Practice_Midterm_2010

# Suppose we have two independent random samples from

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Suppose we have two independent random samples from two normal populations: ( 29 1 2 1 2 1 , , , ~ , n X X X N μ σ K , and ( 29 2 2 1 2 2 , , , ~ , n Y Y Y N μ σ K . At the significance level α, please construct a test to test whether 1 2 2 μ μ = or not. (*Please include the derivation of the pivotal quantity, the proof of its distribution, and the derivation of the rejection region for full credit.) SOLUTION: Here is a simple outline of the derivation of the test: 0 2 : 2 1 0 = - μ μ H versus 0 2 : 2 1 - μ μ a H (a) We start with the point estimator for the parameter of interest ( 29 2 1 2 μ μ - : ( 29 Y X 2 - . Its distribution is ( 29 ( 29 2 1 2 2 1 / 4 / 1 , 2 n n N + - σ μ μ using the mgf for ( 29 2 , σ μ N which is ( 29 ( 29 2 / exp 2 2 t t t M σ μ + = , and the independence properties of the random samples. From this we have ( 29 ( 29 ( 29 1 , 0 ~ / 4 / 1 2 2 2 1 2 1 N n n Y X Z + - - - = σ μ μ . Unfortunately, Z can not serve as the pivotal quantity because σ is unknown. (b) We next look for a way to get rid of the unknown σ following a similar approach in the construction of the pooled-variance t-statistic. We found that ( 29 ( 29 [ ] 2 2 2 2 2 2 2 1 1 2 1 ~ / 1 1 - + - + - = n n S n S n W χ σ using the mgf for 2 k χ which is ( 29 2 / 2 1 k t t M = , and the independence properties of the random samples. (c) Then we found, from the theorem of sampling from the normal population, and the independence properties of the random samples, that Z and W are independent, and therefore, by the definition of the t-distribution, we have obtained our pivotal quantity: ( 29 ( 29 2 2 1 2 1 2 1 ~ / 4 / 1 2 2 - + + - - - = n n p t n n S Y X T μ μ , where ( 29 ( 29 2 1 1 2 1 2 2 2 2 1 1 2 - + - + - = n n S n S n S p is the pooled sample variance.

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(d) The rejection region is derived from ( 29 α = 0 0 | H c T P , where ( 29 2 2 1 0 2 1 0 ~ / 4 / 1 0 2 - + + - - = n n H p t n n S Y X T . Thus 2 / , 2 2 1 α - + = n n t c . Therefore at the significance level of α, we reject 0 H in favor of a H iff 2 / , 2 0 2 1 α - + n n t T 2B (for all non-AMS students) . In order to test the accuracy of speedometers purchased from a subcontractor, the purchasing department of an automaker orders a test of a sample of speedometers at a controlled speed of 55 mph. At this speed, it is estimated that the variance of the readings is 1. (a). How many speedometers need to be tested to have a 95% power to detect a bias of 0.5 mph or greater using a 0.01 level test?
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