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Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 16-1 Business Statistics: A Decision-Making Approach 6 th Edition Chapter 16 Introduction to Nonparametric Statistics

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Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 16-2 Chapter Goals After completing this chapter, you should be able to: Recognize when and how to use the Wilcoxon signed rank test for a population median Recognize the situations for which the Wilcoxon signed rank test applies and be able to use it for decision-making Know when and how to perform a Mann-Whitney U-test Perform nonparametric analysis of variance using the Kruskal-Wallis one-way ANOVA

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Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 16-3 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

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Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 16-4 Wilcoxon Signed Rank Test Used to test a hypothesis about one population median the median is the midpoint of the distribution: 50% below, 50% above A hypothesized median is rejected if sample results vary too much from expectations no highly restrictive assumptions about the shape of the population distribution are needed

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Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 16-5 The W Test Statistic Performing the Wilcoxon Signed Rank Test Calculate the test statistic W using these steps: Step 1: collect sample data Step 2: compute d i = difference between each value and the hypothesized median Step 3: convert d i values to absolute differences

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Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 16-6 The W Test Statistic Performing the Wilcoxon Signed Rank Test Step 4: determine the ranks for each d i value eliminate zero d i values Lowest d i value = 1 For ties, assign each the average rank of the tied observations (continued)

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Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 16-7 The W Test Statistic Performing the Wilcoxon Signed Rank Test Step 5: Create R+ and R- columns for data values greater than the hypothesized median, put the rank in an R+ column for data values less than the hypothesized median, put the rank in an R- column (continued)

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Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 16-8 The W Test Statistic Performing the Wilcoxon Signed Rank Test Step 6: the test statistic W is the sum of the ranks in the R+ column Test the hypothesis by comparing the calculated W to the critical value from the table in appendix P Note that n = the number of non-zero d i values (continued)

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Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 16-9 Example The median class size is claimed to be 40 Sample data for 8 classes is randomly obtained Compare each value to the hypothesized median to find difference Class size = x i Difference d i = x i – 40 | d i |

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Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap Example Rank the absolute differences: | d i |Rank tied (continued)

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Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap Example Put ranks in R+ and R- columns and find sums: Class size = x i Difference d i = x i – 40 | d i |RankR+R = 27 = 9 (continued) These three are below the claimed median, the others are above

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Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap Completing the Test H 0 : Median = 40 H A : Median ≠ 40 Test at the =.05 level: This is a two-tailed test and n = 8, so find W L and W U in appendix P: W L = 3 and W U = 33 The calculated test statistic is W = R+ = 27

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Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap Completing the Test H 0 : Median = 40 H A : Median ≠ 40 W L = 3 and W U = 33 W L < W < W U so do not reject H 0 (there is not sufficient evidence to conclude that the median class size is different than 40) (continued) W L = 3 do not reject H 0 reject H 0 W = R+ = 27 W U = 33 reject H 0

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Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap If the Sample Size is Large The W test statistic approaches a normal distribution as n increases For n > 20, W can be approximated by where W = sum of the R+ ranks d = number of non-zero d i values

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Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap Nonparametric Tests for Two Population Centers Nonparametric Tests for Two Population Centers Wilcoxon Matched-Pairs Signed Rank Test Mann-Whitney U-test Large Samples Small Samples Large Samples Small Samples

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Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, 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 measurement scale used is at least ordinal If they differ, the distributions of the two populations will differ only with respect to the central location

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Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap Consider two samples combine into a singe list, but keep track of which sample each value came from rank the values in the combined list from low to high For ties, assign each the average rank of the tied values separate back into two samples, each value keeping its assigned ranking sum the rankings for each sample Mann-Whitney U-Test (continued)

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Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 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)

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Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap (continued) Mann-Whitney U-Test Mann-Whitney U-Statistics where: n 1 and n 2 are the two sample sizes R 1 and R 2 = sum of ranks for samples 1 and 2

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Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap (continued) Mann-Whitney U-Test Claim: Median class size for Math is larger than the median class size for English A random sample of 9 Math and 9 English classes is selected (samples do not have to be of equal size) Rank the combined values and then split them back into the separate samples

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Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap Suppose the results are: Class size (Math, M)Class size (English, E) (continued) Mann-Whitney U-Test

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Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap SizeRank SizeRank Ranking for combined samples tied (continued) Mann-Whitney U-Test

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Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap Split back into the original samples: Class size (Math, M) Rank Class size (English, E) Rank = 104 = 67 (continued) Mann-Whitney U-Test

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Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap H 0 : Median M ≤ Median E H A : Median M > Median E Claim: Median class size for Math is larger than the median class size for English Note: U 1 + U 2 = n 1 n 2 (continued) Mann-Whitney U-Test Math: English:

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Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap The Mann-Whitney U tables in Appendices L and M give the lower tail of the U-distribution For one-tailed tests like this one, check the alternative hypothesis to see if U 1 or U 2 should be used as the test statistic Since the alternative hypothesis indicates that population 1 (Math) has a higher median, use U 1 as the test statistic (continued) Mann-Whitney U-Test

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Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap Use U 1 as the test statistic: U = 22 Compare U = 22 to the critical value U from the appropriate table For sample sizes less than 9, use Appendix L For samples sizes from 9 to 20, use Appendix M If U < U , reject H 0 (continued) Mann-Whitney U-Test

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Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap Since U U , do not reject H 0 Use U 1 as the test statistic: U = 19 U from Appendix M for =.05, n 1 = 9 and n 2 = 9 is U = 7 (continued) Mann-Whitney U-Test U = 7 U = 19 do not reject H 0 reject H 0

