664 CHAPTER 13 Nonparametric Tests 13. Rank Sums Exercise 12 uses Data Set 33 “Disney World Wait Times” in Appendix B, and the sample size for the 5:00 PM Tower of Terror wait times is n = 50. a. If we have sample paired data with 50 nonzero differences, what are the smallest and largest possible values of T? b. If we have sample paired data with 50 nonzero differences, what is the expected value of T if the population consists of matched pairs with differences having a median of 0? c. If we have sample paired data with 50 nonzero differences and the sum of the positive ranks is 165, find the absolute value of the sum of the negative ranks. d. If we have sample paired data with n nonzero differences and one of the two rank sums is k, find an expression for the other rank sum. 13-3 Beyond the Basics Key Concept This section describes the Wilcoxon rank-sum test, which uses ranks of values from two independent samples to test the null hypothesis that the samples are from populations having equal medians. The Wilcoxon rank-sum test is equivalent to the Mann-Whitney U test (see Exercise 13), which is included in some textbooks and technologies (such as Minitab, StatCrunch, and XLSTAT). Here is the basic idea underlying the Wilcoxon rank-sum test: If two samples are drawn from identical populations and the individual values are all ranked as one combined collection of values, then the high and low ranks should fall evenly between the two samples. If the low ranks are found predominantly in one sample and the high ranks are found predominantly in the other sample, we have an indication that the two populations have different medians. Unlike the parametric t tests for two independent samples in Section 9-2, the Wilcoxon rank-sum test does not require normally distributed populations and it can be used with data at the ordinal level of measurement, such as data consisting of ranks. In Table 13-2 we noted that the Wilcoxon rank-sum test has a 0.95 efficiency rating when compared to the parametric test. Because this test has such a high efficiency rating and involves easier calculations, it is often preferred over the parametric t test, even when the requirement of normality is satisfied. 13-4 Wilcoxon Rank-Sum Test for Two Independent Samples CAUTION Don’t confuse the Wilcoxon rank-sum test for two independent samples with the Wilcoxon signed-ranks test for matched pairs. Use “Internal Revenue Service” as the mnemonic for IRS to remind yourself of “Independent: Rank Sum.” DEFINITION The Wilcoxon rank-sum test is a nonparametric test that uses ranks of sample data from two independent populations to test this null hypothesis: H0: Two independent samples come from populations with equal medians. (The alternative hypothesis H1 can be any one of the following three possibilities: The two populations have different medians, or the first population has a median greater than the median of the second population, or the first population has a median less than the median of the second population.)
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