to implement the independent samples t procedures are satisfied, identify a possible disadvantage to implementing a matched-pairs experiment rather than taking independent samples. 12.13 Exercise 10.76 describes a dental experiment conducted to investigate the effectiveness of an oral rinse used to inhibit the growth of plaque on teeth. Subjects were divided into two groups: One group used a rinse containing the antiplaque agent, and the control group used a rinse **with** only inactive ingredients. Another experiment has been performed to assess the growth of plaque for individuals who have used the rinse **with** the antiplaque agent. For each person in the study, plaque buildup was measured 4 hours after using the rinse and again after 8 hours. If you wanted to compare the mean plaque buildup for the two different times, would you implement an analysis based on a matched-pairs or independent samples procedure? Why? 12.14 Two procedures for sintering copper are to be compared by testing each procedure on six different types of powder. The measurement of interest is the porosity (volume percentage due to voids) of each test specimen. The results of the tests are as shown in the accompanying table. Powder Procedure I Procedure II 1 21 23 2 27 26 3 18 21 4 22 24 5 26 25 6 19 16 Is there sufficient evidence to claim that procedure II produces higher mean porosity values? Give bounds for the p-value. What would you conclude at the α = .05 level?

www.downloadslide.com 650 Chapter 12 Considerations in Designing Experiments 12.15 A plant manager, in deciding whether to purchase a machine of design A or design B, checks the times for completing a certain task on each machine. Eight technicians were used in the experiment, **with** each technician using both machine A and machine B in a randomized order. The times (in seconds) required to complete the task are given in the accompanying table. Technician A B 1 32 30 2 40 39 3 42 42 4 26 23 5 35 36 6 29 27 7 45 41 8 22 21 a Test to see if there is a significant difference between mean completion times, at the 5% significance level. b Do you think pairing on technicians was worthwhile in this case? Explain. c What assumptions are necessary for the test in part (a)? 12.16 “Muck” is the rich, highly organic type of soil that serves as the primary growth medium for vegetation in the Florida Everglades. Because of the high concentration of organic material, muck can be destroyed over time by a variety of natural and human-made causes. Members of the Florida Game and Fresh Water Fish Commission staked out several plots in the Everglades. The depth of muck at each location was measured when each plot was marked and again 6 years later. The following table identifies a portion of the data (given in inches) obtained. Plot Initial Reading Later Reading Plot Initial Reading Later Reading 1 34.5 31.5 9 44.0 35.2 2 44.0 37.9 10 40.5 37.2 3 37.5 35.5 11 27.0 24.7 4 27.0 23.0 12 29.5 25.8 5 37.0 34.5 13 31.5 29.0 6 40.0 31.1 14 35.0 36.8 7 47.2 46.0 15 44.0 36.5 8 35.2 31.0 a b c d e Test to see if there is sufficient evidence to indicate a decrease in average muck depth during the study period. Give bounds on the associated p-value. What would you conclude if you desired to implement an α = .01 level test? (Although you are free to take the necessary differences in any order that you prefer, the answer provided at the back of the book assumes that the differences were formed by taking later readings minus initial readings.) Give a 95% confidence interval for the difference in mean muck depths at the end and beginning of the study. Interpret this interval. [See the remark following part (a).] Give a 95% confidence interval for the initial mean muck depth in the portion of the Everglades in which the study was conducted. Repeat the instructions of part (c) for later readings. What assumptions are necessary to apply the techniques you used in answering parts (a) and (b)? Parts (c) and (d)?

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