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STA 4502 Chapter 4

# STA 4502 Chapter 4 - Chapter 4 The Two Sample Location...

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STA 4502 page 1 Chapter 4 The Two Sample Location Problem With Independent Samples 4.1 Wilcoxon Rank Sum Test This test was first published by Wilcoxon in 1945. That is why some authors refer to this test as the Wilcoxon Rank Sum Test (because it is based on the sum of ranks). Later, in 1947, Mann and Whitney published the same test with more detail and some authors refer to it as the Wilcoxon, Mann-Whitney Rank Sum Test. Data: We have a random sample of m observations from population 1, denoted by X 1 , • • •, X m and another independent random sample of n observations from population 2, denoted by Y 1 , • • •, Y n . The total number of observations is N = m + n. Without loss of generality we take n < m . Assumptions: o A1 : The observations X 1 , • • •, X m are a random sample from population 1, thus they are independent of one another and have the same distribution. The other sample, Y 1 , • • •, Y n are a random sample from population 2, so they also are independent of one another and have the same distribution. o A2 : The two samples are mutually independent of one another. o A3: Population 1 and population 2 are both continuous populations. o A4 : Y j has the same distribution as X i + for all i = 1, • • •, m and j = 1, • ••, n. o The parameter is the treatment effect . An Interpretation of the Parameter in the Two Sample Location Problem • The parameter ∆ denotes the amount of shift, i.e. the separation between the two populations. [Assumption A4 means the two populations have the same distribution, they differ only in location.] The tables that are contained in the text assume that o The sample size from population 2, n < m, the sample size from population 1. o That is, "Population 2" is the one from which the smaller sample is selected. Procedure: 1. Order the combined sample of N = m + n X- and Y-values from smallest to largest. 2. Let S 1 denote the rank of Y 1 , S 2 denote the rank of Y 2 , . .., and S n denote the rank of Y n in this joint ordering.

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STA 4502 page 2 3. The test statistic is W is the sum of the ranks assigned to the Y -values. That is, 1 n j j W S (4.3) a. One-Sided Upper-Tail Test: To test Ho: = 0 vs. Ha: > 0 at α level of significance, the decision rule is: Reject Ho if W ≥ w α , otherwise do not reject. (4.4) Here the constant w α is chosen to make the type I error probability equal to α. Values of w α are given in Table A.6. The table is entered with the sum of the ranks corresponding to the smaller sample, so when naming the samples, call the smaller sample the Y-sample of size n ( n < m). If n = m, either sample can be designated the Y-sample for use of Table A.6. b. One-Sided Lower-Tail Test: To test Ho: = 0 vs. Ha: < 0 at the α level, the decision rule is Reject Ho if W n(m + n + 1) w α , otherwise do not reject. (4.5) c. Two-Sided Test: To test Ho: = 0 vs. Ha: ≠ 0 , at α level of significance, the decision rule is Reject Ho if W cal ≥ w α/2 or W cal ≤ n(m + n +1) – w α/2 , otherwise do not reject. (4.6) Here W cal is the calculated value of the test statistic based on the samples.
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