Lecture Slides

1. Lecture Slides Elementary StatisticsTwelfth Edition and the Triola Statistics Series by Mario F. Triola

2. 13-1 Review and Preview 13-2 Sign Test 13-3 Wilcoxon Signed-Ranks Test for Matched Pairs 13-4 Wilcoxon Rank-Sum Test for Two Independent Samples 13-5 Kruskal-Wallis Test 13-6 Rank Correction 13-7 Runs Test for Randomness Chapter 13Nonparametric Statistics

3. Key Concept The Wilcoxon rank-sum test uses ranks of values from two independent samples to test the null hypothesis that the two populations have equal medians. The basic idea underlying the Wilcoxon rank-sum test is this: If two samples are drawn from identical populations and the individual values are all ranked as one combined collection of values, then the high and low ranks should fall evenly between the two samples. If the low ranks are found predominantly in one sample and the high ranks are found predominantly in the other sample, we suspect that the two populations have different medians.

4. Caution Don’t confuse the Wilcoxon rank-sum test for two independent samples with the Wilcoxon signed-ranks test for matched pairs. Use Internal Revenue Service as the mnemonic for IRS to remind us of “Independent: Rank Sum.”

5. The Wilcoxon rank-sum test is a nonparametric test that uses ranks of sample data from two independent populations. It is used to test the null hypothesis that the two independent samples come from populations with equal medians. Definition

6. n1 = size of Sample 1 n2= size of Sample 2 R1 = sum of ranks for Sample 1 R2 = sum of ranks for Sample 2 R = same as R1 (sum of ranks for Sample 1) μR= mean of the sample R values that is expected when the two populations have equal medians σR= standard deviation of the sample R values that is expected when the two populations have equal medians Notation

7. Requirements 1. There are two independent simple random samples. 2. Each of the two samples has more than 10 values. Note: There is no requirement that the two populations have a normal distribution or any other particular distribution.

8. Test Statistic where n1 = size of the sample from which the rank sum R is found n2 = size of the other sample R = sum of ranks of the sample with size n1

9. P-Values can be found using the z test statistic and Table A-2. Critical values can be found in Table A-2 (because the test statistic is based on the normal distribution). P-Values / Critical Values

10. Procedure for Finding the Value of the Test Statistic 1. Temporarily combine the two samples into one big sample, then replace each sample value with its rank. 2. Find the sum of the ranks for either one of the two samples. 3. Calculate the value of the z test statistic, where either sample can be used as “Sample 1.”

11. Example Table 13-5 lists pulse rates of samples of males and females (from Data Set 1 in Appendix B). Use a 0.05 significance level to test the claim that males and females have the same median pulse rate.

12. Example - Continued Requirement Check: The sample data are two independent random samples, and the sample sizes are 12 and 11, which both exceed 10. The hypotheses are:

13. Example - Continued Rank the combined 23 pulse rates – refer to the table in the previous slide. If we choose the pulse rates of males as Sample 1, we get: Also, n1 = 12, n2 = 11, and we can find the values ofμR, σR, and the test statistic z.

14. Example - Continued

15. Example - Continued Since we have a two-tailed test with α = 0.05, the critical values are ±1.96. The test statistic of z = –1.26 does not fall in the critical region, so we fail to reject the null hypothesis. There is not sufficient evidence to warrant the rejection of the claim that males and females have the same median pulse rate. Based on the available sample data, it appears males and females have pulse rates with the same median.