666 CHAPTER 13 Nonparametric Tests mean of the ranks involved in the tie. See Section 13-1 for a description of ranks and the procedure for handling ties.) EXAMPLE: In Table 13-5, the ranks of the 27 male heights are shown in parentheses. The rank of 1 is assigned to the lowest sample value of 1667, the rank of 2 is assigned to the next lowest value of 1680, and the rank of 3 is assigned to the next lowest value of 1684. The 17th and 18th values are tied at 1780, so we assign the rank of 17.5 to both of those tied values. Step 2: Find the sum of the ranks for either one of the two samples. EXAMPLE: In Table 13-5, the sum of the ranks from the first sample is 127. (That is, R1 = 5 + 8 + 11 + g + 9 = 127.) The sum of the ranks from the second sample is 251. Step 3: Calculate the value of the z test statistic as shown in the preceding Key Elements box, where either sample can be used as “Sample 1.” (If both sample sizes are greater than 10, then the sampling distribution of R is approximately normal with mean mR and standard deviation sR, and the test statistic is as shown in the preceding Key Elements box.) EXAMPLE: Calculations of mR and sR and z are shown in Example 1, which follows. TABLE 13-5 Heights (mm) of Males from ANSUR I and ANSUR II ANSUR I 1988 ANSUR II 2012 1698 (5) 1810 (21) 1727 (8) 1850 (25) 1734 (11) 1777 (16) 1684 (3) 1811 (22) 1667 (1) 1780 (17.5) 1680 (2) 1733 (10) 1785 (19) 1814 (23) 1885 (27) 1861 (26) 1841 (24) 1709 (7) 1702 (6) 1740 (13) 1738 (12) 1694 (4) 1732 (9) 1766 (15) 1748 (14) 1794 (20) 1780 (17.5) n1 = 12 n2 = 15 R1 = 127 R2 = 251 Heights of Males from ANSUR I 1988 and ANSUR II 2012 EXAMPLE 1 Table 13-5 lists samples of heights of males from the ANSUR I 1988 and ANSUR II 2012 data sets. Use a 0.05 significance level to test the claim that the two samples are from populations with the same median. SOLUTION REQUIREMENT CHECK (1) The sample data are two independent simple random samples. (2) The sample sizes are 12 and 15, so both sample sizes are greater than 10. The requirements are satisfied. The null and alternative hypotheses are as follows: H0: The two samples are from populations with the same median. H1: The two samples are from populations with different medians. Rank the combined list of all 27 male heights, beginning with a rank of 1 (assigned to the lowest value of 1667). The ranks corresponding to the individual sample values are shown in parentheses in Table 13-5. R denotes the sum of the ranks for the sample we choose as Sample 1. If we choose the ANSUR I 1988 sample, we get R = 5 + 8 + 11 + g + 9 = 127 Because there are 12 heights in the first sample, we have n1 = 12. Also, n2 = 15 because there are 15 heights in the second sample. The values of mR and sR and the test statistic z can now be found as follows. mR = n11n1 + n2 + 12 2 = 12112 + 15 + 12 2 = 168 sR = Bn1n21n1 + n2 + 12 12 = B11221152112 + 15 + 12 12 = 20.4939 z = R - mR sR = 127 - 168 20.4939 = -2.00
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