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 13-5 Kruskal-Wallis Test

4. Copyright © 2010, 2007, 2004 Pearson Education, Inc. 13.1 - 4 Key Concept This section introduces the Kruskal-Wallis test, which uses ranks of data from three or more independent samples to test the null hypothesis that the samples come from populations with equal medians.

5. Copyright © 2010, 2007, 2004 Pearson Education, Inc. 13.1 - 5 We compute the test statistic, which has a distribution that can be approximated by the chi-square distribution as long as each sample has at least 5 observations. When we use the chi-square distribution in this context, the number of degrees of freedom is, where is the number of samples. Kruskal-Wallis Test

6. Copyright © 2010, 2007, 2004 Pearson Education, Inc. 13.1 - 6 The Kruskal-Wallis test (also called the H test) is a nonparametric test that uses ranks of simple random samples from three or more independent populations. It is used to test the null hypothesis that the independent samples come from populations with the equal medians. Definition :The samples come from populations with equal medians. :The samples come from populations with medians that are not all equal.

7. Copyright © 2010, 2007, 2004 Pearson Education, Inc. 13.1 - 7 = total number of observations in all observations combined = number of samples = sum of ranks for Sample 1 = number of observations in Sample 1 For Sample 2, the sum of ranks is and the number of observations is, and similar notation is used for the other samples. Kruskal-Wallis Test Notation

8. Copyright © 2010, 2007, 2004 Pearson Education, Inc. 13.1 - 8 1.We have at least three independent random samples. 2.Each sample has at least 5 observations. Note: There is no requirement that the populations have a normal distribution or any other particular distribution. Kruskal-Wallis Test Requirements

9. Copyright © 2010, 2007, 2004 Pearson Education, Inc. 13.1 - 9 Kruskal-Wallis Test Test Statistic Critical Values 1.Test is right-tailed. 2. (Because the test statistic H can be approximated by the distribution, use Table A- 4).

10. Copyright © 2010, 2007, 2004 Pearson Education, Inc. 13.1 - 10 Procedure for Finding the Value of the Test Statistic H 1 Temporarily combine all samples into one big sample and assign a rank to each sample value. 2.For each sample, find the sum of the ranks and find the sample size. 3.Calculate H by using the results of Step 2 and the notation and test statistic.

11. Copyright © 2010, 2007, 2004 Pearson Education, Inc. 13.1 - 11 Procedure for Finding the Value of the Test Statistic H The test statistic H is basically a measure of the variance of the rank sums If the ranks are distributed evenly among the sample groups, then H should be a relatively small number. If the samples are very different, then the ranks will be excessively low in some groups and high in others, with the net effect that H will be large.

12. Copyright © 2010, 2007, 2004 Pearson Education, Inc. 13.1 - 12 Example: The chest deceleration measurements resulting from car crash tests are listed in Table 13-6. The data are from the Chapter Problem in Chapter 12. Test the claim that the three samples come from populations with medians that are all equal. Use a 0.05 significance level.

13. Copyright © 2010, 2007, 2004 Pearson Education, Inc. 13.1 - 13 Example:

14. Copyright © 2010, 2007, 2004 Pearson Education, Inc. 13.1 - 14 Are requirements met? Each of the three samples is a simple random independent and sample. Each sample size at least 5. Example: : The populations of chest deceleration measurements from the three categories have the same median. : The populations of chest deceleration measurements from the three categories have medians that are not all the same.

15. Copyright © 2010, 2007, 2004 Pearson Education, Inc. 13.1 - 15 Example: The following statistics come from Table 13-6:

16. Copyright © 2010, 2007, 2004 Pearson Education, Inc. 13.1 - 16 Evaluate the test statistic. Example:

17. Copyright © 2010, 2007, 2004 Pearson Education, Inc. 13.1 - 17 Find the critical value.. Because each sample has at least five observations, the distribution of H is approximately a chi-square distribution. Example: (right-tail) From Table A-4 the critical value = 5.991.

18. Copyright © 2010, 2007, 2004 Pearson Education, Inc. 13.1 - 18 Example:

19. Copyright © 2010, 2007, 2004 Pearson Education, Inc. 13.1 - 19 Example: The test statistic 5.774 is NOT in the critical region bounded by 5.991, so we fail to reject the null hypothesis of equal medians. There is not sufficient evidence to reject the claim that chest deceleration measurements from small cars, medium cars, and large cars all have equal medians. The medians do not appear to be different.

20. Copyright © 2010, 2007, 2004 Pearson Education, Inc. 13.1 - 20 Recap In this section we have discussed: The Kruskal-Wallis Test is the non- parametric equivalent of ANOVA. It tests the hypothesis that three or more populations have equal means. The populations do not have to be normally distributed.