stat425fall10exam1-sln - 1 1 A biologist gathers data of...

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Unformatted text preview: 1 1. A biologist gathers data of the form ( x i , y i ), i = 1 ,.., n , where x i is the amount of time since the start of the experiment and y i is the amount of pollen that arrives in pollen trap i . She considers the regression model i i i y x e β = + where the i e s are uncorrelated random variables with mean 0 and variance 2 σ . a) What is the least-squares estimate of β ? i i x y β ˆ ˆ = , ) ˆ ( ˆ i i i y y e- = ∑ =- = n i i i x y RSS 1 2 ) ( β ∑ = = =-- = ∂ ∂ n i i i i x y x RSS 1 ˆ ) ˆ ( 2 β β β β ∑ = =- n i i i i x y x 1 ) ˆ ( β , ∑ ∑ = = =- n i n i i i i x y x 1 1 2 ˆ β ∑ ∑ = = = n i i n i i i x y x 1 2 1 ˆ β b) Show that least square estimate β ˆ is unbiased. β β β = = = = ∑ ∑ ∑ ∑ ∑ ∑ = = = = = = n i i n i i i n i i n i i i n i i n i i i x x x x y E x x y x E E 1 2 1 1 2 1 1 2 1 ) ( ) ( ) ˆ ( . Thus, it is unbiased....
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This note was uploaded on 12/12/2010 for the course STAT 425 taught by Professor Ma,p during the Fall '08 term at University of Illinois, Urbana Champaign.

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stat425fall10exam1-sln - 1 1 A biologist gathers data of...

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