TA13a - F > F α /2 ( n Y – 1, n X – 1). So...

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ECO2121: Methods of Economic Statistics TA13a 8 April 2009 This is a supplement to TA13. 1. Covariance Cov[ X , Y ] = E [( X μ X )( Y μ Y )] = E [ XY ] μ X μ Y In our example, μ X = 0.4 and μ Y = 5, so E [ XY ] = 1 × 3 × f X , Y (1,3) + 1 × 5 × f X , Y (1,5) + 1 × 7 × f X , Y (1,7) + 0 × 2 × f X , Y (0,2) + 0 × 3 × f X , Y (0,3) + 0 × 5 × f X , Y (0,5) + 0 × 6 × f X , Y (0,6) + 0 × 7 × f X , Y (0,7) = 2. Hence, Cov[ X , Y ] = 2 0.4 × 5 = 0. Therefore, X and Y are uncorrelated , but they are dependent because f X , Y ( x , y ) f X ( x ) × f Y ( y ) due to f Y | X ( y | x ) f Y ( y ). 2. F -test (a) For H 1 : σ X 2 > σ Y 2 , the test statistic F 0 = S X 2 / S Y 2 ~ F ( n X 1, n Y 1), the rejection rule is F 0 > F α /2 ( n X 1, n Y 1). (b) Similarly for H 1 : σ X 2 < σ Y 2 , F 0 = S Y 2 / S X 2 ~ F ( n Y 1, n X 1), the rejection rule is
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Unformatted text preview: F &gt; F α /2 ( n Y – 1, n X – 1). So for (a) and (b), F depends on H 1 , whichever is larger shall put into the numerator. (c) For H 1 : σ X 2 ≠ σ Y 2 , F = larger sample variance/smaller sample variance, the rejection rule is F &gt; F α /2 ( n larger sample variance – 1, n smaller sample variance – 1) or F &lt; F 1-α /2 ( n larger sample variance – 1, n smaller sample variance – 1). Essentially, only the first part is possible. Taylor 8 April 2009...
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