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When the underlying population is normal, either the t-test or the Wilcoxon signed-rank test can be used to test hypotheses about . As mentioned earlier, the t-test is the best test in such situations in the sense that it produces a minimum value of for all tests with signi cance level . However, since it is not always clear that the normal distribution is appropriate, and since in many situations it is inappropriate, it is of interest to compare the two procedures for both normal and nonnormal populations. Unfortunately, such a comparison is not easy. The problem is that for the Wilcoxon signed-rank test is very dif cult to obtain, and for the t-test is dif cult to obtain for nonnormal distributions. Because type II error comparisons are dif cult, other measures of comparison have been developed. One widely used measure is asymptotic relative ef ciency (ARE).

<asp:DataList id=DataList1 runat= server > <headerstyle backcolor= black forecolor= white font-bold= True horizontalalign= Center > </headerstyle> <alternatingitemstyle backcolor= silver > </alternatingitemstyle> <footerstyle backcolor= black forecolor= white font-bold= True horizontalalign= Center > </footerstyle> <headertemplate> Employee List </headertemplate> <itemtemplate> <table>

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The ARE of one test relative to another is the limiting ratio of the sample sizes necessary to obtain identical error probabilities for the two procedures. For example, if the ARE of one test relative to the competitor is 0.5, when sample sizes are large, the rst test will require twice as large a sample as the second one to obtain similar error performance. While this does not tell us anything for small sample sizes, we can say the following: 1. 2. For normal populations, the ARE of the Wilcoxon signed-rank test relative to the t-test is approximately 0.95. For nonnormal populations, the ARE is at least 0.86, and in many cases it will exceed unity.

Although these are large-sample results, we generally conclude that the Wilcoxon signed-rank test will never be much worse than the t-test and that in many cases where the population is nonnormal it may be superior. Thus, the Wilcoxon signed-rank test is a useful alternate to the t-test. EXERCISES FOR SECTION 15-3

<tr><td>Employee ID:</td> <td><b> <%# string.Format( {0:D4} ,Container.DataItem.EID ) %> </b> </td></tr> <tr><td>Employee Name: </td> <td><b> <%# DataBinder.Eval(Container.DataItem, LastName )%>, <%# DataBinder.Eval(Container.DataItem, FirstName )%> </b> </td></tr> <tr><td>Salary: </td> <td><b> <%# DataBinder.Eval(Container.DataItem, Salary , {0:C} )%> </b></td></tr> </table> </ItemTemplate> <footertemplate> End of List </footertemplate> </asp:DataList>

15-20. Consider the data in Exercise 15-1 and assume that the distribution of pH is symmetric and continuous. Use the Wilcoxon signed-rank test with 0.05 to test the hypothesis H0: 7 against H1: 7. 15-21. Consider the data in Exercise 15-2. Suppose that the distribution of titanium content is symmetric and continuous. Use the Wilcoxon signed-rank test with 0.05 to test the hypothesis H0: 8.5 versus H1: 8.5. 15-22. Consider the data in Exercise 15-2. Use the largesample approximation for the Wilcoxon signed-rank test to test the hypothesis H0: 8.5 versus H1: 8.5. Use 0.05. Assume that the distribution of titanium content is continuous and symmetric. 15-23. Consider the data in Exercise 15-3. Use the Wilcoxon signed-rank test to test the hypothesis H0: 2.5 ppm versus H1: 2.5 ppm with 0.05. Assume that the distribution of impurity level is continuous and symmetric. 15-24. Consider the Rockwell hardness test data in Exercise 15-9. Assume that both distributions are continuous and use the Wilcoxon signed-rank test to test that the mean difference in hardness readings between the two tips is zero. Use 0.05. 15-25. Consider the paint drying time data in Exercise 15-10. Assume that both populations are continuous, and use the Wilcoxon signed-rank test to test that the difference in mean drying times between the two formulations is zero. Use 0.01. 15-26. Apply the Wilcoxon signed-rank test to the measurement data in Exercise 15-12. Use 0.05 and assume that the two distributions of measurements are continuous. 15-27. Apply the Wilcoxon signed-rank test to the blood cholesterol data from Exercise 10-39. Use 0.05 and assume that the two distributions are continuous.

15-4

The browser output (see Figure 6.7) shows a very readable list of employees. The data is in the item template, and the formatting is in the style elements.

Suppose that we have two independent continuous populations X1 and X2 with means 1 and 2. Assume that the distributions of X1 and X2 have the same shape and spread and differ only (possibly) in their locations. The Wilcoxon rank-sum test can be used to test the hypothesis H0: 1 2. This procedure is sometimes called the Mann-Whitney test, although the MannWhitney test statistic is usually expressed in a different form.

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