Consider the simple linear regression model y = β0 + β1x + where E[ ] = 0, and var( ) = σ2. a. Show that the least squa

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Consider the simple linear regression modely = β0 + β1x + where E[ ] = 0, andvar( ) =σ2.
a. Show that the least squares estimators are given by

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where

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Hint: Solve the normal equations and use the fact that ∑(xi −̄x)(yi − ̄y) = ∑yi(xi − ̄x).
b. Show that the CMEs ˆβ0 and ˆβ1 areunbiased.
c. Show that

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d. Show that the adjusted mean response at level x,ˆy(x)

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and that, for a new observation at level x,̇y(x):

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3₁ = Sxy. ; Sxx Bо = ÿ - B₁
Sxy = Σ(Xi − Ñ)(Yi — Y), y Sxx = Σ(x; − x)².
var(ŝ₁) = 0² 1 1 (+) ( + ) ; var(B) = 0² Sxx
(x (30+6 ~² (²/1 + (²2=-2) ²)) Sxx n ŷ(x) = ßo + Â₁x ~ No + ₁x, 0² N
1 Sxx n (²) = 8₂ +3₁² + €~²N (3₂+ B₁²₁0²³ ( 1 + = + (x = 2² ) )