## The Wilcoxon Signed Rank Test

The Wilcoxon signed rank test assumes only a continuous and symmetric distribution with mean =median = If we have a sample then we find and rank them from smallest to largest.

The null hypothesis is The test statistic is the sum of the ranks of those with positive.

The alternative hypothesis may be stated as one of those below, with associated rejection region for a level test, where and are obtained from tables.     Either or Example: A manufacturer of electric irons, wishing to test the accuracy of the thermostat control at the 500 degree fahrenheit setting, obtains actual temperatures at that setting for fifteen irons. They are

494.6, 510.8, 487.5, 493.2, 502.6, 485, 495.9, 498.2, 501.6, 497.3, 492.0, 504.3, 499.2, 493.5, 505.8

Assuming a symmetric distribution for the temperature, we can apply the Wilcoxon signed rank test. Subtracting 500 from each gives

-5.4, 10.8, -12.5, -6.8, 2.6, -15, -4.1, -1.8, 1.6, -2.7, -8.0, 4.3,-0.8, -6.5,5.8

The ranks are obtained by ordering these from smallest to largest, obtaining,

 Absolute Value 0.8 1.6 1.8 2.6 2.7 4.1 4.3 5.6 5.8 6.8 6.8 8 10.8 12.5 15 Rank 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Sign - + - + - - + - + - - - + - -

Thus From the Wilcoxon tables, when is true so is rejected if either or neither of which apply here so is not rejected. There is no evidence at this level to suppose the thermostat is defective. 