Business Statistics: A Decision-Making Approach 7th Edition Chapter 11 Hypothesis Tests for One and Two Population Variances Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc Chap 11-1 Chapter Goals After completing this chapter, you should be able to: Formulate and complete hypothesis tests for a single population variance Find critical chi-square distribution values from the chisquare table Formulate and complete hypothesis tests for the difference between two population variances Use the F table to find critical F values Business Statistics: A Decision- Chap 11-2 Hypothesis Tests for Variances Hypothesis Tests for Variances Tests for a Single Population Variance Tests for Two Population Variances Chi-Square test statistic F test statistic Business Statistics: A Decision- Chap 11-3 Single Population Hypothesis Tests for Variances Tests for a Single Population Variance * Chi-Square test statistic Business Statistics: A Decision- H0: σ2 = σ02 HA: σ2 ≠ σ02 Two tailed test H0: σ2 ≥ σ02 HA: σ2 < σ02 Lower tail test H0: σ2 ≤ σ02 HA: σ2 > σ02 Upper tail test Chap 11-4 Chi-Square Test Statistic Hypothesis Tests for Variances The chi-squared test statistic for a Single Population Variance is: Tests for a Single Population Variance Chi-Square test statistic (n − 1)s χ = σ2 * where χ2 = standardized chi-square variable n = sample size s2 = sample variance σ2 = hypothesized variance Business Statistics: A Decision- Chap 11-5 The Chi-square Distribution The chi-square distribution is a family of distributions, depending on degrees of freedom: d.f = n - 12 16 20 24 28 d.f = χ2 12 16 20 24 28 d.f = Business Statistics: A Decision- χ2 12 16 20 24 28 χ2 d.f = 15 Chap 11-6 Finding the Critical Value The critical value, chi-square table , is found from the χ 2α Upper tail test: H0: σ2 ≤ σ02 HA: σ2 > σ02 α χ2 Do not reject H0 χ Business Statistics: A Decision- Reject H0 α Chap 11-7 Example A commercial freezer must hold the selected temperature with little variation Specifications call for a standard deviation of no more than degrees (or variance of 16 degrees2) A sample of 16 freezers is tested and yields a sample variance of s2 = 24 Test to see whether the standard deviation specification is exceeded Use α = 05 Business Statistics: A Decision- Chap 11-8 Finding the Critical Value Use the chi-square table to find the critical value: χ 2α = 24.9958 (α = 05 and 16 – = 15 d.f.) The test statistic is: (n − 1)s (16 − 1)24 χ = = = 22.5 σ 16 Since 22.5 < 24.9958, not reject H0 There is not significant evidence at the α = 05 level that the standard deviation specification is exceeded α = 05 χ2 Do not reject H0 Business Statistics: A Decision- χ 2α Reject H0 = 24.9958 Chap 11-9 Lower Tail or Two Tailed Chi-square Tests Lower tail test: Two tail test: H0: σ2 ≥ σ02 HA: σ2 < σ02 H0: σ2 = σ02 HA: σ2 ≠ σ02 α α/2 α/2 χ2 Reject χ Do not reject H0 χ2 Reject 1-α Business Statistics: A Decision- Do not reject H0 χ 21-α/2 (χ 2L) Reject χ 2α/2 (χ 2U) Chap 11-10 Confidence Interval Estimate for σ2 The confidence interval estimate for σ2 is (n − 1)s2 (n − 1)s ≤ σ ≤ χU χL2 α/2 α/2 χ 21-α/2 (χ 2L) χ 2α/2 (χ 2U) χ2 Business Statistics: A Decision- Where χ2L and χ2U are from the χ2 distribution with n -1 degrees of freedom Chap 11-11 Example A sample of 16 freezers yields a sample variance of s = 24 Form a 95% confidence interval for the population variance Business Statistics: A Decision- Chap 11-12 Example (continued) Use the chi-square table to find χ2L and χ2U : (α = 05 and 16 – = 15 d.f.) α/2=.025 α/2=.025 χ 975 (χ 2L) χ 025 (χ 2U) 6.2621 27.4884 2 (n − 1)s (n − 1)s ≤ σ ≤ χU χL2 (16 − 1)24 (16 − 1)24 ≤σ ≤ 27.4884 6.2621 13.096 ≤ σ ≤ 57.489 We are 95% confident that the population variance is between 13.096 and 57.489 degrees2 (Taking the square root, we are 95% confident that the population standard deviation is between 3.619 and 7.582 degrees.) Business Statistics: A Decision- Chap 11-13 F Test for Difference in