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Chi-Square (  2 ) Test of Variance

Chi-Square (  2 ) Test of Variance. Chi-Square (  2 ) Test for Variance. 1. Tests One Population Variance or Standard Deviation 2. Assumes Population Is Approximately Normally Distributed 3. Null Hypothesis Is H 0 :  2 =  0 2. Chi-Square (  2 ) Test for Variance.

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Chi-Square (  2 ) Test of Variance

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  1. Chi-Square (2) Test of Variance

  2. Chi-Square (2) Testfor Variance • 1. Tests One Population Variance or Standard Deviation • 2. Assumes Population Is Approximately Normally Distributed • 3. Null Hypothesis Is H0: 2 = 02

  3. Chi-Square (2) Testfor Variance • 1. Tests One Population Variance or Standard Deviation • 2. Assumes Population Is Approximately Normally Distributed • 3. Null Hypothesis Is H0: 2 = 02 • 4. Test Statistic 2 (n  1)  S Sample Variance 2   2  Hypothesized Pop. Variance 0

  4. Chi-Square (2) Distribution

  5. Finding Critical Value Example What is the critical 2 value given:Ha: 2 > 0.7n = 3 =.05?

  6. Finding Critical Value Example What is the critical 2 value given:Ha: 2 > 0.7n = 3 =.05? 2 Table (Portion)

  7. Finding Critical Value Example What is the critical 2 value given:Ha: 2 > 0.7n = 3 =.05?  = .05 2 Table (Portion)

  8. Finding Critical Value Example What is the critical 2 value given:Ha: 2 > 0.7n = 3 =.05?  = .05 2 Table (Portion)

  9. Finding Critical Value Example What is the critical 2 value given:Ha: 2 > 0.7n = 3 =.05?  = .05 2 Table (Portion)

  10. Finding Critical Value Example What is the critical 2 value given:Ha: 2 > 0.7n = 3 =.05?  = .05 2 Table (Portion)

  11. Finding Critical Value Example What is the critical 2 value given:Ha: 2 > 0.7n = 3 =.05?  = .05 df = n - 1 = 2 2 Table (Portion)

  12. Finding Critical Value Example What is the critical 2 value given:Ha: 2 > 0.7n = 3 =.05?  = .05 df = n - 1 = 2 2 Table (Portion)

  13. Finding Critical Value Example What is the critical 2 value given:Ha: 2 > 0.7n = 3 =.05?  = .05 df = n - 1 = 2 2 Table (Portion)

  14. Finding Critical Value Example What is the critical 2 value given:Ha: 2 > 0.7n = 3 =.05?  = .05 df = n - 1 = 2 2 Table (Portion)

  15. Finding Critical Value Example What is the critical 2 value given:Ha: 2 > 0.7n = 3 =.05?  = .05 df = n - 1 = 2 2 Table (Portion)

  16. Finding Critical Value Example What is the critical 2 value given:Ha: 2< 0.7n = 3 =.05? What Do You Do If the Rejection Region Is on the Left?

  17. Finding Critical Value Example What is the critical 2 value given:Ha: 2< 0.7n = 3 =.05? 2 Table (Portion)

  18. Finding Critical Value Example What is the critical 2 value given:Ha: 2< 0.7n = 3 =.05? Reject  = .05 2 Table (Portion)

  19. Finding Critical Value Example What is the critical 2 value given:Ha: 2< 0.7n = 3 =.05? Upper Tail Area for Lower Critical Value = 1-.05 = .95 Reject  = .05 2 Table (Portion)

  20. Finding Critical Value Example What is the critical 2 value given:Ha: 2< 0.7n = 3 =.05? Upper Tail Area for Lower Critical Value = 1-.05 = .95 Reject  = .05 2 Table (Portion)

  21. Finding Critical Value Example What is the critical 2 value given:Ha: 2< 0.7n = 3 =.05? Upper Tail Area for Lower Critical Value = 1-.05 = .95 Reject  = .05 df = n - 1 = 2 2 Table (Portion)

  22. Finding Critical Value Example What is the critical 2 value given:Ha: 2< 0.7n = 3 =.05? Upper Tail Area for Lower Critical Value = 1-.05 = .95 Reject  = .05 df = n - 1 = 2 2 Table (Portion)

  23. Finding Critical Value Example What is the critical 2 value given:Ha: 2< 0.7n = 3 =.05? Upper Tail Area for Lower Critical Value = 1-.05 = .95 Reject  = .05 df = n - 1 = 2 2 Table (Portion)

  24. Chi-Square (2) Test Example • Is the variation in boxes of cereal, measured by the variance, equal to 15 grams? A random sample of 25 boxes had a standard deviation of17.7 grams. Test at the .05 level.

