AMS312m_2010 - AMS312.01 Midterm Exam Spring 2010 Name ID...

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AMS312.01 Midterm Exam Spring 2010 ♠♣♥♦ Name: ______________________________________ ID: ___________________________ Signature: ______________________ Instruction: This is a close book exam. Anyone who cheats in the exam shall receive a grade of F. Please provide complete solutions for full credit. The exam goes from 12:50-2:10pm. Good luck! 1. Let X 1 , X 2 , …, X n be a random sample from a normal population N(μ, σ 2 ). Furthermore, the population variance σ 2 is known. For a 1-sided test of H 0 : μ = μ 0 versus Ha: μ < μ 0 , at the significance level α, (a). Derive the one-sample Z-test using the pivotal quantity method. (* Please include the derivation of the pivotal quantity, the proof of its distribution, and the derivation of the rejection region for full credit.) (b). For a given specific alternative hypothesis, Ha: μ = μ a (< μ 0 ) , please derive the formula for the power of the test at the significance level α. (* Please include the derivation of the distribution of the test statistic Z 0 under the alternative hypothesis Ha: μ = μ a for full credit.) Solution (Lectures 11 & 12). (a) First we derive the pivotal quantity for the inference on μ based on a random sample from a normal population N(μ, σ 2 ). 1. Point Estimator for μ : 2 ~ ( , ) X N n σ μ X is NOT a pivotal quantity since its distribution is not entirely known. 2. Let X Z n μ σ - = , we can show that ~ (0,1) Z N using the moment generating function method: ( 29 1 1 2 2 2 exp( ) exp( ) exp( )exp( ) exp( )exp( ) exp( ) exp( ) 1 exp( )exp ( ) ( ) exp 2 2 Z n n i i i i X t t M t tZ t X n n n
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