Comparing Two Variances Section 10.3 Comparing Two Variances

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

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.

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.

Properties of the F-Distribution Chapter 10 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. d.f.N = 1 and d.f.D = 8 F 1 2 3 4 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-Distributions Larson/Farber 4th ed

Finding Critical Values for the F-Distribution Chapter 10 Finding Critical Values for the F-Distribution Specify the level of significance α. 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 α F-table. two-tailed, use the ½α F-table. Larson/Farber 4th ed

Example: Finding Critical F-Values Find the critical F-value for a right-tailed test when α = 0.10, d.f.N = 5 and d.f.D = 28. Solution: The critical value is F0 = 2.06.

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.

Solution: Finding Critical F-Values ½α = 0.025, d.f.N = 4 and d.f.D = 8 The critical value is F0 = 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. The samples must be randomly selected. The samples must be independent. Each population must have a normal distribution.

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

Finding F-statistic Larger variance is always in numerator Chapter 10 Finding F-statistic Larger variance is always in numerator Find: F and dfN and dfD for the following a) from samples: 𝒔 𝟏 𝟐 = 842, n1 = 11; 𝒔 𝟐 𝟐 = 834, n2 = 18 which variance is larger? Larson/Farber 5th ed

Finding F-statistic Larger variance is always in numerator Chapter 10 Finding F-statistic Larger variance is always in numerator a) from samples: 𝒔 𝟏 𝟐 = 842, n1 = 11; 𝒔 𝟐 𝟐 = 834, n2 = 18 which variance is larger? Sample 1 (842 > 834) so, F = 𝟖𝟒𝟐 𝟖𝟑𝟒 dfN = 11-1 = 10 and dfD = 18-1=17 Larson/Farber 5th ed

Finding F-statistic Larger variance is always in numerator Chapter 10 Finding F-statistic Larger variance is always in numerator b) from samples: 𝒔 𝟏 𝟐 = 365, n1 = 15; 𝒔 𝟐 𝟐 = 402, n2 = 9 which variance is larger? Sample 2 (402 > 365) so, F = 𝟒𝟎𝟐 𝟑𝟔𝟓 dfN = 9-1 = 8 and dfD = 15-1=14 Larson/Farber 5th ed

Two-Sample F-Test for Variances Chapter 10 Two-Sample F-Test for Variances In Words In Symbols Identify the claim. State the null and alternative hypotheses. Specify the level of significance. Determine the degrees of freedom. Determine the critical value. State H0 and Ha. Identify α. d.f.N = n1 – 1 d.f.D = n2 – 1 Use Table 7 in Appendix B. Larson/Farber 4th ed

Two-Sample F-Test for Variances Chapter 10 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

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.

Solution: Performing a Two-Sample F-Test Because 400 > 256, H0: Ha: α = d.f.N= d.f.D= Rejection Region: σ12 ≤ σ22 σ12 > σ22 (Claim) Test Statistic: Decision: 0.10 9 20 Fail to Reject H0 There is not enough evidence at the 10% level of significance to convince the manager to switch to the new system. F 1.96 0.10 1.96 1.56

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. Stock A Stock B n2 = 30 n1 = 31 s2 = 3.5 s1 = 5.7

Solution: Performing a Two-Sample F-Test Because 5.72 > 3.52, H0: Ha: ½α = d.f.N= d.f.D= Rejection Region: σ12 = σ22 σ12 ≠ σ22 (Claim) Test Statistic: Decision: 0. 025 30 29 Reject H0 There is enough evidence at the 5% level of significance to support the claim that one of the two stocks is a riskier investment. F 2.09 0.025 2.09 2.652

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