chap10S4PRN

# chap10S4PRN - Chapter 10.4 Testing the Equality of...

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Econ 325 – Chapter 10.4 1 Chapter 10.4 Testing the Equality of Variances Consider two random samples. Assume: independent samples, and normally distributed populations. The samples have x n and y n observations and the population variances are 2 X σ and 2 Y σ . Estimators of the population variances are the random variables 2 X s and 2 Y s . Introduce the random variable: 2 Y 2 Y 2 X 2 X s s F σ σ = 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 ) 1 n ( x - and denominator degrees of freedom ) 1 n ( y - . That is, ) 1 n , 1 n ( y x F ~ F - - ‘is distributed as Econ 325 – Chapter 10.4 2 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 0.5 1 1.5 2 2.5 3 3.5 F(10,20) F(20,20) A textbook Appendix Table lists cutoff points or critical values that give an upper tail area of either 0.05 or 0.01 .

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Econ 325 – Chapter 10.4 3 The application of interest is to test the null hypothesis: 2 Y 2 X 0 : H σ = σ population variances are equal against the two-sided alternative: 2 Y 2 X 1 : H σ σ population variances are not equal When the null hypothesis is true, 2 Y 2 X σ = σ , and the random variable: 2 Y 2 X s s F = has an ) 1 n , 1 n ( y x F - - distribution. From the numeric data set calculate the sample variances 2 x s and 2 y s . Arrange the two samples so that 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 2 x s is bigger than 2 y s . Econ 325 – Chapter 10.4 4 For a chosen significance level α , the decision rule is to reject the null hypothesis of equal variances if: c 2 y 2 x F s s > where c F is the critical value from the F-distribution that satisfies: 2 α = > - - ) F F ( P c ) 1 n , 1 n ( y x The F-distribution Appendix Table lists critical values for upper tail

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