13-4 Wilcoxon Rank-Sum Test for Two Independent Samples 665 Wilcoxon Rank-Sum Test Objective Use the Wilcoxon rank-sum test with samples from two independent populations for the following null and alternative hypotheses: H0: The two samples come from populations with equal medians. H1:The median of the first population is different from (or greater than, or less than) the median from the second population. Notation n1 = size of Sample 1 n2 = size of Sample 2 R1 = sum of ranks for Sample 1 R2 = sum of ranks for Sample 2 R = same as R1 (sum of ranks for Sample 1) mR = mean of the sample R values that is expected when the two populations have equal medians sR = standard deviation of the sample R values that is expected with two populations having equal medians Requirements 1. There are two independent simple random samples. 2. Each of the two samples has more than 10 values. (For samples with 10 or fewer values, special tables are available in special reference books, such as CRC Standard Probability and Statistics Tables and Formulae, published by CRC Press.) Note: There is no requirement that the two populations have a normal distribution or any other particular distribution. Test Statistic z = R - mR sR where mR = n11n1 + n2 + 12 2 and sR = Bn1n21n1 + n2 + 12 12 n1 = size of the sample from which the rank sum R is found n2 = size of the other sample R = sum of ranks of the sample with size n1 P-Values P-values can be found from technology or by using the z test statistic and Table A-2. Critical Values Critical values can be found in Table A-2 (because the test statistic is based on the normal distribution). KEY ELEMENTS Procedure for Finding the Value of the Test Statistic To see how the following steps are applied, refer to the sample data listed in Table 13-5 on the next page. The data are heights (mm) of males randomly selected from Data Set 2 “ANSUR I 1988” and Data Set 3 “ANSUR II 2012.” Both data sets are in Appendix B. Step1: Temporarily combine the two samples into one big sample, then replace each sample value with its rank. (The lowest value gets a rank of 1, the next lowest value gets a rank of 2, and so on. If values are tied, assign to them the continued
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