1. 1 Wilcoxon Wilcoxon Rank Sum Test 1. Wilcoxon with both n 1 and n 2 < 10 2. Wilcoxon with both n 1 and n 2 ≥ 10 3. Examples

2. 2 Wilcoxon Wilcoxon Rank Sum Test Recall from last week:  When we test a hypothesis about the difference between two independent population means, we do so using the difference between two sample means.  When the two sample variances are tested and found not to be equal  we cannot pool the sample variances  thus we cannot use the t-test for independent samples. Instead, we use the Wilcoxon Rank Sum Test.

3. 3 Wilcoxon Population 1Population 2 µ1µ1 µ2µ2 Sample1Sample2 X1X1 X2X2 µ tells us about the population The sample mean tells us about µ

4. 4 Wilcoxon Wilcoxon Rank Sum Test The Z test and the t test are “parametric tests” – that is, they answer a question about the difference between populations by comparing sample statistics (e.g., X 1 and X 2 ) and making an inference to the population parameters (μ 1 and μ 2 ). The Wilcoxon, in contrast, allows inferences about whole populations

5. 5 Wilcoxon X X Distribution B Distribution A μ μ Note that distribution B is shifted to the right of distribution A

6. 6 Wilcoxon 1b. Small samples, independent groups Wilcoxon Rank Sum Test  first, combine the two samples and rank order all the observations.  smallest number has rank 1, largest number has rank N (= sum of n 1 and n 2 ).  separate samples and add up the ranks for the smaller sample. (If n 1 = n 2, choose either one.)  test statistic : rank sum T for smaller sample.

7. 7 Wilcoxon 1b. Small samples, independent groups Wilcoxon – One-tailed Hypotheses H 0 : Prob. distributions for 2 sampled populations are identical. H A : Prob. distribution for Population A shifted to right of distribution for Population B. (Note: could be to the left, but must be one or the other, not both.)

8. 8 Wilcoxon 1b. Small samples, independent groups Wilcoxon – Two-tailed Hypotheses H 0 : Prob. distributions for 2 sampled populations are identical. H A : Prob. distribution for Population A shifted to right or left of distribution for Population B.

9. 9 Wilcoxon 1b. Small samples, independent groups Wilcoxon – Rejection region: (With Sample taken from Population A being smaller than sample for Population B) – reject H 0 if T A ≥ T U or T A ≤ T L

10. 10 Wilcoxon 1b. Small samples, independent groups Wilcoxon for n 1 ≥ 10 and n 2 ≥ 10: Test statistic: Z = T A – n 1 (n 1 + n 2 + 1) 2 n 1 n 2 (n 1 + n 2 + 1) 12

11. 11 Wilcoxon Wilcoxon for n 1 ≥ 10 and n 2 ≥ 10 Rejection region: One-tailedTwo-tailed Z > Z α │Z│ > Z α/2 Note: use this only when n 1 ≥ 10 and n 2 ≥ 10

12. 12 Wilcoxon Example 1 These are small samples, and they are independent (“random samples of Cajun and Creole dishes”). Therefore, we have to begin with the test of equality of variances.

13. 13 Wilcoxon Test of hypothesis of equal variances H 0 :  1 2 =  2 2 H A :  1 2 ≠  2 2 Test statistic:F =S 1 2 S 2 2 Rej. region:F > F α/2 = F (6,6,.025) = 5.82 or F < (1/5.82) =.172

14. 14 Wilcoxon Test of hypothesis of equal variances S 2 Cajun = (385.27) 2 = 148432.14 S 2 Creole = (1027.54) 2 = 1055833.33 F obt = 148432.14= 7.11 1055833.33 Reject H 0 – variances are not equal, so we do the Wilcoxon.

15. 15 Wilcoxon Example 1 – Wilcoxon Rank Sum Test H 0 : Prob. distributions for Cajun and Creole populations are identical. H A : Prob. distribution for Cajun is shifted to right of distribution for Creole. Statistical test:T

16. 16 Wilcoxon Example 1 – Wilcoxon Rank Sum Test Rejection region: Reject H 0 if T Cajun > 66 (or if T Creole < 39) (Note: We shall give lower heat values lower rank values)

17. 17 Wilcoxon Example 1 – Wilcoxon Rank Sum Test CajunCreole 35003100 42004700 41002700 47003500 42002000 37053100 41001550 1 2 3 4.5 6.5 8 9.5 11.5 13.5 Σ 70 35

18. 18 Wilcoxon Example 1 – Wilcoxon Rank Sum Test Calculation check: Sum of the ranks should = (n) (n+1) 2 70 + 35 = 105 = (14)(15) 2

19. 19 Wilcoxon Example 1 – Wilcoxon Rank Sum Test T Cajun = 70 > 66(and T Creole = 35 < 39) Therefore, reject H0 – Cajun dishes are significantly hotter than Creole dishes.

20. 20 Wilcoxon Example 2 – Wilcoxon Rank Sum Test H 0 :  1 2 =  2 2 H A :  1 2 ≠  2 2 Test statistic:F =S 1 2 S 2 2 Rej. region:F > F α/2 = F (7,8,.025) = 4.53 or F < (1/4.90) =.204

21. 21 Wilcoxon Example 2 – Wilcoxon Rank Sum Test F obt = 4.316 = 9.38.46 Reject H 0 – do Wilcoxon

22. 22 Wilcoxon Example 2 – Wilcoxon Rank Sum Test H 0 : Prob. distributions for females and males populations are identical. H A : Prob. distribution for females is shifted to left of distribution for males. Statistical test:T Rejection region: T ♂ > T U = 90 (or T ♀ < T L = 54)

23. 23 Wilcoxon Example 2 – Wilcoxon Rank Sum Test 6.4162.73 1.713.910 3.254.612 5.9153.04 2.023.46.5 3.684.111 5.4143.46.5 7.2174.713 3.89 Σ7875

24. 24 Wilcoxon Example 2 – Wilcoxon Rank Sum Test T ♂ = 78 < T U = 90 Therefore, do not reject H 0 – no evidence that mean distance in females is less than that in males.

25. 25 Wilcoxon Example 3 – Wilcoxon Rank Sum Test H 0 :  1 2 =  2 2 H A :  1 2 ≠  2 2 Test statistic:F =S 1 2 S 2 2 Rej. region:F > F α/2 = F (5,5,.025) = 7.15 or F < (1/7.15) =.140

26. 26 Wilcoxon Example 3 – Wilcoxon Rank Sum Test F obt = (7.563) 2 = 57.20 (2.04) 2 4.16 = 13.74 Reject H 0 – do Wilcoxon

27. 27 Wilcoxon Example 3 – Wilcoxon Rank Sum Test H 0 : Prob. distributions for Hoodoo and Mukluk populations are identical. H A : Prob. distribution for Hoodoos is shifted to right or left of distribution for Mukluks. Statistical test: T Rejection region: T H > 52 or < 26

28. 28 Wilcoxon Example 3 – Wilcoxon Rank Sum Test HoodooMukluk 2165 6589.5 42.577.5 23121011 77.589.5 6542.5 Σ3345

29. 29 Wilcoxon Example 3 – Wilcoxon Rank Sum Test Check: T H + T M = 78 (12)(13) = 78 2 T H = 33 > 26 and < 52 Do not reject H 0 – no evidence for a significant difference between teams.