Applied Statistics and Probability for Engineers

15-3 WILCOXON SIGNED-RANK TEST 583

Ties in the Wilcoxon Signed-Rank Test

Because the underlying population is continuous, ties are theoretically impossible, although

they will sometimes occur in practice. If several observations have the same absolute magni-

tude, they are assigned the average of the ranks that they would receive if they differed slightly

from one another.

15-3.2 Large-Sample Approximation

If the sample size is moderately large, say n 20, it can be shown that W(or W) has

approximately a normal distribution with mean

and variance

Therefore, a test of H 0 :  0 can be based on the statistic

^2 W

n 1 n 1212 n 12

24

W

n 1 n 12

4

Z 0  (15-6)

Wn 1 n 124

2 n 1 n 1212 n 1224

An appropriate critical region for either the two-sided or one-sided alternative hypotheses can

be chosen from a table of the standard normal distribution.

15-3.3 Paired Observations

The Wilcoxon signed-rank test can be applied to paired data. Let (X 1 j, X 2 j),j1, 2,... , nbe

a collection of paired observations from two continuous distributions that differ only with re-

spect to their means. (It is not necessary that the distributions of X 1 and X 2 be symmetric.) This

assures that the distribution of the differences DjX 1 jX 2 jis continuous and symmetric.

Thus, the null hypothesis is H 0 :  1  2 , which is equivalent to H 0 : D0. We initially con-

sider the two-sided alternative H 1 :  1  2 (or H 1 : D0).

To use the Wilcoxon signed-rank test, the differences are first ranked in ascending order of

their absolute values, and then the ranks are given the signs of the differences. Ties are assigned

average ranks. Let Wbe the sum of the positive ranks and Wbe the absolute value of the sum

of the negative ranks, and Wmin(W, W). If the observed value w w*, the null hypoth-

esis H 0 :  1  2 (or H 0 : D0) is rejected where w*is chosen from Appendix Table VIII.

For one-sided tests, if the alternative is H 1 :  1 

 2 (or H 1 : D 0), reject H 0 if ww*;

and if H 1 :  1  2 (or H 1 : D 0), reject H 0 if w w*. Be sure to use the one-sided test

significance levels shown in Appendix Table VIII.

EXAMPLE 15-5 We will apply the Wilcoxon signed-rank test to the fuel-metering device test data used previously

in Example 15-3. The eight-step hypothesis-testing procedure can be applied as follows:

1. The parameters of interest are the mean fuel mileage performance for the two meter-
   ing devices.

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