Chap 15-1 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chapter 15 Nonparametric Statistics Statistics for Business and Economics 6 th Edition

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 15-2 Chapter Goals After completing this chapter, you should be able to: Use the sign test for paired or matched samples Use a sign test for a single population median Recognize when and how to use the Wilcoxon signed rank test for a 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

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 15-3 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

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 15-4 Sign Test and Confidence Interval A sign test for paired or matched samples: Calculate the differences of the paired observations Discard the differences equal to 0, leaving n observations Record the sign of the difference as + or – For a symmetric distribution, the signs are random and + and – are equally likely

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 15-5 Sign Test Define + to be a success and let P = the true proportion of +s in the population The sign test is used for the hypothesis test The test-statistic S for the sign test is S = the number of pairs with a positive difference S has a binomial distribution with P = 0.5 and n = the number of nonzero differences (continued)

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 15-6 Determining the p-value The p-value for a Sign Test is found using the binomial distribution with n = number of nonzero differences, S = number of positive differences, and P = 0.5 For an upper-tail test, H 1 : P > 0.5,p-value = P(x S) For a lower-tail test, H 1 : P < 0.5,p-value = P(x S) For a two-tail test, H 1 : P 0.5,2(p-value)

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 15-7 Sign Test Example Ten consumers in a focus group have rated the attractiveness of two package designs for a new product ConsumerRatingDifferenceSign of Difference Package 1Package 2Rating 1 – ––0+––––+–––0+––––+–

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 15-8 Sign Test Example (continued) Test the hypothesis that there is no overall package preference using = 0.10 The proportion of consumers who prefer package 1 is the same as the proportion preferring package 2 A majority prefer package 2 The test-statistic S for the sign test is S = the number of pairs with a positive difference = 2 S has a binomial distribution with P = 0.5 and n = 9 (there was one zero difference)

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 15-9 The p-value for this sign test is found using the binomial distribution with n = 9, S = 2, and P = 0.5: For a lower-tail test, p-value = P(x 2|n=9, P=0.5) = Since < = 0.10 we reject the null hypothesis and conclude that consumers prefer package 2 Sign Test Example (continued)

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Sign Test: Normal Approximation If the number n of nonzero sample observations is large, then the sign test is based on the normal approximation to the binomial with mean and standard deviation The test statistic is Where S* is the test-statistic corrected for continuity: For a two-tail test, S* = S + 0.5, if S μ For upper-tail test, S* = S – 0.5 For lower-tail test, S* = S + 0.5

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Sign Test for Single Population Median The sign test can be used to test that a single population median is equal to a specified value For small samples, use the binomial distribution For large samples, use the normal approximation

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Wilcoxon Signed Rank Test for Paired Samples Uses matched pairs of random observations Still based on ranks Incorporates information about the magnitude of the differences Tests the hypothesis that the distribution of differences is centered at zero The population of paired differences is assumed to be symmetric

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Conducting the test: Discard pairs for which the difference is 0 Rank the remaining n absolute differences in ascending order (ties are assigned the average of their ranks) Find the sums of the positive ranks and the negative ranks The smaller of these sums is the Wilcoxon Signed Rank Statistic T: T = min(T +, T - ) Where T + = the sum of the positive ranks T - = the sum of the negative ranks n = the number of nonzero differences The null hypothesis is rejected if T is less than or equal to the value in Appendix Table 10 Wilcoxon Signed Rank Test for Paired Samples (continued)

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Signed Rank Test Example T + = 3 T – = 42 ConsumerRatingDifference Package 1Package 2Diff (rank)Rank (+)Rank (–) (5) -4 (7 tie) 0 (-) +1 (2) -6 (9) -4 (7 tie) -1 (3) -4 (7 tie) +3 (1) -2 (4) Ten consumers in a focus group have rated the attractiveness of two package designs for a new product

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Test the hypothesis that the distribution of paired differences is centered at zero, using = 0.10 Conducting the test: The smaller of T + and T - is the Wilcoxon Signed Rank Statistic T: T = min(T +, T - ) = 3 Use Appendix Table 10 with n = 9 to find the critical value: The null hypothesis is rejected if T 4 Since T = 3 < 4, we reject the null hypothesis (continued) Signed Rank Test Example

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Wilcoxon Signed Rank Test Normal Approximation A normal approximation can be used when Paired samples are observed The sample size is large The hypothesis test is that the population distribution of differences is centered at zero

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap The table in Appendix 10 includes T values only for sample sizes from 4 to 20 The T statistic approaches a normal distribution as sample size increases If the number of paired values is larger than 20, a normal approximation can be used Wilcoxon Signed Rank Test Normal Approximation (continued)

