 # Section 10.3 Comparing Two Variances Larson/Farber 4th ed1.

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Section 10.3 Comparing Two Variances Larson/Farber 4th ed1

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 ed2

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 ed3

Properties of the F-Distribution 1. 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 2. F-distributions are positively skewed. 3. The total area under each curve of an F- distribution is equal to 1. Larson/Farber 4th ed4

Properties of the F-Distribution 4. F-values are always greater than or equal to 0. 5. For all F-distributions, the mean value of F is approximately equal to 1. Larson/Farber 4th ed5 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 F 1234

Critical Values for the F-Distribution 1. Specify the level of significance . 2. Determine the degrees of freedom for the numerator, d.f. N. 3. Determine the degrees of freedom for the denominator, d.f. D. 4. Use Table 7 in Appendix B to find the critical value. If the hypothesis test is a.one-tailed, use the  F-table. b.two-tailed, use the ½  F-table. Larson/Farber 4th ed6

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. Larson/Farber 4th ed7 The critical value is F 0 = 2.43. Solution:

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. Larson/Farber 4th ed8 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 ½ α table.

Solution: Finding Critical F-Values ½ α = 0.025, d.f. N = 4 and d.f. D = 8 Larson/Farber 4th ed9 The critical value is F 0 = 5.05.

Two-Sample F-Test for Variances To use the two-sample F-test for comparing two population variances, the following must be true. 1. The samples must be randomly selected. 2. The samples must be independent. 3. Each population must have a normal distribution. Larson/Farber 4th ed10

Two-Sample F-Test for Variances Test Statistic Larson/Farber 4th ed11 where represent the sample variances with The degrees of freedom for the numerator is d.f. N = n 1 – 1 where n 1 is the size of the sample having variance The degrees of freedom for the denominator is d.f. D = n 2 – 1, and n 2 is the size of the sample having variance

Two - Sample F - Test for Variances Larson/Farber 4th ed12 1.Identify the claim. State the null and alternative hypotheses. 2.Specify the level of significance. 3.Identify the degrees of freedom. 4.Determine the critical value. State H 0 and H a. Identify . Use Table 7 in Appendix B. d.f. N = n 1 – 1 d.f. D = n 2 – 1 In WordsIn Symbols

Two - Sample F - Test for Variances Larson/Farber 4th ed13 If F is in the rejection region, reject H 0. Otherwise, fail to reject H 0. 5.Determine the rejection region. 6.Calculate the test statistic. 7.Make a decision to reject or fail to reject the null hypothesis. 8.Interpret the decision in the context of the original claim. In WordsIn Symbols

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 ed14

Solution: Performing a Two-Sample F-Test Larson/Farber 4th ed15 H 0 : H a : α = d.f. N = d.f. D = Rejection Region: Test Statistic: Decision: σ12 ≤ σ22σ12 ≤ σ22 σ12 > σ22σ12 > σ22 0.10 920 0F1.96 0.10 Because 400 > 256, There is not enough evidence to convince the manager to switch to the new system. 1.96 1.56 Fail to Reject H 0

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 ed16 Stock AStock B n 2 = 30n 1 = 31 s 2 = 3.5s 1 = 5.7

Solution: Performing a Two-Sample F-Test Larson/Farber 4th ed17 H 0 : H a : ½α = d.f. N = d.f. D = Rejection Region: Test Statistic: Decision: σ12 = σ22σ12 = σ22 σ12 ≠ σ22σ12 ≠ σ22 0. 025 3029 0F2.09 0.025 Because 5.7 2 > 3.5 2, There is enough evidence to support the claim that one of the two stocks is a riskier investment. 2.09 2.65 Reject H 0

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 ed18