1. Copyright © 2010, 2007, 2004 Pearson Education, Inc. 13.1 - 1 Lecture Slides Elementary Statistics Eleventh Edition and the Triola Statistics Series by Mario F. Triola

2. Copyright © 2010, 2007, 2004 Pearson Education, Inc. 13.1 - 2 Chapter 13 Nonparametric Statistics 13-1Review and Preview 13-2Sign Test 13-3Wilcoxon Signed-Ranks Test for Matched Pairs 13-4Wilcoxon Rank-Sum Test for Two Independent Samples 13-5Kruskal-Wallis Test 13-6Rank Correction 13-7Runs Test for Randomness

3. Copyright © 2010, 2007, 2004 Pearson Education, Inc. 13.1 - 3 Section 13-3 Wilcoxon Signed-Ranks Test for Matched Pairs

4. Copyright © 2010, 2007, 2004 Pearson Education, Inc. 13.1 - 4 Key Concept The Wilcoxon signed-ranks test involves the conversion of the sample data ranks. This test can be used for two different applications.

5. Copyright © 2010, 2007, 2004 Pearson Education, Inc. 13.1 - 5 The Wilcoxon signed-ranks test is a nonparametric test that uses ranks for these applications: Definition 1.Test a null hypothesis that the population of matched pairs has differences with a median equal to zero. 2.Test a null hypothesis that a single population has a claimed value of the median.

6. Copyright © 2010, 2007, 2004 Pearson Education, Inc. 13.1 - 6 Use the Wilcoxon signed-ranks test with matched pairs for the following null and alternative hypotheses: Objective : The matched pairs have differences that come from a population with a median equal to zero. :The matched pairs have differences that come from a population with a nonzero median.

7. Copyright © 2010, 2007, 2004 Pearson Education, Inc. 13.1 - 7 1. The data consist of matched pairs that have been randomly selected. 2. The population of differences (found from the pairs of data) has a distribution that is approximately symmetric, meaning that the left half of its histogram is roughly a mirror image of its right half. (There is no requirement that the data have a normal distribution.) Wilcoxon Signed-Ranks Test Requirements

8. Copyright © 2010, 2007, 2004 Pearson Education, Inc. 13.1 - 8 Notation T = the smaller of the following two sums: 1. The sum of the positive ranks of the nonzero differences d 2. The absolute value of the sum of the negative ranks of the nonzero differences d

9. Copyright © 2010, 2007, 2004 Pearson Education, Inc. 13.1 - 9 Test Statistic for the Wilcoxon Signed-Ranks Test for Matched Pairs For, the test statistic is T. For, the test statistic is

10. Copyright © 2010, 2007, 2004 Pearson Education, Inc. 13.1 - 10 Critical Values for the Wilcoxon Signed-Ranks Test for Matched Pairs For, the critical T value is found in Table A-8. For, the critical z values are found in Table A-2.

11. Copyright © 2010, 2007, 2004 Pearson Education, Inc. 13.1 - 11 Procedure for Finding the Value of the Test Statistic Step 1: For each pair of data, find the difference d by subtracting the second value from the first. Keep the signs, but discard any pairs for which d = 0. Step 2: Ignore the signs of the differences, then sort the differences from lowest to highest and replace the differences by the corresponding rank value. When differences have the same numerical value, assign to them the mean of the ranks involved in the tie.

12. Copyright © 2010, 2007, 2004 Pearson Education, Inc. 13.1 - 12 Procedure for Finding the Value of the Test Statistic Step 3: Attach to each rank the sign difference from which it came. That is, insert those signs that were ignored in step 2. Step 4:Find the sum of the absolute values of the negative ranks. Also find the sum of the positive ranks. Step 5:Let T be the smaller of the two sums found in Step 4. Either sum could be used, but for a simplified procedure we arbitrarily select the smaller of the two sums.

13. Copyright © 2010, 2007, 2004 Pearson Education, Inc. 13.1 - 13 Step 6:Let n be the number of pairs of data for which the difference d is not 0. Step 7:Determine the test statistic and critical values based on the sample size, as shown above. Step 8:When forming the conclusion, reject the null hypothesis if the sample data lead to a test statistic that is in the critical region that is, the test statistic is less than or equal to the critical value(s). Otherwise, fail to reject the null hypothesis. Procedure for Finding the Value of the Test Statistic

14. Copyright © 2010, 2007, 2004 Pearson Education, Inc. 13.1 - 14 Example: The first two rows of Table 13-4 include some of the weights from Data Set 3 in Appendix B. Those weights were measured from college students in September and April of their freshman year. Use the sample data in the first two rows of Table 13-4 to test the claim that there is no difference between the September weights and the April weights. Use the Wilcoxon signed-ranks test with a 0.05 significance level.

15. Copyright © 2010, 2007, 2004 Pearson Education, Inc. 13.1 - 15 Example: Requirements are satisfied: subjects volunteered, but proceed as if simple random sample; STATDISK-generated histogram of the differences (3rd row) does not appear symmetric (next slide), but with only 9 differences it is not extreme so consider this requirement to be satisfied

16. Copyright © 2010, 2007, 2004 Pearson Education, Inc. 13.1 - 16 Example:

17. Copyright © 2010, 2007, 2004 Pearson Education, Inc. 13.1 - 17 Example: : There is no difference between the times of the first and second trials. : There is a difference between the times of the first and second trials.

18. Copyright © 2010, 2007, 2004 Pearson Education, Inc. 13.1 - 18  The ranks of differences in row four of the table are found by ranking the absolute differences, handling ties by assigning the mean of the ranks.  The signed ranks in row five of the table are found by attaching the sign of the differences to the ranks. Example: Freshman Weight Gain  The differences in row three of the table are found by computing the September weight – April weight.

19. Copyright © 2010, 2007, 2004 Pearson Education, Inc. 13.1 - 19 Example: Freshman Weight Gain Step 1:In Table 13- 4, the row of differences is obtained by computing this difference for each pair of data: d = September weight – April weight Step 2:Ignoring their signs, we rank the absolute differences from lowest to highest. Step 3:The bottom row of Table 13- 4 is created by attaching to each rank the sign of the corresponding differences. Calculate the Test Statistic

20. Copyright © 2010, 2007, 2004 Pearson Education, Inc. 13.1 - 20 Example: Freshman Weight Gain If there really is no difference between the weights (as in the null hypothesis), we expect the sum of the positive ranks to be approximately equal to the sum of the absolute values of the negative ranks. Step 4:We now find the absolute of the sum values of the negative ranks, and we also find the sum of the positive ranks. Calculate the Test Statistic

21. Copyright © 2010, 2007, 2004 Pearson Education, Inc. 13.1 - 21 Example: Freshman Weight Gain Absolute value of sum of negative ranks: 40 Sum of positive ranks: 5 Step 5:Let T be the smaller of the two sums found in Step 4, we find that. Step 6:Let n be the number of pairs of data for which the difference d is not 0, we have. Calculate the Test Statistic Step 4 (cont.):

22. Copyright © 2010, 2007, 2004 Pearson Education, Inc. 13.1 - 22 Example: Freshman Weight Gain Step 7: test statistic. From Table A- 8, using and  in two tails, critical value is 6 Step 8: The test statistic is less than or equal to the critical value of 6, so we reject the null hypothesis. We conclude that the September and April weights do not appear to be about the same. Calculate the Test Statistic

23. Copyright © 2010, 2007, 2004 Pearson Education, Inc. 13.1 - 23 Recap In this section we have discussed: The Wilcoxon signed-ranks test which uses matched pairs. The hypothesis is that the matched pairs have differences that come from a population with a median equal to zero.