# Ex03ans - STAT 420 Examples#3 Spring 2008 Y i = 0 1 x i 1 p x i p1 e i E e i = 0 Y = X e Var e i = 2 i = 1 2 n Cov e i e j = 0 i j ij = In 2 E e =

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STAT 420 Examples #3 Spring 2008 Y i = β 0 + β 1 x i 1 + … + β p x i p –1 + e i , i = 1, 2, … , n , E ( e i ) = 0, Var ( e i ) = σ 2 , Cov ( e i , e j ) = 0, i j . Y = X β + e , E ( e ) = 0 , Var ( e ) = (( Cov ( e i , e j ) )) i j = σ 2 I n . Y = ± ² ³ ³ ³ ³ ³ ´ µ n Y ... Y Y 2 1 , X = ± ² ³ ³ ³ ³ ³ ´ µ - - - 1 1 1 ... 1 ... ... ... ... ... ... 1 ... 1 2 1 2 2 2 1 2 1 2 1 1 1 p n n n p p x x x x x x x x x , β = ± ² ³ ³ ³ ³ ³ ´ µ - 1 ... 1 0 p , e = ± ² ³ ³ ³ ³ ³ ´ µ n e e e ... 2 1 . ˆ = ( X T X ) 1 X T Y , E ( ˆ ) = β , Var ( ˆ ) = σ 2 ( X T X ) 1 . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ The (normal) simple linear regression model: y i = β 0 + β 1 x i + ε i , where ε i ’s are independent Normal ( 0 , σ 2 ) ( iid Normal ( 0 , σ 2 ) ). β 0 , β 1 , and σ 2 are unknown model parameters. Suppose x i ’s are fixed (not random). Y i ’s are independent Normal ( β 0 + β 1 x i , σ 2 ) random variables. 1 ˆ = ( ) ( ) · · - - 2 Y x x x x i i i ~ N ( ) ¸ ¸ ¸ ¹ º » » » ¼ ½ - · 2 2 1 ± , x x i 0 ˆ = x ˆ Y 1 - ~ N ( ) ¸ ¸ ¸ ¹ º » » » ¼ ½ - · · 2 2 2 0 ± , x x n x i i = N ( ) ¸ ¸ ¸ ¹ º » » » ¼ ½ ¸ ¸ ¸ ¹ º » » » ¼ ½ - + · 1 , 2 2 2 0 ± x x x n i ( ) 2 1 0 2 ˆ ˆ Y 2 1 S i i e x n · - - - = ( ) 2 2 ± S 2 e n - ~ χ 2 ( n – 2 )

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1. The owner of Momma Leona’s Pizza restaurant chain believes that if a restaurant is located near a college campus, then there is a linear relationship between sales and the size of the student population. Suppose data were collected from a sample of 10 Momma Leona’s Pizza restaurants located near college campuses. For the
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## This note was uploaded on 02/10/2009 for the course STAT 420 taught by Professor Stepanov during the Spring '08 term at University of Illinois at Urbana–Champaign.

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Ex03ans - STAT 420 Examples#3 Spring 2008 Y i = 0 1 x i 1 p x i p1 e i E e i = 0 Y = X e Var e i = 2 i = 1 2 n Cov e i e j = 0 i j ij = In 2 E e =

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