Section 10.3

1. Section 10.3 Comparing Two Variances Larson/Farber 4th ed

2. Section 10.3 Objectives • Interpret the F-distribution and use an F-table to find critical values • Perform a two-sample F-test to compare two variances Larson/Farber 4th ed

3. F-Distribution • Let represent the sample variances of two different populations. • If both populations are normal and the population variances are equal, then the sampling distribution of is called an F-distribution. Larson/Farber 4th ed

4. Properties of the F-Distribution • The F-distribution is a family of curves each of which is determined by two types of degrees of freedom: • The degrees of freedom corresponding to the variance in the numerator, denoted d.f.N • The degrees of freedom corresponding to the variance in the denominator, denoted d.f.D • F-distributions are positively skewed. • The total area under each curve of an F-distribution is equal to 1. Larson/Farber 4th ed

5. d.f.N = 1 and d.f.D = 8 d.f.N = 8 and d.f.D = 26 d.f.N = 16 and d.f.D = 7 d.f.N = 3 and d.f.D = 11 1 2 3 4 F Properties of the F-Distribution • F-values are always greater than or equal to 0. • For all F-distributions, the mean value of F is approximately equal to 1. Larson/Farber 4th ed

6. Critical Values for the F-Distribution • Specify the level of significance . • Determine the degrees of freedom for the numerator, d.f.N. • Determine the degrees of freedom for the denominator, d.f.D. • Use Table 7 in Appendix B to find the critical value. If the hypothesis test is • one-tailed, use the  F-table. • two-tailed, use the ½ F-table. Larson/Farber 4th ed

7. Example: Finding Critical F-Values Find the critical F-value for a right-tailed test when α = 0.05, d.f.N = 6 and d.f.D = 29. Solution: The critical value is F0 = 2.43. Larson/Farber 4th ed

8. Example: Finding Critical F-Values Find the critical F-value for a two-tailed test when α = 0.05, d.f.N = 4 and d.f.D = 8. • Solution: • When performing a two-tailed hypothesis test using the F-distribution, you need only to find the right-tailed critical value. • You must remember to use the ½α table. Larson/Farber 4th ed

9. Solution: Finding Critical F-Values ½α = 0.025, d.f.N = 4 and d.f.D = 8 The critical value is F0 = 5.05. Larson/Farber 4th ed

10. Two-Sample F-Test for Variances To use the two-sample F-test for comparing two population variances, the following must be true. • The samples must be randomly selected. • The samples must be independent. • Each population must have a normal distribution. Larson/Farber 4th ed

11. Two-Sample F-Test for Variances • Test Statistic • where represent the sample variances with • The degrees of freedom for the numerator is d.f.N = n1 – 1 where n1 is the size of the sample having variance • The degrees of freedom for the denominator is d.f.D = n2 – 1, and n2 is the size of the sample having variance Larson/Farber 4th ed

12. Two-Sample F-Test for Variances In Words In Symbols • Identify the claim. State the null and alternative hypotheses. • Specify the level of significance. • Identify the degrees of freedom. • Determine the critical value. State H0 and Ha. Identify . d.f.N = n1 – 1 d.f.D = n2 – 1 Use Table 7 in Appendix B. Larson/Farber 4th ed

13. Two-Sample F-Test for Variances In Words In Symbols Determine the rejection region. Calculate the test statistic. Make a decision to reject or fail to reject the null hypothesis. Interpret the decision in the context of the original claim. If F is in the rejection region, reject H0. Otherwise, fail to reject H0. Larson/Farber 4th ed

14. Example: Performing a Two-Sample F-Test A restaurant manager is designing a system that is intended to decrease the variance of the time customers wait before their meals are served. Under the old system, a random sample of 10 customers had a variance of 400. Under the new system, a random sample of 21 customers had a variance of 256. At α = 0.10, is there enough evidence to convince the manager to switch to the new system? Assume both populations are normally distributed. Larson/Farber 4th ed

15. σ12≤σ22 σ12>σ22 0.10 9 20 Because 400 > 256, Solution: Performing a Two-Sample F-Test • H0: • Ha: • α = • d.f.N= d.f.D= • Rejection Region: • Test Statistic: • Decision: Fail to Reject H0 There is not enough evidence to convince the manager to switch to the new system. 0.10 0 1.96 F 1.96 1.56 Larson/Farber 4th ed

16. Example: Performing a Two-Sample F-Test You want to purchase stock in a company and are deciding between two different stocks. Because a stock’s risk can be associated with the standard deviation of its daily closing prices, you randomly select samples of the daily closing prices for each stock to obtain the results. At α = 0.05, can you conclude that one of the two stocks is a riskier investment? Assume the stock closing prices are normally distributed. Larson/Farber 4th ed

17. σ12=σ22 σ12≠σ22 0. 025 30 29 Because 5.72 > 3.52, Solution: Performing a Two-Sample F-Test • H0: • Ha: • ½α = • d.f.N= d.f.D= • Rejection Region: • Test Statistic: • Decision: Reject H0 There is enough evidence to support the claim that one of the two stocks is a riskier investment. 0.025 0 2.09 F 2.09 2.65 Larson/Farber 4th ed

18. Section 10.3 Summary • Interpreted the F-distribution and used an F-table to find critical values • Performed a two-sample F-test to compare two variances Larson/Farber 4th ed