spr04_1 - STA 6208 Spring 2004 Exam 1 Print Name: UFID: All...

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STA 6208 – Spring 2004 – Exam 1 Print Name: UFID: All questions are based on the following two regression models, where SIMPLE REGRESSION refers to the case where p =1 , and X is of full column rank (no linear dependencies among the predictor variables) Model 1: Y i = β 0 + β 1 X i 1 + ··· + β p X ip + ε i i ,...,n ε i NID (0 2 ) Model 2: Y = X β + ε X n × p ± β p ± × 1 ε N ( 0 2 I ) Cochran’s Theorem Suppose Y is distributed as follows with nonsingular matrix V : Y N ( μ, V σ 2 ) r (V) = n then: 1. Y ± ( 1 σ 2 A ) Y is distributed noncentral χ 2 with: (a) Degrees of freedom = r ( A ) (b) Noncentrality parameter = Ω = 1 2 σ 2 μ ± A μ if AV is idempotent 2. Y ± AY and Y ± BY are independent if AVB = 0 3. Y ± and linear function BY are independent if BVA = 0 1) Based on Model 1 , derive the normal equations for the simple linear regression model. 2) Show that n i =1 ( ˆ Y i - Y ) = 0. You may do this based on either Model 1 or Model 2 . 3) A simple linear regression is Ft, relating Frst weekend revenues ( Y ) to advertising expenditures ( X ) for n = 5 randomly selected horror Flms: ±ilm i Sales Ad Exp Scarier Movie 1 25.0 8.0 I Know What You Did Last Winter 2 15.0 6.0 Rural Legend 3 12.0 4.0 Shout 4 30.0 10.0
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