09_11 - STAT 420 Examples for 09/11/2007 Fall 2007 The...

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STAT 420 Examples for 09/11/2007 Fall 2007 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 ( ) ² ² ² ³ ´ µ µ µ · - ± 2 2 1 ± , x x i 0 ˆ = x ˆ Y 1 - ~ N ( ) ² ² ² ³ ´ µ µ µ · - ± ± 2 2 2 0 ± , x x n x i i = N ( ) ² ² ² ³ ´ µ µ µ · ² ² ² ³ ´ µ µ µ · - + ± 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 ) _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ Confidence interval for β 1 : ( ) ± - ± 2 2 1 t ˆ e x x s i α SXX e s ± 2 1 t ˆ where 2 t is the appropriate value of t-distribution with n – 2 degrees of freedom. Test statistic for H 0 : β 1 = β 10 : T = ( ) SXX i e e s x x s 10 1 2 10 1 ˆ ˆ - = - - ± ( n – 2 degrees of freedom )
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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 04/29/2010 for the course STAT stat 420 taught by Professor Stepanov during the Spring '07 term at University of Illinois at Urbana–Champaign.

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09_11 - STAT 420 Examples for 09/11/2007 Fall 2007 The...

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