Statistics for Business and Economics 7th Edition Chapter 14 Analysis of Categorical Data Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 14-1 Chapter Goals After completing this chapter, you should be able to: Use the chi-square goodness-of-fit test to determine whether data fits specified probabilities Perform tests for the Poisson and Normal distributions Set up a contingency analysis table and perform a chisquare test of association Use the sign test for paired or matched samples Recognize when and how to use the Wilcoxon signed rank test for paired or matched samples Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 14-2 Chapter Goals (continued) After completing this chapter, you should be able to: Use a sign test for a single population median Apply a normal approximation for the Wilcoxon signed rank test Know when and how to perform a Mann-Whitney U-test Explain Spearman rank correlation and perform a test for association Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 14-3 Nonparametric Statistics Nonparametric Statistics Fewer restrictive assumptions about data levels and underlying probability distributions Population distributions may be skewed The level of data measurement may only be ordinal or nominal Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 14-4 Goodness-of-Fit Tests 14.1 Does sample data conform to a hypothesized distribution? Examples: Do sample results conform to specified expected probabilities? Are technical support calls equal across all days of the week? (i.e., calls follow a uniform distribution?) Do measurements from a production process follow a normal distribution? Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 14-5 Chi-Square Goodness-of-Fit Test (continued) Are technical support calls equal across all days of the week? (i.e., calls follow a uniform distribution?) Sample data for 10 days per day of week: Sum of calls for this day: Monday Tuesday Wednesday Thursday Friday Saturday Sunday 290 250 238 257 265 230 192 = 1722 Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 14-6 Logic of Goodness-of-Fit Test If calls are uniformly distributed, the 1722 calls would be expected to be equally divided across the days: 1722 246 expected calls per day if uniform Chi-Square Goodness-of-Fit Test: test to see if the sample results are consistent with the expected results Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 14-7 Observed vs Expected Frequencies Observed Oi Expected Ei Monday Tuesday Wednesday Thursday Friday Saturday Sunday 290 250 238 257 265 230 192 246 246 246 246 246 246 246 TOTAL 1722 1722 Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 14-8 Chi-Square Test Statistic H0: The distribution of calls is uniform over days of the week H1: The distribution of calls is not uniform The test statistic is (O E ) i i Ei i1 K (where d.f K 1) where: K = number of categories Oi = observed frequency for category i Ei = expected frequency for category i Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 14-9 The Rejection Region H0: The distribution of calls is uniform over days of the week H1: The distribution of calls is not uniform (O E ) i i Ei i 1 K Reject H0 if α (with k – degrees of freedom) Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall 2 Do not reject H0 Reject H0 Ch 14-10 Mann-Whitney U-Test Example (continued) Suppose the results are: Class size (Math, M) Class size (English, E) 23 45 34 78 34 66 62 95 81 99 Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall 30 47 18 34 44 61 54 28 40 96 Ch 14-53 Mann-Whitney U-Test Example (continued) Ranking for combined samples tied Size Rank Size Rank 18 47 11 23 54 12 28 61 13 30 62 14 34 66 15 34 78 16 34 81 17 40 95 18 44 96 19 45 10 99 20 Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 14-54 Mann-Whitney U-Test Example (continued) Rank by original sample: Class size (Math, M) Rank Class size (English, E) Rank 23 45 34 78 34 66 62 95 81 99 10 16 15 14 18 17 20 30 47 18 34 44 61 54 28 40 96 11 13 12 19 = 124 Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall = 86 Ch 14-55 Mann-Whitney U-Test Example (continued) Claim: Median class size for Math is larger than the median class size for English H0: MedianM ≤ MedianE (Math median is not greater than English median) HA: MedianM > MedianE U n1n2 n1(n1 1) (10)(11) (Math median is larger) R (10)(10) 124 31 Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 14-56 Mann-Whitney U-Test Example (continued) H0: MedianM ≤ MedianE HA: MedianM > MedianE n1n2 (10)(10) U 31 U μU 2 z 1.436 σU n1n2 (n1 n2 1) (10)(10)(10 10 1) 12 12 The decision rule for this one-sided upper-tailed alternative hypothesis: U μU Reject H0 if z zα σU For = 0.05, -z = -1.645 The calculated z value is not in the rejection region, so we conclude that there is not sufficient evidence of difference in class size medians Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 14-57 Wilcoxon Rank Sum Test Similar to Mann-Whitney U test Results will be the same for both tests Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 14-58 Wilcoxon Rank Sum Test (continued) n1 observations from the first population n2 observations from the second population Pool the samples and rank the observations in ascending order Let T denote the sum of the ranks of the observations from the first population (T in the Wilcoxon Rank Sum Test is the same as R1 in the Mann-Whitney U Test) Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 14-59 Wilcoxon Rank Sum Test (continued) The Wilcoxon Rank Sum Statistic, T, has mean E(T) μT n1(n1 n2 1) And variance Var(T) σ 2T n1n2 (n1 n2 1) 12 Then, for large samples (n1 10 and n2 10) the distribution of the random variable Z T μT σT is approximated by the normal distribution Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 14-60 Wilcoxon Rank Sum Example We wish to test H0: Median1 Median2 H1: Median1 < Median2 Use = 0.05 Suppose two samples are obtained: n1 = 40 , n2 = 50 When rankings are completed, the sum of ranks for sample is R1 = 1475 = T When rankings are completed, the sum of ranks for sample is R2 = 2620 Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 14-61 Wilcoxon Rank Sum Example (continued) Using the normal approximation: n1(n1 n2 1) (40)(40 50 1) T 1475 T μT 2 z 2.80 σT n1n2 (n1 n2 1) (40)(50)(40 50 1) 12 12 Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 14-62 Wilcoxon Rank Sum Example (continued) H0: Median1 Median2 H1: Median1 < Median2 = 05 z 1.645 Reject H0 Do not reject H0 T μT z 2.80 σT Since z = -2.80 < -1.645, we reject H0 and conclude that median is less than median at the 0.05 level of significance Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 14-63 14.6 Spearman Rank Correlation Consider a random sample (x1 , y1), ,(xn, yn) of n pairs of observations Rank xi and yi each in ascending order Calculate the sample correlation of these ranks The resulting coefficient is called Spearman’s Rank Correlation Coefficient If there are no tied ranks, an equivalent formula for computing this coefficient is n rS 1 6 di2 i1 n(n 1) where the di are the differences of the ranked pairs Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 14-64 Spearman Rank Correlation (continued) Consider the null hypothesis H0: no association in the population To test against the alternative of positive association, the decision rule is Reject H0 if rS rS,α To test against the alternative of negative association, the decision rule is Reject H0 if rS rS,α To test against the two-sided alternative of some association, the decision rule is Reject H0 if rS rS,α/2 or rS rS,α/2 Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 14-65 Chapter Summary Used the chi-square goodness-of-fit test to determine whether sample data match specified probabilities Conducted goodness-of-fit tests when a population parameter was unknown Tested for normality using the Jarque-Bera test Used contingency tables to perform a chi-square test for association Compared observed cell frequencies to expected cell frequencies Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 14-66 Chapter Summary (continued) Used the sign test for paired or matched samples, and the normal approximation for the sign test Developed and applied the Wilcoxon signed rank test, and the large sample normal approximation Developed and applied the Mann-Whitney U-test for two population medians Used the Wilcoxon rank-sum test Examined Spearman rank correlation for tests of association Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 14-67 ... Prentice Hall Ch 14- 2 Chapter Goals (continued) After completing this chapter, you should be able to: Use a sign test for a single population median Apply a normal approximation for the Wilcoxon... perform a Mann-Whitney U-test Explain Spearman rank correlation and perform a test for association Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 14- 3 Nonparametric Statistics. . .Chapter Goals After completing this chapter, you should be able to: Use the chi-square goodness-of-fit test to determine whether data fits specified probabilities Perform tests for the