13-3 Wilcoxon Signed-Ranks Test for Matched Pairs 659 Wilcoxon Signed-Ranks Procedure The following example includes the eight steps of the Wilcoxon signed-ranks procedure. This procedure requires that you sort data, then assign ranks. When working with larger data sets, sorting and ranking become tedious, but technology can be used to automate that process. Stemplots can also be very helpful in sorting data. TABLE 13-4 Measured and Reported Weights (kg) Measured Weights 152.6 149.3 174.8 119.5 194.9 180.3 215.4 239.6 Reported Weights 150 148 170 119 185 180 224 239 d (difference) 2.6 1.3 4.8 0.5 9.9 0.3 -8.6 0.6 Rank of |d | 5 4 6 2 8 1 7 3 Signed rank 5 4 6 2 8 1 -7 3 Measured and Reported Weights EXAMPLE 1 The first two rows of Table 13-4 include measured and reported weights from a simple random sample of eight different male subjects (from Data Set 4 “Measured and Reported” in Appendix B). The data are matched, so each measured weight is paired with the corresponding reported weight. Assume that we want to use the Wilcoxon signed-ranks test with a 0.05 significance level to test the claim that there is a significant difference between measured weights and reported weights of males. That is, assume that we want to test the null hypothesis that the matched pairs are from a population of matched pairs with differences having a median equal to zero. SOLUTION REQUIREMENT CHECK (1) The data are a simple random sample, as required. (2) The second requirement is that the population of differences has a distribution that is approximately symmetric, meaning that the left half of its histogram is roughly a mirror image of its right half. A histogram of the differences in the third row of Table 13-4 shows that the difference between the left and right sides is not too extreme, so we will consider this requirement to be satisfied. Wilcoxon Signed-Ranks Procedure Step 1: For each pair of data, find the difference d by subtracting the second value from the first value. Discard any pairs that have a difference of 0. EXAMPLE: The third row of Table 13-4 lists the differences found by subtracting the reported weights from the measured weights. Ignore any differences equal to 0. Step 2: Ignore the signs of the differences, then sort the differences from lowest to highest and replace the differences by the corresponding rank value (as described in Section 13-1). When differences have the same numerical value, assign to them the mean of the ranks involved in the tie. EXAMPLE: The fourth row of Table 13-4 shows the ranks of the values of d . The smallest value of d is 0.3, so it is assigned the rank of 1. The next smallest value of d is 0.5, so it is assigned the rank of 2. If there had been any ties, they would have been assigned the mean of the ranks involved in the tie. continued
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