1. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 1 Understandable Statistics S eventh Edition By Brase and Brase Prepared by: Lynn Smith Gloucester County College Chapter Eleven Part 1 (Section 11.1) Chi-Square and F Distributions

2. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 2 The Chi-Square Distribution The  2 Distribution is not symmetrical and depends on the number of degrees of freedom.

3. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 3  is the Greek Letter Chi.

4. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 4 The  2 Distribution for d.f. = 3 1 2 3 4 5 6 7 8 910n

5. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 5 The  2 Distribution for d.f. = 5 1 2 3 4 5 6 7 8 910n d.f. = 3

6. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 6 The  2 Distribution for d.f. = 10 1 2 3 4 5 6 7 8 910n d.f. = 3 d.f. = 5

7. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 7 The mode or high point occurs over n – 2 for n  3. 1 2 3 4 5 6 7 8 910n d.f. = 10 d.f. = 3 d.f. = 5

8. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 8 As the degrees of freedom increase, the graphs looks more bell-like and symmetric. 1 2 3 4 5 6 7 8 910n d.f. = 3 d.f. = 10 d.f. = 5

9. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 9 Use Table 7 in Appendix II to find Critical Values of  2 Distributions

10. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 10 Area in the Right Tail of the Distribution =   22

11. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 11 Use Table 7 (with d.f. = 8) to find the area to the right of  2 = 2.73.  =

12. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 12 Chi Square: Tests of Independence To test the independence of two factors, use a contingency table.

13. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 13 Contingency Table

14. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 14 Shaded boxes (called “cells”) will contain frequencies.

15. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 15 Horizontal lines of cells are called rows.

16. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 16 Vertical lines of cells are called columns.

17. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 17 The size of a table is given as row X column.

18. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 18 This is a 3 X 3 contingency table.

19. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 19 This is a 3 X 2 contingency table.

20. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 20 This is a 2 X 3 contingency table.

21. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 21 When giving the size of a contingency table, Always give the number of rows first.

22. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 22 Suppose we wish to determine (at 5% level of significance) if the time it takes to complete a given task is independent of gender.

23. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 23 Number and gender of individuals who completed a task in the times indicated.

24. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 24 To test the null hypothesis that gender and the time it takes to complete the task are independent: H 0 :Variables are independent. H 1 :Variables are not independent. Use the null hypothesis to determine the expected frequency of each cell.

25. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 25 Expected Frequency

26. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 26 Finding the Expected Frequencies E = (Row total)(Column total) Sample size Sample size

27. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 27 Finding the Expected Frequencies E = (Row total)(Column total) sample size Sample size

28. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 28 Finding the Expected Frequencies E = (Row total)(Column total) sample size

29. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 29 Finding the Expected Frequencies E = (Row total)(Column total) sample size

30. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 30 Finding the Expected Frequencies E = (Row total)(Column total) sample size

31. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 31 Finding the Expected Frequencies E = (Row total)(Column total) sample size

32. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 32 Finding the Expected Frequencies E = (Row total)(Column total) sample size

33. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 33 The actual frequency which occurred is called the observed frequency, O.

34. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 34 The Sample Statistic  2 Chi square is a measure of the sum of the differences between observed frequency O and expected frequency E in each cell.

35. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 35 Difference Between Observed and Expected Frequencies

36. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 36 The Sum of the (O – E) Column Will Equal Zero.

37. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 37 To calculate Chi Square, we use the values (O – E) 2 /E To reflect the magnitude of the differences between the observed and expected frequencies. To reflect the fact that the small difference between the observed and expected frequencies is more important when the expected frequency is small.

38. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 38 Computing  2

39. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 39 Degrees of Freedom d.f. = (R – 1)(C – 1) R = number of cell rows C = number of cell columns

40. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 40 For our example: R = 2, C = 3 d.f. = (2 – 1)(3 – 1) = 2

41. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 41 Using d.f. = 2 and  = 0.05, find the critical value of  2 from Table 7.

42. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 42 If the sample statistic is larger than the critical value, reject the null hypothesis of independence. In our example, the sample statistic  2 = 10.36. The critical value = 5.99.

43. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 43 Conclusion Reject the null hypothesis of independence. We conclude that the time it takes to complete the task is not independent of gender.

44. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 44 P Value Approach In our example, the sample statistic  2 = 10.36. For d.f. = 2, the sample statistic  2 = 10.36 falls between 9.21 and 10.60 (the critical values for  =.010 and.005 respectively). We conclude that 0.005 < P < 0.010. We would reject H 0 for any   P. We, therefore reject H 0 for  = 0.05.

45. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 45 In order to safely use the critical values of  2 from Table 7, we must assure that all expected frequencies are greater than or equal to five. If this condition is not met, the sample size should be increased.

46. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 46 Using Chi-Square Distribution to Test the Independence of Two Variables Set up the hypotheses H 0 :The variables are independent. H 1 :The variables are not independent. Compute the expected frequency for each cell in the contingency table.

47. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 47 Using Chi-Square Distribution to Test the Independence of Two Variables Compute the statistic  2 for the sample.

48. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 48 Using Chi-Square Distribution to Test the Independence of Two Variables Find the critical value   2 in Table 7. Use the level of significance  and degrees of freedom: d.f. = (R – 1)(C – 1) where R and C are the numbers of rows and columns of cells. The critical region = all values of  2 to the right of the critical value   2.

49. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 49 Using Chi-Square Distribution to Test the Independence of Two Variables Compare the sample statistic  2 with the critical value   2. If the sample statistic is larger than the critical value, reject the null hypothesis of independence. Otherwise, do not reject the null hypothesis.