In the sign test procedure described in this section, we exclude ties. A second approach is to treat half of the 0s as positive signs and half as negative signs. With a third approach, in two-tailed tests make half of the 0s positive and half negative; in one-tailed tests make all 0s either positive or negative, whichever supports the null hypothesis. Repeat Example 4 using the second and third approaches to handling ties. Do the different approaches lead to very different results?

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Table 13-2 Efficiency: Comparison of Parametric and Nonparametric Tests Efficiency Rating of Nonparametric Test with Normal Populations Application Matched pairs of sample data test or z test Nonparametric Test Sign test Wilcoxon signed-ranks test Wilcoxon rank-sum test Parametric Test Two independent samples ttest or z test Several independent samples Analysis of variance (Ftest) Kruskal-Wallis test Correlation Randomness 0.63 0.95 0.95 0.95 0.91 No basis for comparison Linear correlation Rank correlation test No parametric test Runs test