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Unformatted text preview: Chapter 10.4 Testing the Equality of Variances Consider two random samples. Assume: x independent samples, and x normally distributed populations. The samples have and observations and the population variances are and. x n y n 2 X V 2 Y V Estimators of the population variances are the random variables and. 2 X s 2 Y s Introduce the random variable: 2 Y 2 Y 2 X 2 X s s F V V F is the ratio of two independently distributed chi-square random variables. A result from statistical theory is that F has an F-distribution with numerator degrees of freedom and denominator degrees of freedom )1 n( x )1 n( y . That is, )1 n,1 n( y x F~F n is distributed as Econ 325 Chapter 10.4 1 Like the chi-square distribution, the F distribution is defined only for non-negative values and the skewed shape of the probability density function depends on the degrees of freedom. PDF of the F distribution with (10, 20) and (20, 20) degrees of freedom. 0.5 1 1.5 2 2.5 3 3.5 F(10,20) F(20,20) Appendix Table 9, pages 8457 of the textbook, lists cutoff points or critical values that give an upper tail area of either 0.05 or 0.01 . Econ 325 Chapter 10.4 2 The application of interest is to test the null hypothesis: population variances are equal 2 Y 2 X :H V V against the two-sided alternative: population variances are not equal 2 Y 2 X 1 :H VzV When the null hypothesis is true, , and the random variable: 2 Y 2 X V V 2 Y 2 X s s F has an distribution. )1 n,1 n( y x F From the numeric data set calculate the sample variances and. Arrange the two samples so that. 2 x s 2 y s 2 y 2 x s s " A test statistic is calculated as the variance ratio: 2 y 2 x s s The test statistic exceeds one since is bigger than. 2 x s 2 y s Econ 325 Chapter 10.4 3 For a chosen significance level , the decision rule is to reject the null hypothesis of equal variances if: c 2 y 2 x F s s " where is the critical value from the F-distribution that satisfies: c F 5 " )F F(P c )1 n,1 n( y x Appendix Table 9 lists critical values for upper tail probabilities of...
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- Spring '10