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8 - 1 © 2003 Pearson Prentice Hall Chi-Square ( 2 ) Test of Variance
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8 - 2 © 2003 Pearson Prentice Hall 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
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8 - 3 © 2003 Pearson Prentice Hall 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 4.Test Statistic Hypothesized Pop. Variance Sample Variance 2 2 2 1) (nS 0
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8 - 4 © 2003 Pearson Prentice Hall Chi-Square ( 2 ) Distribution
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8 - 5 © 2003 Pearson Prentice Hall Finding Critical Value Example What is the critical 2 value given: H a : 2 > 0.7 n = 3 =.05?
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8 - 6 © 2003 Pearson Prentice Hall Finding Critical Value Example 2 Table (Portion) What is the critical 2 value given: H a : 2 > 0.7 n = 3 =.05?
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8 - 7 © 2003 Pearson Prentice Hall Finding Critical Value Example =.05 2 Table (Portion) What is the critical 2 value given: H a : 2 > 0.7 n = 3 =.05?
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8 - 8 © 2003 Pearson Prentice Hall Finding Critical Value Example =.05 2 Table (Portion) What is the critical 2 value given: H a : 2 > 0.7 n = 3 =.05?
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8 - 9 © 2003 Pearson Prentice Hall Finding Critical Value Example =.05 2 Table (Portion) What is the critical 2 value given: H a : 2 > 0.7 n = 3 =.05?
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8 - 10 © 2003 Pearson Prentice Hall Finding Critical Value Example =.05 2 Table (Portion) What is the critical 2 value given: H a : 2 > 0.7 n = 3 =.05?
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8 - 11 © 2003 Pearson Prentice Hall Finding Critical Value Example =.05 2 Table (Portion) df= n - 1 = 2 What is the critical 2 value given: H a : 2 > 0.7 n = 3 =.05?
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8 - 12 © 2003 Pearson Prentice Hall Finding Critical Value Example =.05 2 Table (Portion) df= n - 1 = 2 What is the critical 2 value given: H a : 2 > 0.7 n = 3 =.05?
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8 - 13 © 2003 Pearson Prentice Hall Finding Critical Value Example =.05 2 Table (Portion) df= n - 1 = 2 What is the critical 2 value given: H a : 2 > 0.7 n = 3 =.05?
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8 - 14 © 2003 Pearson Prentice Hall Finding Critical Value Example =.05 2 Table (Portion) df= n - 1 = 2 What is the critical 2 value given: H a : 2 > 0.7 n = 3 =.05?
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8 - 15 © 2003 Pearson Prentice Hall Finding Critical Value Example =.05 2 Table (Portion) df= n - 1 = 2 What is the critical 2 value given: H a : 2 > 0.7 n = 3 =.05?
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8 - 16 © 2003 Pearson Prentice Hall Finding Critical Value Example What Do You Do If the Rejection Region Is on the Left? What is the critical 2 value given: H a : 2 < 0.7 n = 3 =.05?
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8 - 17 © 2003 Pearson Prentice Hall Finding Critical Value Example 2 Table (Portion) What is the critical 2 value given: H a : 2 < 0.7 n = 3 =.05?
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8 - 18 © 2003 Pearson Prentice Hall What is the critical 2 value given: H a : 2 < 0.7 n = 3 =.05? Finding Critical Value Example =.05 2 Table (Portion) Reject
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8 - 19 © 2003 Pearson Prentice Hall What is the critical 2 value given: H a : 2 < 0.7 n = 3 =.05? Finding Critical Value Example =.05 2 Table (Portion) Reject Upper Tail Area for Lower Critical Value = 1-.05 =.95
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8 - 20 © 2003 Pearson Prentice Hall What is the critical 2 value given: H a : 2 < 0.7 n = 3 =.05? Finding Critical Value Example =.05 2 Table (Portion) Reject Upper Tail Area for Lower Critical Value = 1-.05 =.95
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8 - 21 © 2003 Pearson Prentice Hall What is the critical 2 value given: H a : 2 < 0.7 n = 3 =.05? Finding Critical Value Example =.05 2 Table (Portion) Reject Upper Tail Area for Lower Critical Value = 1-.05 =.95 df= n - 1 = 2
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8 - 22 © 2003 Pearson Prentice Hall What is the critical 2 value given: H a : 2 < 0.7 n = 3 =.05? Finding Critical Value Example =.05 2 Table (Portion) Reject Upper Tail Area for Lower Critical Value = 1-.05 =.95 df= n - 1 = 2
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8 - 23 © 2003 Pearson Prentice Hall What is the critical 2 value given: H a : 2 < 0.7 n = 3 =.05? Finding Critical Value Example =.05 2 Table (Portion) df= n - 1 = 2 Upper Tail Area for Lower Critical Value = 1-.05 =.95 Reject
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8 - 24 © 2003 Pearson Prentice Hall 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 of 17.7 grams. Test at the.05 level.
