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 3 (Sections 11. 4 & 11.5) Chi-Square and F Distributions

2. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 2 Testing Two Variances Use independent samples from two populations to test the claim that the variances are equal.

3. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 3 Assumptions for Testing Two Variances The two populations are independent The two populations each have a normal probability distribution.

4. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 4 Notations Used:

5. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 5 Define population I as the population with the larger (or equal) sample variance

6. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 6 Set Up Hypotheses

7. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 7 Set Up Hypotheses

8. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 8 Equivalent hypotheses may be stated about standard deviations.

9. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 9 Use the F Statistic

10. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 10 The F Distribution Not symmetrical Skewed right Values are always greater than or equal to zero. A specific F distribution is determined from two degrees of freedom.

11. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 11 An F Distribution

12. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 12 Degrees of Freedom for Test of Two Variances Degrees of freedom for the numerator = d.f. N = n 1 - 1 Degrees of freedom for the denominator = d.f. D = n 2 - 1

13. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 13 Values of the F Distribution Given in Table 8 of Appendix II

14. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 14 Some Values of the F Distribution

15. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 15 Find critical value of F from Table 8 Appendix II d.f. N = 3 d.f. D = 5 Right tail area =  = 0.025

16. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 16 F Distribution d.f. N = 3, d.f. D = 5,  = 0.025

17. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 17 Testing Two Variances

18. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 18 Assume we have the following data and wish to test the claim that the population variances are not equal.

19. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 19 Hypotheses

20. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 20 The Sample Test Statistic

21. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 21 Degrees of Freedom for Test of Two Variances Degrees of freedom for the numerator = d.f. N = n 1 - 1 = 9 - 1 = 8 Degrees of freedom for the denominator = d.f. D = n 2 - 1 = 10 - 1 = 9

22. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 22 Critical Values of F Distribution Use  = 0.05 For a two-tailed test, the area in the right tail of the distribution should be  /2 = 0.025. With d.f. N = 8 and d.f. D = 9 the critical value of F is 4.10.

23. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 23 Critical Value of F: Two-Tailed Test F = 4.10 Area =  /2

24. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 24 Our Test Statistic Does not fall in the Critical Region F = 4.10 Area =  /2 F = 1.108

25. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 25 Conclusion At 5% level of significance, we cannot reject the claim that the variances are the same.

26. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 26 P Value Approach Our sample test statistic was F = 1.108 Looking in the block of entries in table 8 where d.f. N = 8 and d.f. D = 9, we find entries ranging from 3.23 to 5.47 for  ranging from 0.050 to 0.010. F = 1.108 is less than even the smallest of these results.

27. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 27 P Value Conclusion For a two-tailed test, double the area in the right tail. Therefore P is greater than 0.100.

28. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 28 Analysis of Variance A technique used to determine if there are differences among means for several groups.

29. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 29 One Way Analysis of Variance Groups are based on values of only one variable.

30. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 30 ANOVA Analysis of Variance

31. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 31 Assumptions for ANOVA Each of k groups of measurements is from a normal population. Each group is randomly selected and is independent of all other groups. Variables from each group come from distributions with approximately the same standard deviation.

32. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 32 Purpose of ANOVA To determine the existence (or nonexistence) of a statistically significant difference among the group means

33. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 33 Null Hypothesis All the group populations are the same. All sample groups come from the same population.

34. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 34 Alternate Hypothesis Not all the group populations are equal.

35. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 35 Hypotheses H 0 :  1 =  2 =... =  k H 1 :At least two of the means  1,  2,...,  k are not equal.

36. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 36 Steps in ANOVA Determine null and alternate hypotheses Find SS TOT = the sum of the squares of entire collection of data Find SS BET which measures variability between groups Find SS W which measures variability within groups

37. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 37 Steps in ANOVA Find the variance estimates within groups: MS W Find the variance estimates between groups: MS BET Find the F ratio and complete the ANOVA test

38. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 38 Data for three groups:

39. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 39 Sample sizes The sample sizes for the groups may be the same or different from one another. In our example, each sample has four items.

40. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 40 Population Means Let  1,  2, and  3 represent the population means of groups 1, 2, and 3.

41. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 41 Hypotheses and Level of Significance H 0 :  1 =  2 =  3 H 1 :At least two of the means  1,  2, and  3 are not equal. Use  = 0.05.

42. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 42 Find SS TOT = the sum of the squares of entire collection of data

43. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 43 Find SS TOT = the sum of the squares of entire collection of data

44. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 44 Find SS TOT = the sum of the squares of entire collection of data

45. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 45 Find SS TOT = the sum of the squares of entire collection of data

46. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 46 Squares and Sums

47. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 47 Finding SS TOT

48. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 48 Find SS TOT

49. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 49 Find SS BET

50. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 50 Finding SS BET

51. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 51 Variability Within ith Group

52. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 52 Variation Within First Group

53. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 53 Variation Within Second Group

54. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 54 Variation Within Third Group

55. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 55 Variability Within All Groups SS W = SS 1 + SS 2 + SS 3

56. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 56 Finding SS W SS W = SS 1 + SS 2 + SS 3 = 4.75 + 2.00 + 2.75 = 9.50

57. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 57 Note: SS TOT = SS BET + SS W

58. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 58 We can check: SS TOT = SS BET + SS W = 94 = 84.5 + 9.5

59. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 59 Degrees of Freedom Degrees of freedom between groups = d.f. BET = k - 1, where k is the number of groups. Degrees of freedom within groups = d.f. W = N - k, where N is the total sample size. d.f. BET + d.f. W = N - 1.

60. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 60 Variance Estimate Between Groups Mean Square Between Groups =

61. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 61 Variance Estimate Within Groups Mean Square Within Groups =

62. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 62 Mean Square Between Groups

63. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 63 Mean Square Within Groups

64. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 64 Find the F Ratio

65. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 65 Finding the F Ratio

66. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 66 Null Hypothesis: All the groups are samples from the same distribution If H 0 is true MS BET and MS W would estimate the same quantity. The F ratio should be approximately equal to one.

67. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 67 Using Table 8, Appendix II d.f. BET = number of groups - 1 = d.f. for numerator. d.f. W = total sample size - number of groups = d.f. for denominator. Rejection region is the right tail of the distribution.

68. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 68 Using Table 8, Appendix II d.f. BET = number of groups - 1 = d.f. for numerator = 2. d.f. W = total sample size - number of groups = d.f. for denominator = 9.  = 0.05. Critical F value = 4.26

69. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 69 Conclusion Since our observed value of F (40.009) is greater than the critical F value (4.26) we reject the null hypothesis. We conclude that not all of the means are equal.

70. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 70 P Value Approach For d.f. BET = d.f. for numerator = 2 and d.f. W = d.f. for denominator = 9, our observed F value (42.009) exceeds even the largest critical F value. Thus P < 0.01. We conclude that we would reject the null hypothesis for any   0.01.