1. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 10-1 8 th Lesson Hypothesis Tests for One and Two Population Variances

2. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 10-2 Hypothesis Tests for Variances Hypothesis Tests for Variances Tests for a Single Population Variances Tests for Two Population Variances Chi-Square test statisticF test statistic

3. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 10-3 Single Population Hypothesis Tests for Variances Tests for a Single Population Variances Chi-Square test statistic H 0 : σ 2 = σ 0 2 H A : σ 2 ≠ σ 0 2 H 0 : σ 2  σ 0 2 H A : σ 2 < σ 0 2 H 0 : σ 2 ≤ σ 0 2 H A : σ 2 > σ 0 2 * Two tailed test Lower tail test Upper tail test

4. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 10-4 Chi-Square Test Statistic Hypothesis Tests for Variances Tests for a Single Population Variances Chi-Square test statistic * The chi-squared test statistic for a Single Population Variance is: where  2 = standardized chi-square variable n = sample size s 2 = sample variance σ 2 = hypothesized variance

5. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 10-5 The Chi-square Distribution The chi-square distribution is a family of distributions, depending on degrees of freedom: d.f. = n - 1 0 4 8 12 16 20 24 28 d.f. = 1d.f. = 5d.f. = 15 22 22 22

6. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 10-6 Finding the Critical Value The critical value,, is found from the chi-square table Do not reject H 0 Reject H 0  22 22 22 H 0 : σ 2 ≤ σ 0 2 H A : σ 2 > σ 0 2 Upper tail test:

7. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 10-7 Example A commercial freezer must hold the selected temperature with little variation. Specifications call for a standard deviation of no more than 4 degrees (or variance of 16 degrees 2 ). A sample of 16 freezers is tested and yields a sample variance of s 2 = 24. Test to see whether the standard deviation specification is exceeded. Use  =.05

8. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 10-8 Finding the Critical Value The the chi-square table to find the critical value: Do not reject H 0 Reject H 0  =.05 22 22 22 = 24.9958 = 24.9958 (  =.05 and 16 – 1 = 15 d.f.) The test statistic is: Since 22.5 < 24.9958, do not reject H 0 There is not significant evidence at the  =.05 level that the standard deviation specification is exceeded H 0 : σ 2 ≤ 16 H A : σ 2 > 16

9. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 10-9 Lower Tail or Two Tailed Chi-square Tests H 0 : σ 2 = σ 0 2 H A : σ 2 ≠ σ 0 2 H 0 : σ 2  σ 0 2 H A : σ 2 < σ 0 2  2  /2 Do not reject H 0 Reject   2 1-  22 Do not reject H 0 Reject  /2  2 1-  /2 22  /2 Reject Lower tail test:Two tail test:

10. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 10-10 Hypothesis Tests for Variances Tests for Two Population Variances F test statistic * F Test for Difference in Two Population Variances H 0 : σ 1 2 – σ 2 2 = 0 H A : σ 1 2 – σ 2 2 ≠ 0 Two tailed test Lower tail test Upper tail test H 0 : σ 1 2 – σ 2 2  0 H A : σ 1 2 – σ 2 2 < 0 H 0 : σ 1 2 – σ 2 2 ≤ 0 H A : σ 1 2 – σ 2 2 > 0

11. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 10-11 Hypothesis Tests for Variances F test statistic * F Test for Difference in Two Population Variances Tests for Two Population Variances The F test statistic is: = Variance of Sample 1 n 1 - 1 = numerator degrees of freedom n 2 - 1 = denominator degrees of freedom = Variance of Sample 2

12. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 10-12 F0 rejection region for a one-tail test (upper tail test) is Finding the Critical Value F0 rejection region for a two-tailed test is  /2 FF F  /2 Reject H 0 Do not reject H 0 Reject H 0 Do not reject H 0 H 0 : σ 1 2 – σ 2 2 = 0 H A : σ 1 2 – σ 2 2 ≠ 0 H 0 : σ 1 2 – σ 2 2  0 H A : σ 1 2 – σ 2 2 < 0 H 0 : σ 1 2 – σ 2 2 ≤ 0 H A : σ 1 2 – σ 2 2 > 0 or  /2

13. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 10-13 F Test: An Example You are a financial analyst for a brokerage firm. You want to compare dividend yields between stocks listed on the NYSE & NASDAQ. You collect the following data : NYSE NASDAQ Number 2125 Mean3.272.53 Std dev1.301.16 Is there a difference in the variances between the NYSE & NASDAQ at the  = 0.1 level?

14. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 10-14 F Test: Example Solution Form the hypothesis test: H 0 : σ 2 1 – σ 2 2 = 0 ( there is no difference between variances) H A : σ 2 1 – σ 2 2 ≠ 0 ( there is a difference between variances) Find the F critical value for  =.1: Numerator: df 1 = n 1 – 1 = 21 – 1 = 20 Denominator: df 2 = n 2 – 1 = 25 – 1 = 24 F.95, 20, 24 = 0,48 or F.05, 20, 24 = 2.03

15. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 10-15 The test statistic is: 0  /2 =.05 F  /2 =2.03 Reject H 0 Do not reject H 0 H 0 : σ 1 2 – σ 2 2 = 0 H A : σ 1 2 – σ 2 2 ≠ 0 F Test: Example Solution Conclusion : do not reject H 0 There is no evidence of a difference in variances at  =.1 (continued)