1. Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. CHAPTER 14: Nonparametric Methods to accompany Introduction to Business Statistics fifth edition, by Ronald M. Weiers Presentation by Priscilla Chaffe-Stengel Donald N. Stengel

2. Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. Chapter 14 - Learning Objectives Differentiate between nonparametric and parametric hypothesis tests. Determine when a nonparametric test should be used instead of its parametric counterpart. Appropriately apply each of the nonparametric methods introduced.

3. Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. Chapter 14 Key Terms Nonparametric tests Wilcoxon signed rank test: –One sample –Paired samples Wilcoxon rank sum test, two independent samples Kruskal-Wallis Test, three or more independent samples Friedman test, randomized block design Sign Test, paired samples Runs test for randomness Lilliefors test for normality

4. Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. Nonparametric Tests Advantages: –Fewer assumptions about the population »Shape »Variance –Valid for small samples –Defined over a range of variables, nominal and ordinal scales included –Calculations simple Disadvantages: –Sample data used less efficiently –Power of nonparametric analysis lower –Places greater reliance on statistical tables if computer statistical package or spreadsheet not being used

5. Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. Wilcoxon Signed Rank Test, One Sample Requirements: –Variable - Continuous data –Scale - Interval or ratio scale of measurement The Research Question (H 1 ): Test the value of a single population median, m { , >, <} m 0 Critical Value/Decision Rule: W, Wilcoxon signed rank test

6. Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. An Example Problem 14.8: According to the director of a county tourist bureau, there is a median of 10 hours of sunshine per day during the summer months. For a random sample of 20 days during the past three summers, the number of hours of sunshine has been recorded below. Use the 0.05 level in evaluating the director’s claim. 8 9 810 9 7 7 9 7 7 9 811 9 10 7 811 812

7. Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. An Example, continued hrs. d i |d i | 8–229–11 9–118–22 8–2211 + 11 10 009–11 9–1110 00 7–337–33 7–338–22 9–1111 + 11 7–338–22 7–3312+22 There are: 7 with rank 1 1, 2, 3, 4, 5, 6, 7 average rank = 4 6 with rank 2 8, 9, 10, 11, 12, 13 average rank = 10.5 5 with rank 3 14, 15, 16, 17, 18 average rank = 16

8. Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. An Example, continued hrs. d i |d i | Rank R+ R– 8–2210.5-10.59–114-4 9–114-48–2210.5-10.5 8–2210.5-10.511 + 1144- 10 00---9–114-4 9–114-410 00--- 7–3316-167–3316-16 7–3316-168–2210.5-10.5 9–114-411+1144- 7–3316-168–2210.5-10.5 7–3316-1612+2210.510.5- So,  R+ = 18.5,  R– = 152.5

9. Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. An Example, continued I. H 0 : m = 10 hours H 1 : m  10 hours II. Rejection Region:  = 0.05, n = 18 data values not equal to the hypothesized median of 10 If  R + 130, reject H 0. III. Test Statistics:  R + = 18.5  R – = 152.5

10. Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. An Example, concluded IV. Conclusion: Since the test statistic of  R + = 18.5 falls below the critical value of W = 41, we reject H 0 with at least 95% confidence. V. Implications: There is enough evidence to dispute the director’s claim that this county has a median of 10 days of sunshine per day during the summer months.

11. Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. Wilcoxon Signed Rank Test for Comparing Paired Samples Requirements: –Variable - Continuous data –Scale - Interval or ratio scale of measurement The Research Question (H 1 ): Test the difference in two population medians, paired samples, m d { , >, <} 0 Critical Value/Decision Rule: W, Wilcoxon signed rank table

12. Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. Wilcoxon Rank Sum Test for Comparing Two Independent Samples Requirements: –Variable - Continuous data –Scale – At least ordinal scale of measurement –Samples independent, populations approximately the same shape The Research Question (H 1 ): Test the difference in two population medians, m 1 { , >, <} m 2 Critical Value/Decision Rule: W, Wilcoxon rank sum table

13. Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. Kruskal-Wallis Test, Comparing Two Independent Samples Requirements: –Scale - Ordinal, interval or ratio scale –Independent samples from populations with identical distributions The Research Question (H 1 ): The medians differs from the others. Critical Value/Decision Rule: H, approximated by the chi-square distribution

14. Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. Friedman Test for the Randomized Block Design Requirements: –Scale - Ordinal, interval or ratio scale The Research Question (H 1 ): At least one of the treatment medians differs from others, where block effect has been taken into account. Critical Value/Decision Rule: F r, approximated by the chi-square distribution

15. Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. The Sign Test for Comparing Paired Samples Requirements: –Scale - Ordinal scale of measurement The Research Question (H 1 ): –One sample: The population median, m { , >, <} a single value. –Two sample: The difference between two populations medians { , >, <} 0. Critical Value/Decision Rule: p -value, the binomial distribution

16. Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. The Runs Test for Randomness Requirements: –Scale - Nominal scale of measurement –Two categories The Research Question (H 1 ): –The sequence in which observations from the two categories appear is not random. Critical Value/Decision Rule: z, the standard normal distribution

17. Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. An Example Problem: For the first 31 Super Bowls, the winner is listed below according to “A” (American Conference) or “N” (National Conference). At the 0.05 level of significance, can this sequence be considered as other than random?

18. Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. An Example, continued n A = 12, n N = 19, T = 9, n = 31 Compute the appropriate z statistic:

19. Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. An Example, continued I. H 0 : The sequence is random. H 1 : The sequence is not random. II. Rejection Region:  = 0.05 If z > 1.96 or z < –1.96, reject H 0. III. Test Statistic: z = – 2.59

20. Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. An Example, concluded IV. Conclusion: Since the test statistic of z = –2.59 falls below the critical bound of z = –1.96, we reject H 0 with at least 95% confidence. V. Implications: There is enough evidence to show that the sequence is not random.

21. Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. Lilliefors Test for Normality Requirements: –Scale - Interval or ratio scale –Hypothesized distribution must be completely specified. The Research Question (H 1 ): The sample was not drawn from a normal distribution. Critical Value/Decision Rule: D = max| F i – E i |