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We will now suppose, as if often reasonable, that the population distribution of the paired differences is symmetric. The null hypothesis to be tested is that the center of this distribution is 0. In our example, then, we are assuming that differences in the ratings of the two products have a symmetric distribution, and we want to test whether that distribution is centered on 0—that is, no difference between ratings. We would be suspicious of the null hypothesis if the sum of the ranks for positive differences was very different from that for negative differences. Hence, the null hypothesis will be rejected for low values of the statistic T. Cutoff points for the distribution of this random variable are given in Table 8 of the Appendix for tests against a one-sided alternative that the population distribution of the paired differences is specified either to be centered on some number bigger than 0 or to be centered on some number less than 0. In our example, we might want to take the alternative hypothesis that the new product tends to be preferred over the original. This would imply that the distribution of the paired differences in ratings is centered on some number less than 0. From Table 8, we see that for a sample size of n — 7, the cutoff points are 3 for a 2.5%-level test and 4 for a 5%-level test. Hence, the null hypothesis is rejected against the one-sided alternative at the 5% level and is just rejected at the 2.5% level. The evidence against the hypothesis that the population differences in ratings for these products is centered on 0 is quite strong. It appears likely that overall, ratings are higher for the new product.

If the alternative hypothesis is two-sided—that is, that the population differences are centered on some number other than 0—the appropriate significance levels are twice those for the one-sided alternative. Hence, for these data, the null hypothesis is rejected against a two-sided alternative at the 10% level and is just rejected at the 5% level of significance.

Notice that using the additional information provided by the ranks allows the rejection of the null hypothesis at a much lower level of significance than was possible for the sign test.

The Wilcoxon Test

The Wilcoxon test can be employed when a random sample of matched pairs of observations is available. We assume that the population distribution of the differences in matched pairs is symmetric, and we want to test the null hypothesis that this distribution is centered on 0. Discarding pairs for which the difference is 0, we rank the remaining n absolute differences in ascending order. The sums of the ranks corresponding to positive and negative differences are calculated, and the smaller of these sums is the Wilcoxon test statistic T. The null hypothesis is rejected if T is less than or equal to the value in Table 8 of the Appendix.

When the number n of nonzero differences in the sample is large,4 the normal distribution provides a good approximation to the distribution of the Wilcoxon statistic T under the null hypothesis. It can be shown that when the null hypothesis that the population differences are centered on 0 is true, the mean and variance of this distribution are

4 The approximation is adequate for twenty or more observations. SOME NONPARAMETRIC TESTS

Then, for large n, the distribution of the random variable

CTt is approximately standard normal, and tests can be based on this result, as indicated in the box.

The Wilcoxon Test: Large Samples

If the number n of nonzero differences is large and T is the observed value of the Wilcoxon statistic, the following tests have significance level a:

(i) If the alternative hypothesis is one-sided, reject the null hypothesis if

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