# HW7 - Solution to HW7 Problem 1 a x x h = so 2 1 2 1 − =...

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Unformatted text preview: Solution to HW7: Problem 1: a) x x h = ) ( , so 2 / 1 2 1 ) ( ' − = x x h and . 4 1 ) ( ' ' 2 / 3 − − = x x h 2 2 / 3 2 / 1 2 ) ( 8 1 ) ( 2 1 ) ( ) )( ( ' ' 2 1 ) )( ( ' ) ( ) ( x y x x y x x h x y x h x y x h x h y h − − − + = − + − + ≈ − − Take 2 s y = and 2 σ = x , so 2 2 2 3 2 2 1 ) ( 8 1 ) ( 2 1 σ σ σ σ σ − − − + ≈ − − s s s Since 2 2 ) ( σ = s E , taking the expected value of the equation yields ) ( 8 1 ) ( 8 1 ) ( 2 1 ) ( 2 3 2 2 2 3 2 2 1 s Var s E s E s E − − − − = − − − + ≈ σ σ σ σ σ σ σ The bias of s is 1 4 1 1 2 8 1 ) ( 8 1 ) ( 4 3 2 3 − − = − − = − ≈ − − − n n s Var s E σ σ σ σ σ b) 48 . = s and 8 = n . Use s to estimate σ . The bias of s is 0.01714286 1 8 48 . 4 1 1 4 1 ) ( − = − − = − − ≈ − n s s E σ 0.4628571 48 . 1 8 48 . 4 1 1 4 1 ) ( = + − − = + − − ≈ s n s s E 0.01616327 4628571 . 48 . )) ( ( )) ( ( ) ( ) ( 2 2 2 2 2 2 = − ≈ − = − = s E s E s E s Var σ 0.1271348 01616327 . ) ( = = s se Compared with the standard deviation of s , the bias is not a serious problem. The approximate )% 1 ( 100 α − confidence interval for σ is ) ( ) ( 2 / 2 / s se z s s se z s α α σ + ≤ ≤ − 1271348 . 96 . 1 4628571 . 1271348 . 96 . 1 4628571 . × + ≤ ≤ × − σ 0.7120413 0.2136729 ≤ ≤ σ 9-1 a) 25 : , 25 : 1 ≠ = μ μ H H Yes, because the hypothesis is stated in terms of the parameter of interest, inequality is in the alternative hypothesis, and the value in the null and alternative hypotheses matches....
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HW7 - Solution to HW7 Problem 1 a x x h = so 2 1 2 1 − =...

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