## The Wilcoxon Signed Rank Test

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

The null hypothesis is

The test statistic isthe sum of the ranks of thosewithpositive.

The alternative hypothesis may be stated as one of those below, with associated rejection region for a leveltest, whereandare obtained from tables.

Eitheror |

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 | - | + | - | + | - | - | + | - | + | - | - | - | + | - | - |

ThusFrom the Wilcoxon tables,when is true sois rejected if eitherorneither of which apply here sois not rejected. There is no evidence at this level to suppose the thermostat is defective.