1. Chi-Square and F Distributions Chapter 11 Understandable Statistics Ninth Edition By Brase and Brase Prepared by Yixun Shi Bloomsburg University of Pennsylvania

2. Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.11 | 2 The Chi-Square Distribution The χ 2 distribution has the following features: –All possible values are positive. –The distribution is determined solely by the degrees of freedom. –The graph of the distribution is skewed right, but as the degrees of freedom increase, the distribution becomes more bell-shaped. –The mode of the distribution is at n – 2 (for sample sizes greater than 3).

3. Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.11 | 3 The Chi-Square Distribution

4. Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.11 | 4 Table 7 in Appendix II The table gives critical values for area that falls to the right of the critical value.

5. Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.11 | 5 Chi-Square: Test of Independence The goal of the test is to determine if one qualitative variable is independent of another qualitative variable. The hypotheses of the test: H 0 : The variables are independent H 1 : The variables are not independent

6. Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.11 | 6 Chi-Square: Test of Independence The data will be presented in a contingency table in which the rows will represent one variable and the columns will represent the other variable.

7. Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.11 | 7 Chi-Square: Test of Independence First, we need to compute the expected frequency, E, in each cell.

8. Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.11 | 8 Chi-Square: Test of Independence The sample statistic for the test will be: O = the observed frequency in each cell E = the expected frequency in each cell n = the total sample size

9. Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.11 | 9 Chi-Square: Test of Independence To use Table 7 to estimate the P-value of the test, the degrees of freedom are:

10. Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.11 | 10 Review of Chi-Square Test for Independence

11. Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.11 | 11 Review of Chi-Square Test for Independence

12. Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.11 | 12 Review of Chi-Square Test for Independence

13. Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.11 | 13 Test of Homogeneity A test of homogeneity tests the claim that different populations share the same proportions of specified characteristics. Test of homogeneity is also conducted using contingency tables and the chi-square distribution.

14. Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.11 | 14 Test of Homogeneity Obtain random samples from each of the population. For each population, determine the numbers that share a distinct specified characteristic. Make a contingency table with the different populations as the rows (or columns) and the characteristics as the columns (or rows). The values recorded in the cells of the table are the observed value O taken from the samples. 1.Set the level of significance and use the hypotheses H 0 : The proportion of each population sharing specified characteristics is the same for all populations. H 1 : The proportion of each population sharing specified characteristics is not the same for all populations. 2. Follow steps 2—5 of the procedure used to test for independence.

15. Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.11 | 15 Chi-Square: Goodness of Fit We will test whether a given data set “fits” a given distribution. H 0 : The population fits the given distribution. H 1 : The population has a different distribution.

16. Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.11 | 16 Chi-Square: Goodness of Fit

17. Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.11 | 17 Chi-Square: Goodness of Fit

18. Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.11 | 18 Chi-Square: Goodness of Fit

19. Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.11 | 19 Testing σ 2 If we have a normal population with variance σ 2 and a random sample of n measurements with sample variance s 2, then has a chi-square distribution with n – 1 degrees of freedom.

20. Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.11 | 20 Working with the Chi-Square Distribution Use Table 7 in Appendix II.

21. Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.11 | 21 P-Values for Chi-Square Tests

22. Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.11 | 22 P-Values for Chi-Square Tests

23. Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.11 | 23 P-Values for Chi-Square Tests

24. Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.11 | 24 Test Procedure for σ 2

25. Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.11 | 25 Test Procedure for σ 2

26. Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.11 | 26 Confidence Intervals for σ 2 Confidence Intervals for σ

27. Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.11 | 27 Confidence Intervals for σ 2 or σ

28. Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.11 | 28 Using the Chi-Square Distribution for C.I.

29. Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.11 | 29 Testing Two Variances Assumptions: –The two populations are independent of each other. –Both populations have a normal distribution (not necessarily the same).

30. Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.11 | 30 Notation We choose population 1 to have the larger sample variance, i.e. s 2 1 ≥ s 2 2.

31. Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.11 | 31 More Notation

32. Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.11 | 32 Compute the Test Statistic We will compare the test statistic to an F distribution, found in Table 8 of Appendix II.

33. Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.11 | 33 A Typical F Distribution

34. Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.11 | 34 Properties of the F Distribution

35. Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.11 | 35 Testing Two Variances Using F Distribution Use the F-distribution and the type of test to find or estimate the P-value with Table 8 of Appendix II. Conclude the test. If P-value, then reject H 0. Otherwise do not reject H 0. Interpret your conclusion in the context of the application.

36. Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.11 | 36 Using Table 8 Estimate the P-Value for F = 55.2 with d.f. N = 3 and d.f. D = 2

37. Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.11 | 37 One-Way ANOVA ANOVA is a method of comparing the means of multiple populations at once instead of completing a series of 2-population tests.

38. Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.11 | 38 ANOVA Assumptions

39. Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.11 | 39 Establishing the Hypotheses in ANOVA In ANOVA, there are k groups and k group means. The general problem is to determine if there exists a difference among the group means.

40. Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.11 | 40 One-Way ANOVA Procedure

41. Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.11 | 41 One-Way ANOVA Procedure

42. Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.11 | 42 One-Way ANOVA Procedure

43. Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.11 | 43 One-Way ANOVA Procedure

44. Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.11 | 44 One-Way ANOVA Procedure

45. Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.11 | 45 One-Way ANOVA Procedure

46. Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.11 | 46 Two-Way ANOVA Two-Way ANOVA is a statistical technique to study two variables simultaneously. Each variable is called a factor. Each factor can have multiple levels. We can study the different means of the factors as well as the interaction between the factors.

47. Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.11 | 47 Two-Way ANOVA Assumptions

48. Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.11 | 48 Steps For Two-Way ANOVA 1)Establish the hypotheses. 2)Compute the Sums of Squares Values. 3)Compute the Mean Squares Values. 4)Compute the F Statistic for each factor and the interaction. 5)Conclude the test.

49. Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.11 | 49 Two-Way ANOVA Hypotheses

50. Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.11 | 50 Compute the Mean Squares

51. Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.11 | 51 Compute the F Statistics

52. Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.11 | 52 Concluding the Test Check the P-Values on the computer output. Always perform the test for interaction first. If you reject the test for interaction, you should not test for a row or column variable effect.

53. Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.11 | 53 Experimental Design Completely Randomized Design: Independent random samples determine the individuals or objects for each treatment group. Block Design: One of the factors is predetermined and not randomized.