9-4 Two Variances or Standard Deviations 481 Hypothesis Test with Two Variances or Standard Deviations Objective Conduct a hypothesis test of a claim about two population variances or standard deviations. (Any claim made about two population standard deviations can be restated with an equivalent claim about two population variances, so the same procedure is used for two population standard deviations or two population variances.) Notation KEY ELEMENTS s2 1 = larger of the two sample variances n1 = size of the sample with the larger variance s 2 1 = variance of the population from which the sample with the larger variance was drawn The symbols s2 2, n2, and s 2 2 are used for the other sample and population. Requirements 1. The two populations are independent. 2. The two samples are simple random samples. 3. Each of the two populations must be normally distributed, regardless of their sample sizes. This F test is not robust against departures from normality, so it performs poorly if one or both of the populations have a distribution that is not normal. The requirement of normal distributions is quite strict for this F test. (If the third requirement is not satisfied, alternative methods include the count five method, the LeveneBrown-Forsythe test, and resampling methods.) Test Statistic for Hypothesis Tests with Two Variances (with H0: S2 1 = S2 2) F = s2 1 s2 2 (where s2 1 is the larger of the two sample variances) P-Values: P-values are automatically provided by technology. If technology is not available, use the computed value of the F test statistic with Table A-5 to find a range for the P-value. Critical Values: Use Table A-5 to find critical F values that are determined by the following: 1. The significance level a (Table A-5 includes critical values for a = 0.025 and a = 0.05.) 2. Numerator degrees of freedom = n1 − 1 (determines column of Table A-5) 3. Denominator degrees of freedom = n2 − 1 (determines row of Table A-5) For significance level a = 0.05, refer to Table A-5 and use the right-tail area of 0.025 or 0.05, depending on the type of test, as shown below: • Two-tailed test: Use Table A-5 with 0.025 in the right tail. (The significance level of 0.05 is divided between the two tails, so the area in the right tail is 0.025.) • One-tailed test: Use Table A-5 with a = 0.05 in the right tail. Find the critical F value for the right tail: Because we are stipulating that the larger sample variance is s2 1, all one-tailed tests will be right-tailed and all two-tailed tests will require that we find only the critical value located to the right. (We have no need to find the critical value at the left tail, which is not very difficult. See Exercise 19 “Finding Lower Critical F Values.”) Explore the Data! Because the F test requirement of normal distributions is quite strict, be sure to examine the distributions of the two samples using histograms and normal quantile plots, and confirm that there are no outliers. (See “Assessing Normality” in Section 6-5.)
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