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Statistical Inference: A Short Course by Michael J. Panik

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Appendix 10.B Wilcoxon Signed Rank Test (of a Median)

Suppose we have evidence that points to a symmetric population distribution. Then a Wilcoxon signed rank test for the median is in order. Additionally, with the population distribution symmetric, this test for the median is “equivalent to a test for the mean.” To execute the Wilcoxon signed rank test, let us extract a random sample of size n with values X1, X2, . . ., Xn from a continuous symmetric population and test the following hypotheses pertaining to the population median MED:

Case 1 Case 2 Case 3
H0: MED = MEDo H0: MED = MEDo H0: MED = MEDo
H1: MEDMEDo H1: MED > MEDo H1: MED < MEDo

where MEDo is the null value of MED.

To perform the Wilcoxon signed rank test, let us consider the following sequence of steps:

1. Subtract the null value MEDo from each Xi to obtain Yi = XiMEDo, i = 1,. . .,n. (If any Yi = 0, then eliminate Xi and reduce n accordingly.)
2. Rank the Yi's in order of increasing absolute value. (If any of the nonzero Yi's are tied in value, then these tied Yi's are given the average rank.)
3. Restore to the rank values 1, . . ., n the algebraic sign of the associated difference Yi. Then the ranks with the appropriate signs attached are called the signed ranks Ri, i = 1, . . .,n, where img denotes the rank carrying a positive sign.

Let us specify our test statistic as

(10.B.1)

the sum of the ...

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