1. 16 - 1 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved.

2. 16 - 2 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. 1. Conduct the sign test for single and dependent samples using the binomial and standard normal distributions as the test statistics 2. Conduct a test of hypothesis for dependent samples using the Wilcoxon signed-rank test When you have completed this chapter, you will be able to: 3. Conduct and interpret the Wilcoxon rank-sum test for independent samples

3. 16 - 3 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. 4. 5. 6. Conduct and interpret the Kruskal-Wallis test for several independent samples Compute and interpret Spearman’s coefficient of rank correlation Conduct a test of hypothesis to determine whether the correlation among the ranks in the population is different from zero

4. 16 - 4 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. T erminology Range …is the difference between the largest and the smallest value Only two values are used in its calculation It is influenced by an extreme value It is easy to compute and understand Only two values are used in its calculation It is influenced by an extreme value It is easy to compute and understand

5. 16 - 5 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. The Sign Test The Sign Test is based on the sign of a difference between two related observations... no assumption is necessary regarding the shape of the population of differences … the binomial distribution is the test statistic for small samples and the standard normal (z) for large samples … the test requires dependent (related) samples

6. 16 - 6 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Determine the sign of the difference between related pairs 1. Determine the number of usable pairs 2.Compare the number of positive (or negative) differences to the critical value 3.n is the number of usable pairs (without ties), x is the number of pluses or minuses, and the binomial probability p=.5 The Sign Test …continued Procedure to conduct the test:

7. 16 - 7 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. …if the number of pluses or minuses is more than n/2, then …if the number of pluses or minuses is less than n/2, then Normal Approximation z xn n   (.).. 55 5 z xn n   (.).. 55 5 …if both and are greater than 5, the z distribution is appropriate n pp n () 1  p

8. 16 - 8 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. The Gagliano Research Institute for Business Studies is comparing the Research and Development expense (R&D) as a percent of income for a sample of glass manufacturing firms for 2000 and 2001 At the.05 significance level has the R&D expense declined? Use the sign test Normal Approximation

9. 16 - 9 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Company2000 2001 DifferenceSign Savoth Glass 20 16 4 + Ruisi Glass 14 13 1 + Rubin Inc.. 23 20 3 + Vaught 24 17 7 + Lambert Glass 31 22 9 + Pimental 22 20 2 + Olson Glass 14 20 - 6 - Flynn Glass 18 11 7 + Normal Approximation Continued…

10. 16 - 10 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Step 1: H 0 : p=.5 H 1 : p<.5 Step 2: H 0 : is rejected if the number of negative signs is 0 or 1 Step 3: There is one negative difference, i.e. there was an increase in the percent for one company Step 4: H 0 is rejected R&D expense as a percent of income declined from 2000 to 2001 Normal Approximation Continued…

11. 16 - 11 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Testing a Hypothesis About a Median When testing the value of the median, we use the normal approximation to the binomial distribution The z distribution is used as the test statistic

12. 16 - 12 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. The Gordon Travel Agency claims that their median airfare for all their clients to all destinations is $450. This claim is being challenged by a competing agency, who believe the median is different from $450. A random sample of 300 tickets revealed 170 tickets were below $450. Use the 0.05 level of significance.

13. 16 - 13 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Continued… H 0 is rejected if z is > than –1.96 or < than 1.96 … the value of z is 2.252 H 0 is rejected. We conclude that the median is not $450 450$ median :H $450 =median : 10  H 252.2 3005. ) (50.)5.170(    5. 50.)5.(   n nx z

14. 16 - 14 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Wilcoxon Signed-Rank Test … the test requires the ordinal scale of measurement … the observations must be related or dependent Use the Wilcoxon Signed-Rank if the assumption of normality is violated for the paired-t test Note ontinued…

15. 16 - 15 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Wilcoxon Signed-Rank Test Steps 1. Compute the differences between related observations 2. Rank the absolute differences from low to high 3. Return the signs to the ranks and sum positive and negative ranks 4. Compare the smaller of the two rank sums with the T value, obtained from the Appendix of Wilcoxon T values.

