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Lesson 15 - 5

Lesson 15 - 5. Inferences about the Differences between Two Medians: Independent Samples. Objectives. Test a claim about the difference between the medians of two independent samples. Vocabulary.

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Lesson 15 - 5

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  1. Lesson 15 - 5 Inferences about the Differences between Two Medians: Independent Samples

  2. Objectives • Test a claim about the difference between the medians of two independent samples

  3. Vocabulary • Mann–Whitney Test – nonparametric procedure used to test the equality of two population medians from independent samples • Population “X” – is the population of interest from the problem statement

  4. Mann-Whitney Test • Similar in structure to Wilcoxon Matched-Pairs Signed-Ranks Test • Because samples are independent different test statistics and critical values are used

  5. Hypothesis Tests Using Mann–Whitney Test Step 0 Requirements: 1. the samples are independent random samples and 2. the shape of the distributions are the same (assume to met in our problems) Step 1 Box Plots:Draw a side-by-side boxplot to compare the sample data from the two populations. This helps to visualize the difference in the medians. Step 2 Hypotheses: Step 3 Ranks:Rank all sample observations from smallest to largest. Handle ties by finding the mean of the ranks for tied values. Find the sum of the ranks for the sample from population X. Step 4 Level of Significance:(level of significance determines the critical value) Determine a level of significance, based on the seriousness of making a Type I error Small-sample case: Use Table XII. Large-sample case: Use Table IV. Step 5 Compute Test Statistic: Step 6 Critical Value Comparison: Reject H0 if test statistic more extreme (further away from 0) than the critical value Step 7 Conclusion:Reject or Fail to Reject

  6. Test Statistic for the Mann–Whitney Test n1 n2 T – ---------- 2 z0= ----------------------------- n1n2 (n1 + n2 + 1) ------------------------ 12 The test statistic will depend on the size of the samples from each population. Let n1 represent the sample size for population X and represent the n2 sample size for population Y. Small-Sample Case:(n1≤ 20 and n2 ≤ 20 ) If S is the sum of the ranks corresponding to the sample from population X, then the test statistic, T, is given by n1 (n1 - 1) T = S – -------------- 2 Note: The value of S is always obtained by summing the ranks of the sample data that correspond to Mx , the median of population X, in the hypothesis. Large-Sample Case:(n1> 20 or n2 > 20 ) From the Central Limit Theorem, the test statistic is given by where T is the test statistic from the small-sample case.

  7. Critical Value for Mann–Whitney Test Small-Sample Case: (n1≤ 20 and n2 ≤ 20 ) Using αas the level of significance, the critical value is obtained from Table XII in Appendix A. Large-Sample Case: (n1 > 20 or n2 > 20 ) Using αas the level of significance, the critical value is obtained from Table IV in Appendix A.

  8. Example 1 from 15.5

  9. Example Cont. • Hypothesis: H0: Med ill = Med healthy Ha: Med ill > Med healthy • Test Statistic: T = S – ½ n1(n1+1) = 152 – ½ 11(12) = 86where S is the sum of the ranks obtained from the sample observations from ill population and n1 is the number of observations from sample of ill people • Critical Value: w1-α= n1n2 –wα= 121 – 41 = 80 • Conclusion: Since T > w1-α, we reject the null hypothesis and conclude that there is a difference between the median values for ill and healthy people

  10. Summary and Homework • Summary • The Mann-Whitney test is a nonparametric test for comparing the median of two independent samples • This test is a sum of ranks of the combined dataset • The critical values for small samples are given in tables • The critical values for large samples can be approximated by a calculation with the normal distribution • Homework • problems 4, 5, 9, 10, 12 from the CD

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