Two Population Variances Hypothesis Tests for Variances H0: σ12 = σ22 HA: σ12 ≠ σ22 Two tailed test H0: σ12 ≥ σ22 HA: σ12 < σ22 Lower tail test H0: σ12 ≤ σ22 HA: σ12 > σ22 * Tests for Two Population Variances F test statistic Upper tail test Business Statistics: A Decision- Chap 11-14 F Test for Difference in Two Population Variances Hypothesis Tests for Variances The F test statistic is: 2 s F= s s12 Where F has D1 numerator and D2 denominator degrees of freedom = Variance of Sample Tests for Two Population Variances * F test statistic D1 = n1 - = numerator degrees of freedom s22 = Variance of Sample D2 = n2 - = denominator degrees of freedom Business Statistics: A Decision- Chap 11-15 The F Distribution The F critical value is found from the F table The are two appropriate degrees of freedom: D1 (numerator) and D2 (denominator) s12 F= s2 where D1 = n1 – ; D2 = n2 – In the F table, numerator degrees of freedom determine the row denominator degrees of freedom determine the column Business Statistics: A Decision- Chap 11-16 Formulating the F Ratio s12 F= s2 where D1 = n1 – ; D2 = n2 – For a two-tailed test, always place the larger sample variance in the numerator For a one-tailed test, consider the alternative hypothesis: place in the numerator the sample variance for the population that is predicted (based on HA) to have the larger variance Business Statistics: A Decision- Chap 11-17 Finding the Critical Value H0: σ12 ≥ σ22 HA: σ12 < σ22 H0: σ12 = σ22 HA: σ12 ≠ σ22 H0 : σ12 ≤ σ22 HA : σ12 > σ22 α Do not reject H0 Fα Reject H0 α/2 F rejection region for a one-tail test is s12 F = > Fα s2 Do not reject H0 Fα/2 Reject H0 F rejection region for a two-tailed test is s12 F = > Fα / s2 (where the larger sample variance in the numerator) Business Statistics: A Decision- Chap 11-18 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 Mean 3.272.53 Std dev 1.301.16 Is there a difference in the variances between the NYSE & NASDAQ at the α = 0.05 level? Business Statistics: A Decision- Chap 11-19 F Test: Example Solution Form the hypothesis test: H0: σ21 = σ22 (there is no difference between variances) HA: σ21 ≠ σ22 (there is a difference between variances) Find the F critical value for α = 05: Numerator: D1 = n1 – = 21 – = 20 Denominator: D = n – = 25 – = 24 2 F.05/2, 20, 24 = 2.327 Business Statistics: A Decision- Chap 11-20 F Test: Example Solution (continued) The test statistic is: H0: σ12 = σ22 HA: σ12 ≠ σ22 s12 1.30 F= = = 1.256 s2 1.16 α/2 = 025 F = 1.256 is not greater than the critical F value of 2.327, so we not reject H0 Conclusion: There is no evidence of a difference in variances at α = 05 Business Statistics: A Decision- Do not reject H0 Reject H0 Fα/2 =2.327 Chap 11-21 Using EXCEL and PHStat EXCEL F test for two variances: Data | Data Analysis | F-test: Two Sample for Variances PHStat Chi-square test for the variance: PHStat | One-sample Tests | Chi-square Test for the Variance F test for two variances: PHStat | Two-sample Tests | F Test for Differences in Two Variances Business Statistics: A Decision- Chap 11-22 Chapter Summary Performed chi-square tests for the variance Used the chi-square table to find chi-square critical values Performed F tests for the difference between two population variances Used the F table to find F critical values Business Statistics: A Decision- Chap 11-23 ... F test statistic Business Statistics: A Decision- Chap 11-3 Single Population Hypothesis Tests for Variances Tests for a Single Population Variance * Chi-Square test statistic Business Statistics:... for the difference between two population variances Use the F table to find critical F values Business Statistics: A Decision- Chap 11-2 Hypothesis Tests for Variances Hypothesis Tests for Variances... standardized chi-square variable n = sample size s2 = sample variance σ2 = hypothesized variance Business Statistics: A Decision- Chap 11-5 The Chi-square Distribution The chi-square distribution