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8 - 25 © 2003 Pearson Prentice Hall 2 0 Chi-Square ( 2 ) Test Solution H 0 : H a : = df = Critical Value(s): Test Statistic: Decision:Conclusion:
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8 - 26 © 2003 Pearson Prentice Hall 2 0 Chi-Square ( 2 ) Test Solution H 0 : 2 = 15 H a : 2 15 = df = Critical Value(s): Test Statistic: Decision:Conclusion:
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8 - 27 © 2003 Pearson Prentice Hall 2 0 Chi-Square ( 2 ) Test Solution H 0 : 2 = 15 H a : 2 15 =.05 df = 25 - 1 = 24 Critical Value(s): Test Statistic: Decision:Conclusion:
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8 - 28 © 2003 Pearson Prentice Hall 2 0 Chi-Square ( 2 ) Test Solution H 0 : 2 = 15 H a : 2 15 =.05 df = 25 - 1 = 24 Critical Value(s): Test Statistic: Decision:Conclusion: /2 =.025
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8 - 29 © 2003 Pearson Prentice Hall 2 039.36412.401 Chi-Square ( 2 ) Test Solution H 0 : 2 = 15 H a : 2 15 =.05 df = 25 - 1 = 24 Critical Value(s): Test Statistic: Decision:Conclusion: /2 =.025
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8 - 30 © 2003 Pearson Prentice Hall 2 039.36412.401 Chi-Square ( 2 ) Test Solution H 0 : 2 = 15 H a : 2 15 =.05 df = 25 - 1 = 24 Critical Value(s): Test Statistic: Decision:Conclusion: /2 =.025 2 2 2 2 2 1) (25 - 1)177 15 3342 (nS 0..
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8 - 31 © 2003 Pearson Prentice Hall 2 039.36412.401 Chi-Square ( 2 ) Test Solution H 0 : 2 = 15 H a : 2 15 =.05 df = 25 - 1 = 24 Critical Value(s): Test Statistic: Decision:Conclusion: Do Not Reject at =.05 /2 =.025 2 2 2 2 2 1) (25 - 1)177 15 3342 (nS 0..
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8 - 32 © 2003 Pearson Prentice Hall 2 039.36412.401 Chi-Square ( 2 ) Test Solution H 0 : 2 = 15 H a : 2 15 =.05 df = 25 - 1 = 24 Critical Value(s): Test Statistic: Decision:Conclusion: Do Not Reject at =.05 There Is No Evidence 2 Is Not 15 /2 =.025 2 2 2 2 2 1) (25 - 1)177 15 3342 (nS 0..
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8 - 33 © 2003 Pearson Prentice Hall Calculating Type II Error Probabilities
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8 - 34 © 2003 Pearson Prentice Hall Power of Test 1.Probability of Rejecting False H 0 Correct Decision Correct Decision 2.Designated 1 - 3.Used in Determining Test Adequacy 4.Affected by True Value of Population Parameter True Value of Population Parameter Significance Level Significance Level Standard Deviation & Sample Size n Standard Deviation & Sample Size n
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8 - 35 © 2003 Pearson Prentice Hall X 0 = 368 = 368 Reject Do Not Reject Finding Power Step 1 Hypothesis: H 0 : 0 368 H 1 : 0 < 368 =.05 n = 15/ 25 Draw
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8 - 36 © 2003 Pearson Prentice Hall X 1 = 360 = 360 X 0 = 368 = 368 Reject Do Not Reject Finding Power Steps 2 & 3 Hypothesis: H 0 : 0 368 H 1 : 0 < 368 ‘True’ Situation: 1 = 360 =.05 n = 15/ 25 Draw Draw Specify 1-
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8 - 37 © 2003 Pearson Prentice Hall X 1 = 360 = 360363.065 X 0 = 368 = 368 Reject Do Not Reject Finding Power Step 4 Hypothesis: H 0 : 0 368 H 1 : 0 < 368 ‘True’ Situation: 1 = 360 =.05 n = 15/ 25 Draw Draw Specify
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8 - 38 © 2003 Pearson Prentice Hall X 1 = 360 = 360363.065 X 0 = 368 = 368 Reject Do Not Reject Finding Power Step 5 Hypothesis: H 0 : 0 368 H 1 : 0 < 368 ‘True’ Situation: 1 = 360 =.05 n = 15/ 25 =.154 1- =.846 Draw Draw Specify Z Table
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8 - 39 © 2003 Pearson Prentice Hall Power Curves PowerPower Power Possible True Values for 1 H 0 : 0 H 0 : 0 H 0 : = 0 = 368 in Example
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8 - 40 © 2003 Pearson Prentice Hall 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
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