1. Lesson 15 - 4 Inferences about the Differences between Two Medians: Dependent Samples

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

3. Vocabulary Wilcoxon Matched-Pairs Signed-Ranks Test -- a nonparametric procedure that is used to test the equality of two population medians by dependent sampling. Ranks – 1 through n with ties award the sum of the tied ranks divided by the number tied. For example, if 4 th and 5 th observation was tied then they would both receive a 4.5 ranking. Signed-Ranks – ranks are constructed with absolute values and then the sign of the value (positive or negative) is applied to the ranking.

4. Parametric vs Nonparametric ●For our parametric test for matched-pairs (dependent samples), we  Compared the corresponding observations by subtracting one from the other  Performed a test of whether the mean is 0 ●For our nonparametric case for matched- pairs (dependent samples), we will  Compare the corresponding observations by subtracting one from the other  Perform a test of whether the median is 0

5. Hypothesis Tests Using Wilcoxon Test Step 0: Compute the differences in the matched-pairs observations. Rank the absolute value of all sample differences from smallest to largest after discarding those differences that equal 0. Handle ties by finding the mean of the ranks for tied values. Assign negative values to the ranks where the differences are negative and positive values to the ranks where the differences are positive. Step 1 Hypotheses: Step 2 Box Plot: Draw a boxplot of the differences to compare the sample data from the two populations. This helps to visualize the difference in the medians. Step 3 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 XI. Large-sample case: Use Table IV. Step 4 Compute Test Statistic: Step 5 Critical Value Comparison: Reject H 0 if test statistic < critical value Step 6 Conclusion: Reject or Fail to Reject Left-TailedTwo-TailedRight-Tailed H 0 : M D = 0 H 1 : M D < 0 H 0 : M D = 0 H 1 : M D ≠ 0 H 0 : M D = 0 H 1 : M D > 0

6. Test Statistic for the Wilcoxon Matched-Pairs Signed-Ranks Test Small-Sample Case: (n ≤ 30) Large-Sample Case: (n > 30) n (n + 1) T – -------------- 4 z 0 = ----------------------------- n (n + 1)(2n + 1) ------------------------ 24 Left-TailedTwo-TailedRight-Tailed H 0 : M D = 0 H 1 : M D < 0 H 0 : M D = 0 H 1 : M D ≠ 0 H 0 : M D = 0 H 1 : M D > 0 T = T + T = smaller of T + or |T - |T = |T - | where T is the test statistic from the small-sample case. Note: M D is the median of the differences of matched pairs. T + is the sum of the ranks of the positive differences T - is the sum of the ranks of the negative differences

7. Small-Sample Case (n ≤ 30): Using α as the level of significance, the critical value is obtained from Table XI in Appendix A. Large-Sample Case (n > 30): Using α as the level of significance, the critical value is obtained from Table IV in Appendix A. The critical value is always in the left tail of the standard normal distribution. Left-TailedTwo-TailedRight-Tailed -z α -z α/2 -z α Left-TailedTwo-TailedRight-Tailed -T α -T α/2 -T α Critical Value for Wilcoxon Matched-Pairs Signed-Ranks Test

8. Example 1 from 15.4 XYD = Y-X|D|Signed Rank 11.526.014.5 +7.5 14.126.212.1 +5 19.324.65.3 +3 35.030.8-4.24.2-2 15.937.521.6 +11 21.536.014.5 +7.5 11.725.914.2 +6 17.116.9-0.20.2 27.350.222.9 +12 13.833.119.3 +10 43.299.956.7 +14 11.226.114.9 +9 34.243.89.6 +4 26.763.837.1 +13 T + = 88T - = |-3| = 3

9. Example continued H 0 : Med dif = 0 H a : Med dif > 0 (Right tailed test) Test Statistic: |T - | = 3 From Table XI: T critcal = T 0.05 = 25 The test statistic is less than the critical value (3 < 25) so we reject the null hypothesis.

10. Summary and Homework Summary –The Wilcoxon sign test is a nonparametric test for comparing the median of two dependent samples –This test is a weighted count of the differences in signs between the paired observations –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 2, 4, 5, 8, 9, 15 from the CD