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Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap Mann-Whitney U-Test for Large Samples The table in Appendix M includes U values only for sample sizes between 9 and 20 The U statistic approaches a normal distribution as sample sizes increase If samples are larger than 20, a normal approximation can be used

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Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap Mann-Whitney U-Test for Large Samples The mean and standard deviation for Mann-Whitney U Test Statistic: (continued) Where n 1 and n 2 are sample sizes from populations 1 and 2

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Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap Mann-Whitney U-Test for Large Samples Normal approximation for Mann-Whitney U Test Statistic: (continued)

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Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap Large Sample Example We wish to test 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 When rankings are completed, the sum of ranks for sample 2 is R 2 = 2620 H 0 : Median 1 Median 2 H A : Median 1 < Median 2

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Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap U statistic is found to be U = 655 Since the alternative hypothesis indicates that population 2 has a higher median, use U 2 as the test statistic Compute the U statistics: Large Sample Example (continued)

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Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap Since z = < , we reject H 0 Reject H 0 =.05 Do not reject H 0 0 Large Sample Example (continued)

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Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap Wilcoxon Matched-Pairs Signed Rank Test The Mann-Whitney U-Test is used when samples from two populations are independent If samples are paired, they are not independent Use Wilcoxon Matched-Pairs Signed Rank Test with paired samples

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Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap The Wilcoxon T Test Statistic Performing the Small-Sample Wilcoxon Matched Pairs Test (for n < 25) Calculate the test statistic T using these steps: Step 1: collect sample data Step 2: compute d i = difference between the sample 1 value and its paired sample 2 value Step 3: rank the differences, and give each rank the same sign as the sign of the difference value

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Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap The Wilcoxon T Test Statistic Performing the Small-Sample Wilcoxon Matched Pairs Test (for n < 25) Step 4: The test statistic is the sum of the absolute values of the ranks for the group with the smaller expected sum Look at the alternative hypothesis to determine the group with the smaller expected sum For two tailed tests, just choose the smaller sum (continued)

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Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap Small Sample Example Paired samples, n = 9: Value (before)Value (after) Claim: Median value is smaller after than before

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Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap Small Sample Example Paired samples, n = 9: Value (before) Value (after) Difference d Rank of d Ranks with smaller expected sum = T = 13 (continued)

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Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap The calculated T value is T = 13 Complete the test by comparing the calculated T value to the critical T-value from Appendix N For n = 9 and =.025 for a one-tailed test, T = 6 Since T T , do not reject H 0 T = 6 T = 13 do not reject H 0 reject H 0 Small Sample Example (continued)

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Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap Wilcoxon Matched Pairs Test for Large Samples The table in Appendix N includes T values only for sample sizes from 6 to 25 The T statistic approaches a normal distribution as sample size increases If the number of paired values is larger than 25, a normal approximation can be used

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Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap The mean and standard deviation for Wilcoxon T : (continued) where n is the number of paired values Wilcoxon Matched Pairs Test for Large Samples

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Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap Mann-Whitney U-Test for Large Samples Normal approximation for the Wilcoxon T Test Statistic: (continued)

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Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap Tests the equality of more than 2 population medians Assumptions: variables have a continuous distribution. the data are at least ordinal. samples are independent. samples come from populations whose only possible difference is that at least one may have a different central location than the others. Kruskal-Wallis One-Way ANOVA

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Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap Kruskal-Wallis Test Procedure Obtain relative rankings for each value In event of tie, each of the tied values gets the average rank Sum the rankings for data from each of the k groups Compute the H test statistic

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Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap Kruskal-Wallis Test Procedure The Kruskal-Wallis H test statistic: (with k – 1 degrees of freedom) where: N = Sum of sample sizes in all samples k = Number of samples R i = Sum of ranks in the i th sample n i = Size of the i th sample (continued)

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Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap Complete the test by comparing the calculated H value to a critical 2 value from the chi-square distribution with k – 1 degrees of freedom (The chi-square distribution is Appendix G) Decision rule Reject H 0 if test statistic H > 2 Otherwise do not reject H 0 (continued) Kruskal-Wallis Test Procedure

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Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap Do different departments have different class sizes? Kruskal-Wallis Example Class size (Math, M) Class size (English, E) Class size (History, H)

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Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap Do different departments have different class sizes? Kruskal-Wallis Example Class size (Math, M) Ranking Class size (English, E) Ranking Class size (History, H) Ranking = 44 = 56 = 20

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Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap The H statistic is (continued) Kruskal-Wallis Example

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Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap Since H = 6.72 < do not reject H 0 (continued) Kruskal-Wallis Example Compare H = 6.72 to the critical value from the chi-square distribution for 5 – 1 = 4 degrees of freedom and =.05: There is not sufficient evidence to reject that the population medians are all equal

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Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap Kruskal-Wallis Correction If tied rankings occur, give each observation the mean rank for which it is tied The H statistic is influenced by ties, and should be corrected Correction for tied rankings: where: g = Number of different groups of ties t i = Number of tied observations in the i th tied group of scores N = Total number of observations

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Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap H Statistic Corrected for Tied Rankings Corrected H statistic:

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Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap Chapter Summary Developed and applied the Wilcoxon signed rank W-test for a population median Small Samples Large sample z approximation Developed and applied the Mann-Whitney U-test for two population medians Small Samples Large Sample z approximation Used the Wilcoxon Matched-Pairs T-test for paired samples Small Samples Large sample z approximation Applied the Kruskal-Wallis H-test for multiple population medians

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