proj4 - , : , and : , 2 1 2 1 2 1 2 1 = = = Q.2. Give the...

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STA 4322/5328 – Spring 2010 – Project 4 – Due 4/20/10 Likelihood Ratio Tests and Wald Tests A large metropolitan school district is considering a long-term contract with a light bulb manufacturer. Two companies have placed bids, and the school district wishes to test whether the mean lifetimes of the two brands differ (the prices are the same). Suppose that Brand A’s bulbs lifetimes are Exponentially distributed with mean of θ 1 and Brand B’s are Exponentially distributed with mean θ 2 . The school district’s engineer tests n 1 =20 from brand A and n 2 =20 from brand B, observing the lifetime of each bulb. The density functions for the brands are: ( 29 ( 29 = = - - otherwise 0 0 , 0 1 | otherwise 0 0 , 0 1 | 2 2 2 1 1 1 2 1 θ y e y f x e x f y x She observes X 1 ,…,X 20 from Brand A and Y 1 ,…,Y 20 from Brand B and wishes to test: H 0 : θ 1 = θ 2 versus H A : θ 1 θ 2 at the α = 0.05 significance level. Q.1. Give the Likelihood function under the two parameter spaces: ( 29 { } ( 29 { }
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Unformatted text preview: , : , and : , 2 1 2 1 2 1 2 1 = = = Q.2. Give the ML estimators (formulas) for 1 , 2 for , and = 1 = 2 for Q.3. Give the Likelihood Ratio Test Statistic (Formula) and Rejection Region for this test. Q.4. Give the Wald (Z) Test Statistic (Formula) and Rejection Region for this test (based on large-sample normal approximation). Q.5. Generate 10000 random samples of size 20 from Exp( 1 ) and Exp( 2 ) and obtain the following quantities: ( 29 obs Y X Z S Y S X , ln 2 , , , , , , , ^ 2 ^ 1 ^ 2 2 -Q.6. Give histograms of your test statistics (for the Wald test, give a histogram of (Z obs ) 2 Chi-square (df=1)) Complete this for the following values for 1 , 2 1 2 Pr{Reject H } LR Test Pr{Reject H } Wald Test 1000 1000 1100 1000 1200 1000...
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