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 X_{1}, X_{2}, . . ., X_{n} from a continuous symmetric population and test the following hypotheses pertaining to the population median MED:

Case 1 | Case 2 | Case 3 |

H_{0}: MED = MED_{o} |
H_{0}: MED = MED_{o} |
H_{0}: MED = MED_{o} |

H_{1}: MED ≠ MED_{o} |
H_{1}: MED > MED_{o} |
H_{1}: MED < MED_{o} |

where MED_{o} 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 MED_{o} from each X_{i} to obtain Y_{i} = X_{i} − MED_{o}, i = 1,. . .,n. (If any Y_{i} = 0, then eliminate X_{i} and reduce n accordingly.)

2. Rank the Y_{i}'s in order of increasing absolute value. (If any of the nonzero Y_{i}'s are tied in value, then these tied Y_{i}'s are given the average rank.)

3. Restore to the rank values 1, . . ., n the algebraic sign of the associated difference Y_{i}. Then the ranks with the appropriate signs attached are called the signed ranks R_{i}, i = 1, . . .,n, where 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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