13-3 Wilcoxon Signed-Ranks Test for Matched Pairs 657 Appendix B Data Sets. In Exercises 13–16, refer to the indicated data set in Appendix B and use the sign test for the claim about the median of a population. 13.Body Temperatures Data Set 5 “Body Temperatures” in Appendix B includes measured body temperatures of adults. Use the body temperatures listed for 12 AM on Day 1 with the sign test to test the claim that the median is equal to 98.6°F. Use a 0.01 significance level. 14.Peanut Butter Cups Data Set 38 “Candies” includes weights (grams) of randomly selected Reese’s Peanut Butter Cup Miniatures. They are from a package of 38 cups, and the package label states that the total weight is 12 oz, or 340.2 g. If the 38 cups have a total weight of 340.2 g, then the cups should have a median weight of 340.2 g>38 = 8.953 g. Use the listed sample data to test the claim that the sample is from a population with a median weight equal to 8.953 g. Use a significance level of a = 0.05. 15.Cotinine in Smokers Data Set 15 “Passive and Active Smoke” includes cotinine measurements from 902 smokers. Cotinine is a biomarker of nicotine in the body. Use a 0.01 significance level to test the claim that smokers have cotinine levels with a median of 2.84 ng/mL, which is the median for nonsmokers not exposed to tobacco smoke. 16.Tower of Terror Data Set 33 “Disney World Wait Times” includes wait times (minutes) for the Tower of Terror ride at 5:00 PM. Use those times to test the claim that the median of all such wait times is equal to 30 minutes. Use a 0.01 significance level. 17.Procedures for Handling Ties In the sign test procedure described in this section, we exclude ties (represented by 0 instead of a sign of + or -). A second approach is to treat half of the 0s as positive signs and half as negative signs. (If the number of 0s is odd, exclude one so that they can be divided equally.) With a third approach, in two-tailed tests make half of the 0s positive and half negative; in one-tailed tests make all 0s either positive or negative, whichever supports the null hypothesis. Repeat Example 4 “Body Temperatures” using the second and third approaches to handling ties, and use a significance level of 0.05. Do the different approaches lead to very different test statistics, P-values, and conclusions? 18. Finding Critical Values Table A-7 lists critical values for limited choices of a. Use Table A-1 to add a new column in Table A-7 (from n = 1 to n = 8) that represents a significance level of 0.03 in one tail or 0.06 in two tails. For any particular n, use p = 0.5, because the sign test requires the assumption that P(positive sign) = P(negative sign) = 0.5. The probability of x or fewer like signs is the sum of the probabilities for values up to and including x. 13-2 Beyond the Basics Key Concept This section introduces the Wilcoxon signed-ranks test, which begins with the conversion of the sample data into ranks. This test can be used for the two different applications described in the following definition. 13-3 Wilcoxon Signed-Ranks Test for Matched Pairs DEFINITION The Wilcoxon signed-ranks test is a nonparametric test that uses ranks for these applications: 1. Testing a claim that a population of matched pairs has the property that the matched pairs have differences with a median equal to zero 2. Testing a claim that a single population of individual values has a median equal to some claimed value
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