1. 9 - 1 © 2003 Pearson Prentice Hall F-Test for Two Variances 1.Tests for Differences in 2 Population Variances 2.Assumptions Both Populations Are Normally Distributed Both Populations Are Normally Distributed Test Is Not Robust to Violations Test Is Not Robust to Violations Independent, Random Samples Independent, Random Samples

2. 9 - 2 © 2003 Pearson Prentice Hall F-Test for Variances Hypotheses 1.Hypotheses H 0 :  1 2 =  2 2 OR H 0 :  1 2   2 2 (or  ) H 0 :  1 2 =  2 2 OR H 0 :  1 2   2 2 (or  ) H a :  1 2   2 2 H a :  1 2  2 2 (or >) H a :  1 2   2 2 H a :  1 2  2 2 (or >) 2.Test Statistic F = s 1 2 /s 2 2 F = s 1 2 /s 2 2 Two Sets of Degrees of Freedom Two Sets of Degrees of Freedom 1 = n 1 - 1; 2 = n 2 - 1 1 = n 1 - 1; 2 = n 2 - 1 Follows F Distribution Follows F Distribution

3. 9 - 3 © 2003 Pearson Prentice Hall F Distribution

4. 9 - 4 © 2003 Pearson Prentice Hall F Distribution Population 1  1 1   1 1

5. 9 - 5 © 2003 Pearson Prentice Hall F Distribution Population 1 2  1 1   1 1   2 2 2 2

6. 9 - 6 © 2003 Pearson Prentice Hall F Distribution Select simple random sample, size n 1 Compute S 1 2 Population 1 2  1 1   1 1   2 2 2 2

7. 9 - 7 © 2003 Pearson Prentice Hall F Distribution Select simple random sample, size n 1 Compute S 1 2 Population 1 2  1 1   1 1   2 2 2 2 Select simple random sample, size n 2 ComputeS 2 2

8. 9 - 8 © 2003 Pearson Prentice Hall F Distribution Select simple random sample, size n 1 Compute S 1 2 Compute F =  S 1 2 /S 2 2 for every pair of n 1 & n 2 size samples Population 1 2  1 1   1 1   2 2 2 2 Select simple random sample, size n 2 ComputeS 2 2

9. 9 - 9 © 2003 Pearson Prentice Hall F Distribution Select simple random sample, size n 1 Compute S 1 2 Compute F =  S 1 2 /S 2 2 for every pair of n 1 & n 2 size samples Astronomical number ofS 1 2 /S 2 2 values Population 1 2  1 1   1 1   2 2 2 2 Select simple random sample, size n 2 ComputeS 2 2

10. 9 - 10 © 2003 Pearson Prentice Hall F Distribution

11. 9 - 11 © 2003 Pearson Prentice Hall F-Test for 2 Variances Critical Values Note!  /2

12. 9 - 12 © 2003 Pearson Prentice Hall F-Test for Variances Example You’re a financial analyst for Charles Schwab. You want to compare dividend yields between stocks listed on the NYSE & NASDAQ. You collect the following data: NYSE NASDAQ Number 2125 NYSE NASDAQ Number 2125 Mean3.272.53 Std Dev1.301.16 Is there a difference in variances between the NYSE & NASDAQ at the.05 level? © 1984-1994 T/Maker Co.

13. 9 - 13 © 2003 Pearson Prentice Hall F-Test for 2 Variances Solution

14. 9 - 14 © 2003 Pearson Prentice Hall F-Test for 2 Variances Solution H 0 : H a :   1  2  1  2  Critical Value(s): Test Statistic: Decision:Conclusion:

15. 9 - 15 © 2003 Pearson Prentice Hall F-Test for 2 Variances Solution H 0 :  1 2 =  2 2 H a :  1 2   2 2   1  2  1  2  Critical Value(s): Test Statistic: Decision:Conclusion:

16. 9 - 16 © 2003 Pearson Prentice Hall F-Test for 2 Variances Solution H 0 :  1 2 =  2 2 H a :  1 2   2 2  .05 1  20 2  24 1  20 2  24 Critical Value(s): Test Statistic: Decision:Conclusion:

17. 9 - 17 © 2003 Pearson Prentice Hall F-Test for 2 Variances Solution H 0 :  1 2 =  2 2 H a :  1 2   2 2  .05 1  20 2  24 1  20 2  24 Critical Value(s): Test Statistic: Decision:Conclusion:

18. 9 - 18 © 2003 Pearson Prentice Hall F-Test for 2 Variances Solution  /2 =.025

19. 9 - 19 © 2003 Pearson Prentice Hall F-Test for 2 Variances Solution H 0 :  1 2 =  2 2 H a :  1 2   2 2  .05 1  20 2  24 1  20 2  24 Critical Value(s): Test Statistic: Decision:Conclusion:

20. 9 - 20 © 2003 Pearson Prentice Hall F-Test for 2 Variances Solution H 0 :  1 2 =  2 2 H a :  1 2   2 2  .05 1  20 2  24 1  20 2  24 Critical Value(s): Test Statistic: Decision:Conclusion: Do not reject at  =.05

21. 9 - 21 © 2003 Pearson Prentice Hall F-Test for 2 Variances Solution H 0 :  1 2 =  2 2 H a :  1 2   2 2  .05 1  20 2  24 1  20 2  24 Critical Value(s): Test Statistic: Decision:Conclusion: Do not reject at  =.05 There is no evidence of a difference in variances

22. 9 - 22 © 2003 Pearson Prentice Hall F-Test for Variances Thinking Challenge You’re an analyst for the Light & Power Company. You want to compare the electricity consumption of single-family homes in 2 towns. You compute the following from a sample of homes : Town 1Town 2 Number 25 21 Town 1Town 2 Number 25 21 Mean$ 85$ 68 Std Dev $ 30 $ 18 At the.05 level, is there evidence of a difference in variances between the two towns?

23. 9 - 23 © 2003 Pearson Prentice Hall F-Test for 2 Variances Solution* H 0 :  1 2 =  2 2 H a :  1 2   2 2  .05 1  24 2  20 1  24 2  20 Critical Value(s): Test Statistic: Decision:Conclusion: Reject at  =.05 There is evidence of a difference in variances

24. 9 - 24 © 2003 Pearson Prentice Hall Critical Values Solution*  /2 =.025