Wilcoxon signed rank test on a population mean
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Wilcoxon signed rank test on a population median
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This is a non ; ;parametric test to test a hypothesis about a population median. As with so many of these non ; ;parametric tests it is best introduced through a worked example.Example 1 In a Spanish town each year there is an annual bull festival. Young men run from one end of the town to another being chased by a bull. The young men regularly train and a scientifically minded elderly lady has observed that the median time for running across town when not being chased by a bull is 12 minutes. The same lady observed the bull festival from the same distance and noted the following times for running across town from a random sample of eight men ; ; 10. 9.6 . 1 .1 12. 7.2 6.9 9. By means of a single ; ;sample Wilcoxon signed rank test, test the hypothesis that the median time for running across town is 12 minutes regardless of whether you are chased by a bull or not. Why might a t ; ;test for the mean running time not be appropriate? The hypotheses are ; ;[Equation goes here ; ; download the original to see it.]This is a one ; ;tailed test, [Equation goes here ; ; download the original to see it.]We begin by calculating the differences of the sample values from the expected median; we then rank the absolute values of the differences according to the magnitudes ; ; from smallest to largest. We calculate the signed rank. [Table goes here ; ; download the original to see it.]Let denote the sum of the ranks of the positive differences, and denote the sum of the negative differencesThen [Equation goes here ; ; download the original to see it.][Equation goes here ; ; download the original to see it.]The test statistic can be either[Equation goes here ; ; download the original to see it.]. If is the small then the positive differences are small compared with the negatives ones. This is pushing us towards acceptance of the alternative hypothesis[Equation goes here ; ; download the original to see it.]Thus, in this case the critical region is[Equation goes here ; ; download the original to see it.]where the critical value is drawn from the distribution of . This is given as Wilcoxon ; 9;s T. Tables give critical values as a function of the sample size, the level of significance and whether the test is one ; ; or two ; ;tailed.Here [Equation goes here ; ; download the original to see it.]Since [Equation goes here ; ; download the original to see it.]we reject and accept .Being chased by a bull really does improve your ability to run.The t ; ;test in situation like the one here night not be appropriate because we can not guarantee the normality of the underlying population. The Wilcoxon signed rank test needs a much weaker assumption, namely, that the underlying population has a continuous probability distribution.The next example shows how to deal with a two ; ;sided alternative hypothesis.Example 2The median of the average monthly sales of an agent in an insurance company is 2 . In a certain town the director of the local office of the insurance company recorded the average monthly sales of 10 randomly chosen agents.The results were[Equation goes here ; ; download the original to see it.]By means of a single ; ;sample Wilcoxon signed rank test, test whether or not the median of the average monthly sales in this particular town differs from 2 . Use the % level of significance.SolutionThe hypotheses are[Equation goes here ; ; download the original to see it.]This is a two ; ;tailed test, .[Table goes here ; ; download the original to see it.][Equation goes here ; ; download the original to see it.]As in the single ; ;sample case, the test statistic can be either [Equation goes here ; ; download the original to see it.]. In the case of the two ; ;tailed test, both the large and small values of should make us suspicious about the validity of the null hypothesis. The critical region is this case has the form[Equation goes here ; ; download the original to see it.]where T is either [Equation goes here ; ; download the original to see it.].Here [Equation goes here ; ; download the original to see it.]Since [Equation goes here ; ; download the original to see it.]We accept and reject . No evidence that the median of the average monthly sales differs from 2 .
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Contents of
Wilcoxon signed rank test on a population mean
1 Wilcoxon signed rank test on a population median
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