CHAPTER 14: Nonparametric Methods to accompany Introduction to Business Statistics fourth edition, by Ronald M. Weiers Presentation by Priscilla Chaffe-Stengel Donald N. Stengel © 2002 The Wadsworth Group

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. © 2002 The Wadsworth Group

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 © 2002 The Wadsworth Group

Nonparametric Tests Advantages: Disadvantages: 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 © 2002 The Wadsworth Group

Wilcoxon Signed Rank Test, One Sample Requirements: Variable - Continuous data Scale - Interval or ratio scale of measurement The Research Question (H1): Test the value of a single population median, m {¹, >, <} m0 Critical Value/Decision Rule: W, Wilcoxon signed rank test © 2002 The Wadsworth Group

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 8 10 9 7 7 9 7 7 9 8 11 9 10 7 8 11 8 12 © 2002 The Wadsworth Group

An Example, continued There are: hrs. di|di| hrs. di|di| 8 –2 2 9 –1 1 8 –2 2 9 –1 1 9 –1 1 8 –2 2 8 –2 2 11 +1 1 10 0 0 9 –1 1 9 –1 1 10 0 0 7 –3 3 7 –3 3 7 –3 3 8 –2 2 9 –1 1 11 +1 1 7 –3 3 12 +2 2 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 © 2002 The Wadsworth Group

An Example, continued hrs. di|di| Rank R+ R– hrs. di|di| Rank R+ R– 8 –2 2 10.5 - 10.5 9 –1 1 4 - 4 9 –1 1 4 - 4 8 –2 2 10.5 - 10.5 8 –2 2 10.5 - 10.5 11 +1 1 4 4 - 10 0 0 - - - 9 –1 1 4 - 4 9 –1 1 4 - 4 10 0 0 - - - 7 –3 3 16 - 16 7 –3 3 16 - 16 7 –3 3 16 - 16 8 –2 2 10.5 - 10.5 9 –1 1 4 - 4 11 +1 1 4 4 - 7 –3 3 16 - 16 12 +2 2 10.5 10.5 - So, SR+ = 18.5, SR– = 152.5 © 2002 The Wadsworth Group

An Example, continued I. H0: m = 10 hours H1: m ¹ 10 hours II. Rejection Region: a = 0.05, n = 18 data values not equal to the hypothesized median of 10 If SR+ < 41 or SR+ > 130, reject H0. III. Test Statistics: SR+ = 18.5 SR– = 152.5 © 2002 The Wadsworth Group

An Example, concluded IV. Conclusion: Since the test statistic of SR+ = 18.5 falls below the critical value of W = 41, we reject H0 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. © 2002 The Wadsworth Group

Wilcoxon Signed Rank Test for Comparing Paired Samples Requirements: Variable - Continuous data Scale - Interval or ratio scale of measurement The Research Question (H1): Test the difference in two population medians, paired samples, md {¹, >, <} 0 Critical Value/Decision Rule: W, Wilcoxon rank sum test © 2002 The Wadsworth Group

Kruskal-Wallis Test, Comparing Two Independent Samples Requirements: Scale - Ordinal, interval or ratio scale Independent samples from populations with identical distributions The Research Question (H1): At least one of the medians differs from the others. Critical Value/Decision Rule: H, approximated by the chi-square distribution © 2002 The Wadsworth Group

Friedman Test for the Randomized Block Design Requirements: Scale - Ordinal, interval or ratio scale The Research Question (H1): At least one of the treatment medians differs from others, where block effect has been taken into account. Critical Value/Decision Rule: Fr, approximated by the chi-square distribution © 2002 The Wadsworth Group

The Sign Test Requirements: The Research Question (H1): Scale - Ordinal scale of measurement The Research Question (H1): One sample: The population median, m {¹, >, <} a single value. Two sample: The difference between two populations medians {¹, >, <} 0. Critical Value/Decision Rule: p-value , the binomial distribution © 2002 The Wadsworth Group

The Runs Test for Randomness Requirements: Scale - Nominal scale of measurement Two categories The Research Question (H1): The sequence in which observations from the two categories appear is not random. Critical Value/Decision Rule: z, the standard normal distribution © 2002 The Wadsworth Group

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? © 2002 The Wadsworth Group

An Example, continued nA = 12, nN = 19, T = 9, n = 31 Compute the appropriate z statistic: © 2002 The Wadsworth Group

An Example, continued I. H0: The sequence is random. H1: The sequence is not random. II. Rejection Region: a = 0.05 If z > 1.96 or z < –1.96, reject H0. III. Test Statistic: z = – 2.59 © 2002 The Wadsworth Group

An Example, concluded IV. Conclusion: Since the test statistic of z = –2.59 falls below the critical bound of z = –1.96, we reject H0 with at least 95% confidence. V. Implications: There is enough evidence to show that the sequence is not random. © 2002 The Wadsworth Group

Lilliefors Test for Normality Requirements: Scale - Interval or ratio scale Hypothesized distribution must be completely specified. The Research Question (H1): The sample was not drawn from a normal distribution. Critical Value/Decision Rule: D = max|Fi – Ei| © 2002 The Wadsworth Group