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Unformatted text preview: MA2B HOMEWORK 6 SOLUTIONS Problem 1. We have that X 1 ,...,X n are i.i.d. with normal distribution N ( μ,σ 2 ) where σ = 10. We want to test the null hypothesis H : μ = μ against K : μ = μ , where μ = 0. From example A on page 308–309 of Rice, we know that such a test re jects the null hypothesis when  ¯ X n μ  ≥ σ √ n · z ( α 2 ), where z ( α ) is the inverse of Φ as defined on page 305, namely P ( Z > z ( α )) = α . For n = 25, α = . 1, σ = 10, and μ = 0, this gives z ( . 05) = 1 . 64485 (by table or mathematica) and so  ¯ X n  ≥ 10 5 (1 . 64485) = 3 . 28971 For n = 100, this gives  ¯ X n  ≥ 10 10 (1 . 64485) = 1 . 64485 The power curve with n = 100 should depart more rapidly away than the power curve for n = 25. Problem 2. (a) As defined in 9.1, the likelihood ratio is given by Λ( x ) = P ( x  H ) P ( x  H A ) . For each x j we compute: Λ( x 1 ) = . 2 . 1 = 2 Λ( x 2 ) = . 3 . 4 = 0 . 75 Λ( x 3 ) = . 3 . 1 = 3 Λ( x 4 ) = . 2 . 4 = 0 . 5 . We relabel the ordering of x i according to Λ by y 1 = x 3 ,y 2 = x 1 ,y 3 = x 2 ,y 4 = x 4 and we let the random variable j = J...
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 Winter '08
 Makarov,N
 Differential Equations, Statistics, Equations, Central Limit Theorem, Normal Distribution, Probability, Variance, likelihood ratio, MA2B HOMEWORK

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