midterm_sol - Stat 331 Midterm Solution Spring 2012 1[11...

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Stat 331 : Midterm Solution - Spring 2012 1. [11 marks] Consider the simple linear regression model: y i = β 0 + β 1 x i + i , i iid N (0 , σ 2 ) , i = 1 , 2 , . . . , n Let ˆ β 0 and ˆ β 1 denote the least squares estimates of β 0 and β 1 respectively. (a) [4 marks] You are given that ˆ β 0 = ¯ y - ˆ β 1 ¯ x where ˆ β 1 = s xy s xx . Show that V ( ˆ β 0 ) = σ 2 1 n + ¯ x 2 s xx , where s xx = n X i =1 ( x i - ¯ x ) 2 .
(b) [2 marks] Assume the sample size, n is fixed. Consider an experimental plan where the x i ’s can be controlled. How can you choose the x i ’s so that the estimate of β 0 is more reliable (i.e. so that there is less uncertainty in the estimate)? Support your answer with a theoretical argument.
(c) [2 marks] Suppose we predict the value of y p at x = x p . Why are our predictions more accurate when x p is closer to the sample mean, ¯ x ? Support your answer with 1
a theoretical argument.

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