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www.downloadslide.com 15.6 The Mann–Whitney U Test: Independent Random Samples 759 specifically if U ≥ n 1 n 2 − U 0 , where U 0 is such that α = P(U ≥ n 1 n 2 − U 0 ) = P(U ≤ U 0 ) is of acceptable size. Table 8, Appendix 3, gives the probability that an observed value of U is less than various values, U 0 . This is the value of α for a one-tailed test. To conduct a two-tailed test—that is, to detect difference in the locations of populations I and II—reject H 0 if U ≤ U 0 or U ≥ n 1 n 2 − U 0 , where P(U ≤ U 0 ) = α/2. To see how to locate the rejection region for the Mann–Whitney U test, suppose that n 1 = 4 and n 2 = 5. Then, you would consult the third section of Table 8, Appendix 3 (the one corresponding to n 2 = 5). Notice that the table is constructed assuming that n 1 ≤ n 2 . That is, you must always identify the smaller sample as sample I. From the table we see, for example, P(U ≤ 2) = .0317 and P(U ≤ 3) = .0556. So if you want to conduct a lower-tail Mann–Whitney U test **with** n 1 = 4 and n 2 = 5 for α near .05, you should reject the null hypothesis of equality of population relative frequency distributions when U ≤ 3. The probability of a type I error for the test is α = .0556. When applying the test to a set of data, you may find that some of the observations are of equal value. Ties in the observations can be handled by averaging the ranks that would have been assigned to the tied observations and assigning this average to each. Thus, if three observations are tied and are due to receive ranks 3, 4, and 5, we assign rank 4 to all three. The next observation in the sequence receives rank 6, and ranks 3 and 5 do not appear. Similarly, if two observations are tied for ranks 3 and 4, each receives rank 3.5, and ranks 3 and 4 do not appear. Table 8, Appendix 3, can also be used to find the observed significance level for a test. For example, if n 1 = 5, n 2 = 5, and U = 4, the p-value for a one-tailed test that the distribution of population I is shifted the right of the distribution of population II is P{U ≤ 4} =.0476. If the test is two-tailed, the p-value is 2(.0476), or .0952. The Mann–Whitney U Test Population I is the population from which the smaller sample was taken. Null hypothesis: H 0 : The distributions of populations I and II are identical. Alternative hypothesis: (1) H a : The distributions of populations I and II have different locations (a two-tailed test), or (2) the distribution of population I is shifted to the right of the distribution of population II, or (3) the distribution of population I is shifted to the left of the distribution of population II. Test statistic: U = n 1 n 2 + [n 1 (n 1 + 1)]/2 − W. Rejection region: (1) For the two-tailed test and a given value of α, reject H 0 if U ≤ U 0 or U ≥ n 1 n 2 −U 0 , where P(U ≤ U 0 ) = α/2. [Note: Observe that U 0 is the value such that P(U ≤ U 0 ) is equal to half of α.] (2) To test that population I is shifted to the right of population II **with** a given value

760 Chapter 15 Nonparametric **Statistics** www.downloadslide.com of α, reject H 0 if U ≤ U 0 , where P(U ≤ U 0 ) = α. (3) To test that population I is shifted to the left of population II **with** a given value of α, reject H 0 if U ≥ n 1 n 2 − U 0 , where P(U ≤ U 0 ) = α. Assumptions: Samples have been randomly and independently selected from their respective populations. Ties in the observations can be handled by averaging the ranks that would have been assigned to the tied observations and assigning this average rank to each. Thus, if three observations are tied and are due to receive ranks 3, 4, and 5, we assign rank 4 to all three. EXAMPLE 15.5 Solution Test the hypothesis that there is no difference in the locations of the population distributions for the bacteria count data of Example 15.4. We have already noted that the Mann–Whitney U test and the Wilcoxon rank-sum test are equivalent, so we should reach the same conclusions here as we did in Example 15.4. Recall that the alternative hypothesis was that the distributions of bacteria counts for cultures I and II differed and that this implied a two-tailed test. Thus, because Table 8, Appendix 3, gives values of P(U ≤ U 0 ) for specified sample sizes and values of U 0 , we must double the tabulated value to find α. Suppose, as in Example 15.4, that we desire a value of α near .05. Checking Table 8 for n 1 = n 2 = 4, we find P(U ≤ 1) = .0286. The appropriate rejection region for the two-tailed test is U ≤ 1orU ≥ n 1 n 2 − 1 = 16 − 1 = 15, for which α = 2 (.0286) = .0572 or, rounding to three decimal places, α = .057 (the same value of α obtained for Example 15.4). For the bacteria data, the rank sum is W = 12. Then, U = n 1 n 2 + n 1(n 1 + 1) 2 − W = (4)(4) + 4(4 + 1) 2 − 12 = 14. The calculated value of U does not fall in the rejection region. Hence, there is not sufficient evidence to show a difference in the locations of the population distributions of bacteria counts for cultures I and II. The p-value is given by 2P(U ≥ 14) = 2P(U ≤ 2) = 2 (.0571) = .1142. EXAMPLE 15.6 Solution An experiment was conducted to compare the strengths of two types of kraft papers, one a standard kraft paper of a specified weight and the other the same standard kraft paper treated **with** a chemical substance. Ten pieces of each type of paper, randomly selected from production, produced the strength measurements shown in Table 15.5. Test the hypothesis of no difference in the distributions of strengths for the two types of paper against the alternative hypothesis that the treated paper tends to be stronger. Both samples are of size 10, so either population (standard or treated) may be designated as population I. We have identified the standard paper measurements as coming

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