15-3 WILCOXON SIGNED-RANK TEST 58115-3 WILCOXON SIGNED-RANK TESTThe sign test makes use only of the plus and minus signs of the differences between the ob-
servations and the median (or the plus and minus signs of the differences between the
observations in the paired case). It does not take into account the size or magnitude of these
differences. Frank Wilcoxon devised a test procedure that uses both direction (sign) and mag-
nitude. This procedure, now called the Wilcoxon signed-rank test,is discussed and illus-
trated in this section.
The Wilcoxon signed-rank test applies to the case of symmetric continuous distribu-
tions.Under these assumptions, the mean equals the median, and we can use this procedure to
test the null hypothesis that m 5 m0. We now show how to do this.15-3.1 Description of the TestWe are interested in testing H 0 : 0 against the usual alternatives. Assume that X 1 ,
X 2 ,... , Xnis a random sample from a continuous and symmetric distribution with mean (and
median) . Compute the differences Xi 0 , i1, 2,... , n. Rank the absolute differences
in ascending order, and then give the ranks the signs of their
corresponding differences. Let W be the sum of the positive ranks and Wbe the absolute
value of the sum of the negative ranks, and let Wmin(W, W). Appendix Table VIII con-
tains critical values of W, say w*. If the alternative hypothesis is , then if the ob-
served value of the statistic ww*, the null hypothesis H 0 : 0 is rejected. Appendix
Table VIII provides significance levels of 0.10, 0.05, 0.02, 0.01 for the
two-sided test.
For one-sided tests, if the alternative is H 1 : 0 , reject H 0 : 0 if ww*; and if
the alternative is H 1 : 0 , reject H 0 : 0 if ww*. The significance levels for one-
sided tests provided in Appendix Table VIII are 0.05, 0.025, 0.01, and 0.005.H 1 :
00 Xi 0 0 , i1, 2,... , n 015-13. Consider the blood cholesterol data in Exercise
10-39. Use the sign test to determine whether there is any dif-
ference between the medians of the two groups of measure-
ments, with 0.05. What practical conclusion would you
draw from this study?
15-14. Use the normal approximation for the sign test for
the data in Exercise 15-12. With 0.05, what conclusions
can you draw?
15-15. Use the normal approximation to the sign test for
the data in Exercise 15-13. With 0.05, what conclusions
can you draw?
15-16. The distribution time between arrivals in a telecom-
munication system is exponential, and the system manager
wishes to test the hypothesis that minutes versus
minutes.
(a) What is the value of the mean of the exponential distribu-
tion under?
(b) Suppose that we have taken a sample of n10 observa-
tions and we observe r3. Would the sign test reject H 0
at 0.05?H 0 : 3.5H 1 : 3.5H 0 : 3.5(c) What is the type II error probability of this test if
?
15-17. Suppose that we take a sample of n10 measure-
ments from a normal distribution with 1. We wish to test
H 0 : 0 against H 1 : 0. The normal test statistic is
( ), and we decide to use a critical region of 1.96
(that is, reject H 0 if z 0 1.96).
(a) What is for this test?
(b) What is for this test, if 1?
(c) If a sign test is used, specify the critical region that gives
an value consistent with for the normal test.
(d) What is the value for the sign test, if 1? Compare
this with the result obtained in part (b).
15-18. Consider the test statistic for the sign test in
Exercise 15-9. Find the P-value for this statistic.
15-19. Consider the test statistic for the sign test in Exercise
15-10. Find the P-value for this statistic. Compare it to the
P-value for the normal approximation test statistic computed
in Exercise 15-11.Z 0 X (^) 1 n
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