16. 16 - 16 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Wilcoxon Signed-Rank Test Use the Wilcoxon matched-pair signed-rank test to determine if the R&D expenses as a percent of income (EXAMPLE 1) have declined. Use the.05 significance level Step 1: H 0 : The percent stayed the same H 1 : The percent declined Step 2: H 0 is rejected if the smaller of the rank sums is less than or equal to 5 See the Appendix of Wilcoxon T Values ontinued…

17. 16 - 17 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Company2000 2001 Difference ABS-Diff Rank R+ R- Savoth Glass 20 16 4 4 4 4 * Ruisi Glass 14 13 1 1 1 1 * Rubin Inc.. 23 20 3 3 3 3 * Vaught 24 17 7 7 7 7 * Lambert Glass 31 22 9 8 8 8 * Pimental 22 20 2 2 2 2 * Olson Glass 14 20 - 6 6 5 * 5 Flynn Glass 18 11 7 7 6 6 * ontinued… Wilcoxon Signed-Rank Test

18. 16 - 18 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. The smaller rank sum is 5, which is equal to the critical value of T. H 0 is rejected The percent has declined from one year to the next. Company2000 2001 Difference ABS-Diff Rank R+ R- Savoth Glass 20 16 4 4 4 4 * Ruisi Glass 14 13 1 1 1 1 * Rubin Inc.. 23 20 3 3 3 3 * Vaught 24 17 7 7 7 7 * Lambert Glass 31 22 9 8 8 8 * Pimental 22 20 2 2 2 2 * Olson Glass 14 20 - 6 6 5 * 5 Flynn Glass 18 11 7 7 6 6 * Wilcoxon Signed-Rank Test

19. 16 - 19 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved....used to determine if two independent samples came from the same or equal populations …no assumption about the shape of the population is required …the data must be at least ordinal scale …each sample must contain at least eight observations Wilcoxon Rank-Sum Test

20. 16 - 20 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. …to determine the value of the test statistic W, all data values are ranked from low to high as if they were from a single population …the sum of ranks for each of the two samples is determined Wilcoxon Rank-Sum Test The smaller of the two sums W is used to compute the test statistic from: 12 )1( 2 )1( 2121 211     nnnn nnn W z

21. 16 - 21 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Your school purchased two vehicles, a Ford and a Chevrolet, for the administration’s use when travelling. The repair costs for the two cars over the last three years is shown on the next slide. At the.05 significance level is there a difference in the two distributions? Wilcoxon Rank-Sum Test ontinued…

22. 16 - 22 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Ford ($) Rank Chevrolet($) Rank 25.31 3.0 14.89 1.0 33.68 5.5 20.31 2.0 46.89 7.0 25.97 4.0 51.83 8.0 33.68 5.5 87.65 13.0 68.98 9.0 87.90 14.0 78.2310.0 90.89 15.0 80.3111.0 120.67 16.0 81.7512.0 157.9017.0 81.571.5 Wilcoxon Rank-Sum Test ontinued…

23. 16 - 23 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Step 1: H 0 : The percent stayed the same H 1 : The percent declined Step 2: H 0 is rejected if z >1.96 or z is less than –1.96 Wilcoxon Rank-Sum Test Step 3: The value of the test statistic is 0.914. 12 )198)(9(8 2 )198(8 5.81     12 )1( 2 )1( 2121 211     nnnn nnn W z = 0.914 ontinued…

24. 16 - 24 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Wilcoxon Rank-Sum Test Step 4: The value of the test statistic is 0.914. We do not reject the null hypothesis ! We cannot conclude that there is a difference in the distributions of the repair costs of the two vehicles!

25. 16 - 25 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Kruskal-Wallis Test: Analysis of Variance by Ranks …used to compare three or more samples to determine if they came from equal populations …the ordinal scale of measurement is required …it is an alternative to the one-way ANOVA …the chi-square distribution is the test statistic …each sample should have at least five observations …the sample data is ranked from low to high as if it were a single group Note ontinued…

26. 16 - 26 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. The test statistic is given by: Kruskal-Wallis Test: Analysis of Variance by Ranks continued… ) H NN R n R n R n N k k          12 1 31 1 2 1 2 2 2 2 () ()()... () ( 

