STA 6208 – Spring 2004 – Exam 1Print Name:UFID:All questions are based on the following two regression models, where SIMPLE REGRESSIONrefers to the case wherep=1, and X is of full column rank (no linear dependencies among thepredictor variables)Model 1:Yi=β0+β1Xi1+···+βpXip+εii,...,n εi∼NID(0,σ2)Model 2: Y=Xβ+εX≡n×p±β≡p±×1ε∼N(02I)Cochran’s TheoremSupposeYis distributed as follows with nonsingular matrixV:Y∼N(μ,Vσ2)r(V) =nthen:1.Y±(1σ2A)Yis distributed noncentralχ2with:(a) Degrees of freedom =r(A)(b) Noncentrality parameter = Ω =12σ2μ±AμifAVis idempotent2.Y±AYandY±BYare independent ifAVB=03.Y±and linear functionBYare independent ifBVA=01) Based onModel 1,derivethe normal equations for the simple linear regression model.2) Show that∑ni=1(ˆYi-Y) = 0. You may do this based on eitherModel 1orModel 2.3) A simple linear regression is Ft, relating Frst weekend revenues (Y) to advertising expenditures (X) forn= 5 randomly selected horror Flms:±ilmiSalesAd ExpScarier Movie125.08.0I Know What You Did Last Winter215.06.0Rural Legend312.04.0Shout430.010.0
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