8 - 1 © 2003 Pearson Prentice Hall Chi-Square (  2 ) Test of Variance

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

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 =  Test Statistic Hypothesized Pop. Variance Sample Variance   )  (nS 0

8 - 4 © 2003 Pearson Prentice Hall Chi-Square (  2 ) Distribution

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?

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?

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?

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?

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?

© 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?

© 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?

© 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?

© 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?

© 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?

© 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?

© 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?

© 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?

© 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

© 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 = =.95

© 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 = =.95

© 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 = =.95 df= n - 1 = 2

© 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 = =.95 df= n - 1 = 2

© 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 = =.95 Reject

© 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.

© 2003 Pearson Prentice Hall  2 0 Chi-Square (  2 ) Test Solution H 0 : H a :  = df = Critical Value(s): Test Statistic: Decision:Conclusion:

© 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:

© 2003 Pearson Prentice Hall  2 0 Chi-Square (  2 ) Test Solution H 0 :  2 = 15 H a :  2  15  =.05 df = = 24 Critical Value(s): Test Statistic: Decision:Conclusion:

© 2003 Pearson Prentice Hall  2 0 Chi-Square (  2 ) Test Solution H 0 :  2 = 15 H a :  2  15  =.05 df = = 24 Critical Value(s): Test Statistic: Decision:Conclusion:  /2 =.025

© 2003 Pearson Prentice Hall  Chi-Square (  2 ) Test Solution H 0 :  2 = 15 H a :  2  15  =.05 df = = 24 Critical Value(s): Test Statistic: Decision:Conclusion:  /2 =.025

© 2003 Pearson Prentice Hall  Chi-Square (  2 ) Test Solution H 0 :  2 = 15 H a :  2  15  =.05 df = = 24 Critical Value(s): Test Statistic: Decision:Conclusion:  /2 =.025   ) (25 - 1)      (nS 0..

© 2003 Pearson Prentice Hall  Chi-Square (  2 ) Test Solution H 0 :  2 = 15 H a :  2  15  =.05 df = = 24 Critical Value(s): Test Statistic: Decision:Conclusion: Do Not Reject at  =.05  /2 =.025   ) (25 - 1)      (nS 0..

© 2003 Pearson Prentice Hall  Chi-Square (  2 ) Test Solution H 0 :  2 = 15 H a :  2  15  =.05 df = = 24 Critical Value(s): Test Statistic: Decision:Conclusion: Do Not Reject at  =.05 There Is No Evidence  2 Is Not 15  /2 =.025   ) (25 - 1)      (nS 0..

© 2003 Pearson Prentice Hall Calculating Type II Error Probabilities

© 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

© 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

© 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- 

© 2003 Pearson Prentice Hall X  1 = 360 = 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

© 2003 Pearson Prentice Hall X  1 = 360 = 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  =  =.846     Draw Draw Specify  Z Table

© 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

© 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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