# Many times these assumptions cannot be met and in

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distributed with equal variances. Many times these assumptions cannot be met and in such cases, we can use two non-parametric tests, neither of which depends on the normality assumptions. Both of these tests are called rank sum tests because the test depends on the ranks of the sample observations. Rank sum tests are a whole family of tests. We shall concentrate on just two members of this family the mann-whiney U test and the kruskal-wallis test. We will use Mann-whitney test when only two populations are involved and the kruskal- wallin test when more than two populations are involved. Use of these tests will enable us to determine whether independent samples have been drawn from the same population. The use of ranking information rather than pluses and minuses is less wasteful of data than the sign test. 15.2. OBJECTIVES To use a Mann-whitney U test to see if two independent samples come from the same population. To use a Kruskal-wallis test to see if three or more independent samples come from the same population. 15.3. CONTENTS 15.3.1. The Mann-Whitney U-Test 15.3.2. Assumptions for mann-whitness U test 15.3.3. The Kruskal Wallis Test 15.3.4. Ranking for the Kruskal Wallis Test 15.3.1. THE MANN-WHITNEY U-TEST The Mann-Whitney U test is used to test whether two independent samples have been drawn from the same population. This is the most powerful non- parametric test when the measurements of the variables are based on a scale. Let 1 n be the number of elements in the smaller of the two independent samples and let 2 n be the number of elements in the other sample. These elements of the two samples are combined together and arranged in the ascending order of magnitude, the least value occupying the first position and the highest value the last position in the ordering. The value of U (the test statistic used here) is given as follows: Focusing on the sample with a smaller size (i.e., 1 n , in number), U is given by the number of times that a score in the group with 2 n elements precedes a score in the group with 1 n elements in the ranking.

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123 The sampling distribution of U is known under 0 H . For small samples. Statistical table used to determine the exact probability associated with the occurrence of any U under 0 H . If this probability is smaller than , 0 H is rejected. For adequately large 1 n and 2 n the preceding method for computation of U discarded as it is tedious. Alternatively, U is calculated in the following way: 1 1 1 2 1 ( 1) 2 n n U n n S or equivalently 1 1 1 2 2 ( 1) 2 n n U n n S Where 1 S is the sum of ranks of elements of the firs sample of size 1 n and 2 S is the sum of ranks of elements of the second sample of size 2 n . For large samples, under the null hypothesis that the two samples have been drawn from the same population, U be a sampling distribution (which approaches the normal) with the mean equal to 1 2 ( )/2 n n and 1 2 1 2 ( 1) standard deviation 12 n n n n Thus, the test statistic 1 2 1 2 1 2 2 ( 1) 12 n n U Z n n n n Follows a normal distribution with mean zero and unit standard deviation.
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• Spring '12
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