27. 16 - 27 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Keely Ambrose, director of Human Resources for Miller Industries, wishes to study the percent increase in salary for middle managers at the four manufacturing plants. She gathers a sample of managers and determines the percent increase in salary from last year to this year. At the 5% significance level, can Keely conclude that there is a difference in the percent increases for the various plants? Kruskal-Wallis Test: Analysis of Variance by Ranks ontinued…

28. 16 - 28 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Kruskal-Wallis Test: Analysis of Variance by Ranks Millville Rank Camden Rank Eaton Rank Cliff Rank 2.2 2.01.9 1 3.7 6.0 5.7 9.0 3.6 5.02.7 3 4.5 7.0 6.8 10.5 4.9 8.03.1 4 7.1 13.5 8.9 16.0 6.8 10.56.912 9.3 17.0 11.6 18.5 7.1 13.58.315 11.6 18.5 13.9 20.0 39.0 35 62.0 74.0 Plants ontinued…

29. 16 - 29 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Step 1: H 0 : The populations are the same H 1 : The populations are not the same Step 2: H 0 is rejected if x 2 is greater than 7.185. There are 3 degrees of freedom at the.05 significance level. Kruskal-Wallis Test: Analysis of Variance by Ranks ontinued…

30. 16 - 30 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. )120(3 5 74 5 62 5 35 5 39 )120( 12 2222              )1(3 )( )( )()( )1( 12 2 4 2 2 3 2 2 2 1 2 1                    N n R n R n R n R NN H k = 5.949 Kruskal-Wallis Test: Analysis of Variance by Ranks Millville Rank Camden Rank Eaton Rank Cliff Rank 2.2 2.01.9 1 3.7 6.0 5.7 9.0 3.6 5.02.7 3 4.5 7.0 6.8 10.5 4.9 8.03.1 4 7.1 13.5 8.9 16.0 6.8 10.56.912 9.3 17.0 11.6 18.5 7.1 13.58.315 11.6 18.5 13.9 20.0 39.035 62.0 74.0 ontinued…

31. 16 - 31 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. The null hypothesis is not rejected Kruskal-Wallis Test: Analysis of Variance by Ranks There is no difference in the percent increases in the four plants

32. 16 - 32 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Rank-Order Correlation … Spearman’s coefficient of rank correlation reports the association between two sets of ranked observations Features …it can range from –1.00 up to 1.00 …it is similar to Pearson’s coefficient of correlation, but is based on ranked data ontinued…

33. 16 - 33 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Spearman’s Rank-Order Correlation Formula (to find the coefficient of rank correlation) d is the difference in the ranks and n is the number of observations. r d nn s   1 6 1 2 2  () ontinued…

34. 16 - 34 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Spearman’s Rank-Order Correlation Testing the significance of r s State the null hypothesis: Rank correlation in population is 0 State the alternate hypothesis: Rank correlation in population is not 0. The value of the test statistic is computed from … trsrs n rssrss    2 1

35. 16 - 35 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Spearman’s Rank-Order Correlation The preseason football rankings for the Atlantic Coast Conference by the coaches and sports writers are shown below. What is the coefficient of rank correlation between the two groups? School Coaches Writers Maryland23 NC State34 NC66 Virginia55 Clemson42 Wake Forest78 Duke87 Florida State11 ontinued…

36. 16 - 36 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. School Coaches Writers d d 2 Maryland23 -1 1 NC State34 -1 1 NC66 0 0 Virginia55 0 0 Clemson42 2 4 Wake Forest78 -1 1 Duke87 1 1 Florida State11 0 0 Spearman’s Rank-Order Correlation Total 8 ontinued…

37. 16 - 37 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Spearman’s Rank-Order Correlation School Coaches Writers d d 2 Maryland23 -1 1 NC State34 -1 1 NC66 0 0 Virginia55 0 0 Clemson42 2 4 Wake Forest78 -1 1 Duke87 1 1 Florida State11 0 0 Total 8 = 0.905 )18(8 )8(6 1 2    )1( 6 1 2 2    nn d  r s There is a strong correlation between the ranks of the coaches and the sports writers!

38. 16 - 38 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. Test your learning … www.mcgrawhill.ca/college/lind Click on… Online Learning Centre for quizzes extra content data sets searchable glossary access to Statistics Canada’s E-Stat data …and much more!

39. 16 - 39 Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. This completes Chapter 16