Many times these assumptions cannot be met and in

Info icon This preview shows pages 128–130. Sign up to view the full content.

View Full Document Right Arrow Icon
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.
Image of page 128

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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.
Image of page 129
Image of page 130
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern