13-4 Wilcoxon Rank-Sum Test for Two Independent Samples 667 The test is two-tailed because a large positive value of z would indicate that disproportionately more higher ranks are found in Sample 1, and a large negative value of z would indicate that disproportionately more lower ranks are found in Sample 1. In either case, we would have strong evidence against the claim that the two samples come from populations with equal medians. The significance of the test statistic z can be treated as in previous chapters. We are testing (with a = 0.05) the hypothesis that the two populations have equal medians, so we have a two-tailed test. P-Value: Using the unrounded z score, the P-value is 0.045, so we reject the null hypothesis that the two samples are from populations with the same median. Critical Values: If we use the critical values of z = {1.96, we see that the test statistic of z = -2.00 does fall within the critical region, so we reject the null hypothesis that the two samples are from populations with the same median. YOUR TURN. Do Exercise 5 “Heights of Females from ANSUR I and ANSUR II.” INTERPRETATION There is sufficient evidence to warrant rejection of the claim that the sample of male heights from ANSUR I 1988 and the sample of male heights from ANSUR II 2012 are from populations with the same median. It appears that the medians are different. Based on the listed sample data, it appears that the heights from 1988 are different than the heights from 2012. Because the heights from 2012 have a larger median, it appears that males became taller as time passed from 1988 to 2012, although a one-sided hypothesis test would be better for testing that conclusion. A one-sided test would lead to the conclusion that the 2012 heights are significantly larger than the 1988 heights. In Example 1, if we interchange the two sets of sample values and consider the ANSUR II 2012 heights to be the first sample, then R = 251, mR = 210, sR = 20.4939, and z = 2.00, so the conclusion is exactly the same. Large Samples of Heights of Males from ANSUR I (in 1988) and ANSUR II (in 2012) EXAMPLE 2 Example 1 uses samples of sizes 12 and 15. Data Set 2 “ANSUR I 1988” includes 1774 heights of males and Data Set 3 “ANSUR II 2012” includes 4082 heights of males. Repeating the manual calculations of Example 1 using these much larger samples would not be much fun. But alas, technology comes to the rescue. If we use Statdisk to repeat Example 1 with the much larger data sets, we get the accompanying display. We can see that the test statistic is z = -0.11 rounded. The unrounded test statistic of z = -0.10546 can be used to find that the P-value in this two-tailed test is 0.9160. Although Example 1 led to a conclusion of a difference between the two population medians, the larger data sets lead to a conclusion of no significant difference between the two population medians. YOUR TURN. Do Exercise 9 “Queues.” Statdisk
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