  25. Chi-Square (2) Test Solution • H0: • Ha: •  = • df = • Critical Value(s): Test Statistic: Decision: Conclusion: 2  0

  26. Chi-Square (2) Test Solution • H0: 2 = 15 • Ha: 2 15 •  = • df = • Critical Value(s): Test Statistic: Decision: Conclusion: 2  0

  27. Chi-Square (2) Test Solution • H0: 2 = 15 • Ha: 2 15 •  = .05 • df = 25 - 1 = 24 • Critical Value(s): Test Statistic: Decision: Conclusion: 2  0

  28. Chi-Square (2) Test Solution • H0: 2 = 15 • Ha: 2 15 •  = .05 • df = 25 - 1 = 24 • Critical Value(s): Test Statistic: Decision: Conclusion:  /2 = .025 2  0

  29. Chi-Square (2) Test Solution • H0: 2 = 15 • Ha: 2 15 •  = .05 • df = 25 - 1 = 24 • Critical Value(s): Test Statistic: Decision: Conclusion:  /2 = .025 2  0 12.401 39.364

  30. Chi-Square (2) Test Solution • H0: 2 = 15 • Ha: 2 15 •  = .05 • df = 25 - 1 = 24 • Critical Value(s): Test Statistic: Decision: Conclusion: 2 2 (n  1)  S (25 - 1)  17 . 7 2    2 2 15  0  33 . 42  /2 = .025 2  0 12.401 39.364

  31. Chi-Square (2) Test Solution • H0: 2 = 15 • Ha: 2 15 •  = .05 • df = 25 - 1 = 24 • Critical Value(s): Test Statistic: Decision: Conclusion: 2 2 (n  1)  S (25 - 1)  17 . 7 2    2 2 15  0  33 . 42 Do Not Reject at  = .05  /2 = .025 2  0 12.401 39.364

  32. Chi-Square (2) Test Solution • H0: 2 = 15 • Ha: 2 15 •  = .05 • df = 25 - 1 = 24 • Critical Value(s): Test Statistic: Decision: Conclusion: 2 2 (n  1)  S (25 - 1)  17 . 7 2    2 2 15  0  33 . 42 Do Not Reject at  = .05  /2 = .025 There Is No Evidence 2 Is Not 15 2  0 12.401 39.364

  33. Calculating Type II Error Probabilities

  34. Power of Test • 1. Probability of Rejecting False H0 • Correct Decision • 2. Designated 1 -  • 3. Used in Determining Test Adequacy • 4. Affected by • True Value of Population Parameter • Significance Level  • Standard Deviation & Sample Size n

  35. Finding PowerStep 1 Reject  n =15/25 Hypothesis:H0: 0 368H1: 0 < 368 Do Not Draw Reject  = .05  = 368  X 0

  36. Finding PowerSteps 2 & 3 Reject  n =15/25 Hypothesis:H0: 0 368H1: 0 < 368 Do Not Draw Reject  = .05  = 368  X 0  ‘True’ Situation:1 = 360 Draw   1- Specify  X  = 360 1

  37. Finding PowerStep 4 Reject  n =15/25 Hypothesis:H0: 0 368H1: 0 < 368 Do Not Draw Reject  = .05  = 368  X 0  ‘True’ Situation:1 = 360 Draw   Specify  X  = 360 363.065 1

  38. Finding PowerStep 5 Reject  n =15/25 Hypothesis:H0: 0 368H1: 0 < 368 Do Not Draw Reject  = .05  = 368  X 0  ‘True’ Situation:1 = 360 Draw  = .154    1- =.846 Specify Z Table  X  = 360 363.065 1

  39. Power Curves H0: 0 H0: 0 Power Power Possible True Values for 1 Possible True Values for 1 H0:  =0 Power  = 368 in Example Possible True Values for 1

  40. Conclusion • 1. Distinguished Types of Hypotheses • 2. Described Hypothesis Testing Process • 3. Explained p-Value Concept • 4. Solved Hypothesis Testing Problems Based on a Single Sample • 5. Explained Power of a Test

  41. End of Chapter Any blank slides that follow are blank intentionally.

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