1. Section 13.5-1 Copyright © 2014, 2012, 2010 Pearson Education, Inc. Lecture Slides Elementary Statistics Twelfth Edition and the Triola Statistics Series by Mario F. Triola

2. Section 13.5-2 Copyright © 2014, 2012, 2010 Pearson Education, Inc. 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. Section 13.5-3 Copyright © 2014, 2012, 2010 Pearson Education, Inc. 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. This test is the complement to ANOVA, but it does not require normal distributions.

4. Section 13.5-4 Copyright © 2014, 2012, 2010 Pearson Education, Inc. We compute the test statistic H, 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 k – 1, where k is the number of samples. The H test statistic is basically a measure of the variance of the rank sums R 1, R 2,..., R k. Kruskal-Wallis Test

5. Section 13.5-5 Copyright © 2014, 2012, 2010 Pearson Education, Inc. N= total number of observations in all observations combined k= number of samples R 1 = sum of ranks for Sample 1 n 1 = number of observations in Sample 1 For Sample 2, the sum of ranks is R 2 and the number of observations is n 2, and similar notation is used for the other samples. Kruskal-Wallis Test Notation

6. Section 13.5-6 Copyright © 2014, 2012, 2010 Pearson Education, Inc. 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

7. Section 13.5-7 Copyright © 2014, 2012, 2010 Pearson Education, Inc. Kruskal-Wallis Test Test Statistic Critical Values 1.Test is right-tailed. 2.df = k – 1 (Because the test statistic H can be approximated by the chi-square distribution, use Table A- 4). P-values are often found using technology

8. Section 13.5-8 Copyright © 2014, 2012, 2010 Pearson Education, Inc. 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.

9. Section 13.5-9 Copyright © 2014, 2012, 2010 Pearson Education, Inc. Example Table 13-6 lists IQ scores from a sample of subjects with low, medium, and high lead exposure. Use a 0.05 level of significance to test the claim that the three sample medians come from populations with medians that are all equal.

10. Section 13.5-10 Copyright © 2014, 2012, 2010 Pearson Education, Inc. Example Table 13-6 lists IQ scores from a sample of subjects with low, medium, and high lead exposure. Use a 0.05 level of significance to test the claim that the three sample medians come from populations with medians that are all equal.

11. Section 13.5-11 Copyright © 2014, 2012, 2010 Pearson Education, Inc. Example Requirement Check: Each of the three samples is a simple random sample and each sample size is at least 5. The hypotheses are:

12. Section 13.5-12 Copyright © 2014, 2012, 2010 Pearson Education, Inc. Example - Continued We first rank the data, as noted in Table 13-6. The test statistic is:

13. Section 13.5-13 Copyright © 2014, 2012, 2010 Pearson Education, Inc. Example - Continued Because each sample has at least five observations, the distribution of H is approximately chi-square with k – 1 degrees of freedom (3 – 1 = 2 df). Refer to Table A-4 to find the critical value of 5.991. As shown on the next slide, the test statistic of H = 0.694 does not fall in the rejection region, so we fail to reject the null hypothesis of equal population medians.

14. Section 13.5-14 Copyright © 2014, 2012, 2010 Pearson Education, Inc. Example - Continued There is not sufficient evidence to reject the claim that IQ scores from subjects with low, medium, and high levels of lead exposure all have the same median.