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap The mean and standard deviation for Wilcoxon T : where n is the number of paired values Wilcoxon Matched Pairs Test for Large Samples

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Normal approximation for the Wilcoxon T Statistic: (continued) If the alternative hypothesis is one-sided, reject the null hypothesis if If the alternative hypothesis is two-sided, reject the null hypothesis if Wilcoxon Matched Pairs Test for Large Samples

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Mann-Whitney U-Test Used to compare two samples from two populations Assumptions: The two samples are independent and random The value measured is a continuous variable The two distributions are identical except for a possible difference in the central location The sample size from each population is at least 10

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Consider two samples Pool the two samples (combine into a singe list) but keep track of which sample each value came from rank the values in the combined list in ascending order For ties, assign each the average rank of the tied values sum the resulting rankings separately for each sample If the sum of rankings from one sample differs enough from the sum of rankings from the other sample, we conclude there is a difference in the population medians Mann-Whitney U-Test (continued)

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Mann-Whitney U Statistic Consider n 1 observations from the first population and n 2 observations from the second Let R 1 denote the sum of the ranks of the observations from the first population The Mann-Whitney U statistic is

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Mann-Whitney U Statistic The null hypothesis is that the central locations of the two population distributions are the same The Mann-Whitney U statistic has mean and variance Then for large sample sizes (both at least 10), the distribution of the random variable is approximated by the normal distribution (continued)

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Decision Rules for Mann-Whitney Test The decision rule for the null hypothesis that the two populations have the same central location: For a one-sided upper-tailed alternative hypothesis: For a one-sided lower-tailed hypothesis: For a two-sided alternative hypothesis:

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Mann-Whitney U-Test Example Claim: Median class size for Math is larger than the median class size for English A random sample of 10 Math and 10 English classes is selected (samples do not have to be of equal size) Rank the combined values and then determine rankings by original sample

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Suppose the results are: Class size (Math, M)Class size (English, E) (continued) Mann-Whitney U-Test Example

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap SizeRank SizeRank Ranking for combined samples tied (continued) Mann-Whitney U-Test Example

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Rank by original sample: Class size (Math, M) Rank Class size (English, E) Rank = 124 = 86 (continued) Mann-Whitney U-Test Example

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap H 0 : Median M Median E (Math median is not greater than English median) H A : Median M > Median E (Math median is larger) Claim: Median class size for Math is larger than the median class size for English (continued) Mann-Whitney U-Test Example

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap (continued) Mann-Whitney U-Test Example H 0 : Median M Median E H A : Median M > Median E The decision rule for this one-sided upper-tailed alternative hypothesis: For = 0.05, -z = 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

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Wilcoxon Rank Sum Test Similar to Mann-Whitney U test Results will be the same for both tests

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Wilcoxon Rank Sum Test n 1 observations from the first population n 2 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 R 1 in the Mann-Whitney U Test) (continued)

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Wilcoxon Rank Sum Test The Wilcoxon Rank Sum Statistic, T, has mean And variance Then, for large samples (n 1 10 and n 2 10) the distribution of the random variable is approximated by the normal distribution (continued)

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Wilcoxon Rank Sum Example We wish to test Use = 0.05 Suppose two samples are obtained: n 1 = 40, n 2 = 50 When rankings are completed, the sum of ranks for sample 1 is R 1 = 1475 = T When rankings are completed, the sum of ranks for sample 2 is R 2 = 2620 H 0 : Median 1 Median 2 H 1 : Median 1 < Median 2

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Using the normal approximation: (continued) Wilcoxon Rank Sum Example

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Since z = < , we reject H 0 and conclude that median 1 is less than median 2 at the 0.05 level of significance Reject H 0 =.05 Do not reject H 0 0 (continued) Wilcoxon Rank Sum Example H 0 : Median 1 Median 2 H 1 : Median 1 < Median 2

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Spearman Rank Correlation Consider a random sample (x 1, y 1 ),...,(x n, y n ) of n pairs of observations Rank x i and y i each in ascending order Calculate the sample correlation of these ranks The resulting coefficient is called Spearmans Rank Correlation Coefficient. If there are no tied ranks, an equivalent formula for computing this coefficient is where the d i are the differences of the ranked pairs

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Consider the null hypothesis H 0 : no association in the population To test against the alternative of positive association, the decision rule is To test against the alternative of negative association, the decision rule is To test against the two-sided alternative of some association, the decision rule is Spearman Rank Correlation (continued)

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Chapter Summary 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