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

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. Testing Two Variances.

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

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  1. Understandable StatisticsSeventh EditionBy Brase and BrasePrepared by: Lynn SmithGloucester County College Chapter Eleven Part 3 (Sections 11. 4 & 11.5) Chi-Square and F Distributions

  2. Testing Two Variances Use independent samples from two populations to test the claim that the variances are equal.

  3. Assumptions for Testing Two Variances • The two populations are independent • The two populations each have a normal probability distribution.

  4. Notations Used:

  5. Define population I as the population with the larger (or equal) sample variance

  6. Set Up Hypotheses

  7. Set Up Hypotheses

  8. Equivalent hypotheses may be stated about standard deviations.

  9. Use the F Statistic

  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. An F Distribution

  12. Degrees of Freedom for Test of Two Variances • Degrees of freedom for the numerator = d.f. N = n1 - 1 • Degrees of freedom for the denominator = d.f. D = n2 - 1

  13. Values of the F Distribution Given in Table 8 of Appendix II

  14. Some Values of the F Distribution

  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. F Distributiond.f.N = 3, d.f.D = 5,  = 0.025

  17. Testing Two Variances

  18. Assume we have the following data and wish to test the claim that the population variances are not equal.

  19. Hypotheses

  20. The Sample Test Statistic

  21. Degrees of Freedom for Test of Two Variances • Degrees of freedom for the numerator = d.f. N = n1 - 1 = 9 - 1 = 8 • Degrees of freedom for the denominator = d.f. D = n2 - 1 = 10 - 1 = 9

  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. Critical Value of F: Two-Tailed Test Area = /2 F = 4.10

  24. Our Test Statistic Does not fall in the Critical Region Area = /2 F = 1.108 F = 4.10

  25. Conclusion At 5% level of significance, we cannot reject the claim that the variances are the same.

  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. P Value Conclusion For a two-tailed test, double the area in the right tail. Therefore P is greater than 0.100.

  28. Analysis of Variance A technique used to determine if there are differences among means for several groups.

  29. One Way Analysis of Variance Groups are based on values of only one variable.

  30. ANOVA Analysis of Variance

  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. Purpose of ANOVA To determine the existence (or nonexistence) of a statistically significant difference among the group means

  33. Null Hypothesis All the group populations are the same. All sample groups come from the same population.

  34. Alternate Hypothesis Not all the group populations are equal.

  35. Hypotheses • H0: 1 = 2 = . . . = k • H1: At least two of the means 1, 2, . . . , k are not equal.

  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. 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. Data for three groups:

  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. Population Means Let 1, 2, and 3 represent the population means of groups 1, 2, and 3.

  41. Hypotheses and Level of Significance • H0: 1 = 2 = 3 • H1: At least two of the means 1, 2, and 3 are not equal. Use  = 0.05.

  42. Find SS TOT= the sum of the squares of entire collection of data

  43. Find SS TOT= the sum of the squares of entire collection of data

  44. Find SS TOT= the sum of the squares of entire collection of data

  45. Find SS TOT= the sum of the squares of entire collection of data

  46. Squares and Sums

  47. Finding SS TOT

  48. Find SS TOT

  49. Find SS BET

  50. Finding SS BET

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