The Wilcoxon Signed Rank Test: Normal Approximation (Large Samples) Under the null hypothesis that the population differences are centered on 0. the Wilcoxon signed rank test has mean and variance given by Chapter 1 Anal Catalan and + 1)( 21) Varl) (14701 Then, for large, the distribution of the random variable, Z. is approximately standard normal where T- (14201 If the number, M. of nonzero differences is large and is the observed value of the Wilcoxon statistic, then the following tests have significance level a 1. If the alternative hypothesis is one-sided, reject the null hypothesi 2. If the alternative hypothesis is two-sided, reject the null hypothesis if T-R T u30 Example 14.11 Postaudit Procedures (Wilcoxon Signed Rank Test) Suppose that a study compared firms with and without sophisticated postaudit pro cedures. A random sample of 31 matched pair of firms was examined. For each firm the ratio of market valuation to replacement cost of assets was computed as a measure of firm performance In each of the 31 paits, one firm employed sophisticated pasta dit procedures and the other did not. The 31 differences in ratios were calculated, and the absolute differences were ranked. The smaller of the rank sums, 189, was for those paits where the ratio was higher for the firm without sophisticated postaudit proces dures Test the null hypothesis that the distribution of differences in ratios is centered on against the alternative that the ratio of market valuation to replacement cost of as sets tends to be lower for firms without sophisticated postaudit procedures