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ex03ans - STAT 420 Examples#3 Spring 2008 Y i = 0 1 x i 1 p...

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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 i th restaurant in the sample, x i is the size of the student population (in thousands) and y i is the quarterly sales (in thousands of dollars). The values of x i and y i
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