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www.downloadslide.com Exercises 747 stated: The differences in paired observations are normally distributed **with** a common variance σD 2 . The sign test might be more efficient when these assumptions are not satisfied. Sign Test for Large Samples: n > 25 Null hypothesis: H 0 : p = .5 (neither treatment is preferred to the other). Alternative hypothesis: H a : p ≠ .5 for a two-tailed test (Note: We use the two-tailed test for an example. Many analyses require a one-tailed test.) Test statistic: Z = [M − n/2]/[(1/2) √ n]. Rejection region: Reject H 0 if z ≥ z α/2 or if z ≤−z α/2 , where z α/2 is obtained from Table 3, Appendix 3. The sign test actually tests the null hypothesis that the median of the variables D i is zero versus the alternative that it is different from zero. [The median of the variables D i being zero does imply that P(D i < 0) = P(D i > 0).] If the variables X i and Y i have the same distribution, the median of the variables D i will be zero, as previously discussed. However, for models other than the shift model, there are other situations in which the median of the variables D i is zero. In these instances, the null hypothesis for the sign test is slightly more general than the statement that X i and Y i have the same distribution. Summarizing, the sign test is an easily applied nonparametric procedure for comparing two populations. No assumptions are made concerning the underlying population distributions. The value of the test statistic can be obtained quickly by a visual count, and the rejection region (or p-value) can be found easily by using a table of binomial probabilities. Furthermore, we need not know the exact values of pairs of responses, just whether X i > Y i for each pair (X i , Y i ). Exercise 15.5 provides an example of the use of the sign test for data of this sort. Exercises 15.1 What significance levels between α = .01 and α = .15 are available for a two-tailed sign test **with** 25 paired observations? (Make use of tabulated values in Table 1, Appendix 3, n = 25.) What are the corresponding rejection regions? 15.2 A study reported in the American Journal of Public Health (Science News)—the first to follow lead levels in blood for law-abiding handgun hobbyists using indoor firing ranges—documents a considerable risk of lead poisoning. 1 Lead exposure measurements were made on 17 members of a law enforcement trainee class before, during, and after a 3-month period of firearm instruction at a state-owned indoor firing range. No trainees had elevated lead levels in their blood before training, but 15 of the 17 ended training **with** blood lead levels deemed “elevated” by the Occupational Safety and Health Administration (OSHA). Is there sufficient evidence to claim that indoor firing range use increases blood-level readings? 1. Source: Science News, 136 (August 1989): 126.

www.downloadslide.com 748 Chapter 15 Nonparametric **Statistics** a b c Give the associated p-value. What would you conclude at the α = .01 significance level? Use the normal approximation to give the approximate p-value. Does the normal approximation appear to be adequate when n = 17? 15.3 Clinical data concerning the effectiveness of two drugs for treating a disease were collected from ten hospitals. The number of patients treated **with** the drugs differed for the various hospitals. The data are given in the table that follows. Drug A Drug B Number Number Percentage Number Number Percentage Hospital Treated Recovered Recovered Treated Recovered Recovered 1 84 63 75.0 96 82 85.4 2 63 44 69.8 83 69 83.1 3 56 48 85.7 91 73 80.2 4 77 57 74.0 47 35 74.5 5 29 20 69.0 60 42 70.0 6 48 40 83.3 27 22 81.5 7 61 42 68.9 69 52 75.4 8 45 35 77.8 72 57 79.2 9 79 57 72.2 89 76 85.4 10 62 48 77.4 46 37 80.4 a b Do the data indicate a difference in the recovery rates for the two drugs? Give the associated p-value. Why might it be inappropriate to use the t test to analyze the data? 15.4 For a comparison of the academic effectiveness of two junior high schools A and B, an experiment was designed using ten sets of identical twins, each twin having just completed the sixth grade. In each case, the twins in the same set had obtained their previous schooling in the same classrooms at each grade level. One child was selected at random from each set and assigned to school A. The other was sent to school B. Near the end of the ninth grade, an achievement test was given to each child in the experiment. The results are shown in the accompanying table. Twin Pair A B Twin Pair A B 1 67 39 6 50 52 2 80 75 7 63 56 3 65 69 8 81 72 4 70 55 9 86 89 5 86 74 10 60 47 a b Using the sign test, test the hypothesis that the two schools are the same in academic effectiveness, as measured by scores on the achievement test, against the alternative that the schools are not equally effective. Give the attained significance level. What would you conclude **with** α = .05? Suppose it is suspected that junior high school A has a superior faculty and better learning facilities. Test the hypothesis of equal academic effectiveness against the alternative that school A is superior. What is the p-value associated **with** this test? 15.5 New food products are frequently subjected to taste tests by a panel of judges. The judges are usually asked to state a preference for one food over another so that no quantitative scale

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