# 09_11ans - STAT 420 Examples for Fall 2007 The(normal...

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Unformatted text preview: STAT 420 Examples for 09/11/2007 Fall 2007 The (normal) simple linear regression model: y i = β + β 1 x i + ε i , where ε i ’s are independent Normal ( , σ 2 ) ( iid Normal ( , σ 2 ) ). β , β 1 , and σ 2 are unknown model parameters. Suppose x i ’s are fixed (not random). & Y i ’s are independent Normal ( β + β 1 x i , σ 2 ) random variables. 1 & ˆ = ( ) ( ) ¡ ¡-- 2 Y x x x x i i i ~ N ( ) ¢ ¢ ¢ £ ¤ ¥ ¥ ¥ ¦ §- ¡ 2 2 1 ¡ & , x x i & ˆ = x & ˆ Y 1- ~ N ( ) ¢ ¢ ¢ £ ¤ ¥ ¥ ¥ ¦ §- ¡ ¡ 2 2 2 ¡ & , x x n x i i = N ( ) ¢ ¢ ¢ £ ¤ ¥ ¥ ¥ ¦ § ¢ ¢ ¢ £ ¤ ¥ ¥ ¥ ¦ §- + ¡ 1 , 2 2 2 ¡ & x x x n i ( ) 2 1 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 α where 2 t α is the appropriate value of t-distribution with n – 2 degrees of freedom. Test statistic for H : 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 ) 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 for the 10 restaurants in the sample are summarized in the following table:...
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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_11ans - STAT 420 Examples for Fall 2007 The(normal...

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