sta108_handout2

# sta108_handout2 - Handout 2 The observations are X 1,Y 1 X...

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Unformatted text preview: Handout 2 The observations are ( X 1 ,Y 1 ) ,..., ( X n ,Y n ) and the model is Y i = β + β 1 X i + ε i ,i = 1 ,...,n, where ε 1 ,...,ε n are independent N (0 ,σ 2 ). We will use the following simplifying notations S XX = X ( X i- ¯ X ) 2 ,S Y Y = X ( Y i- ¯ Y ) 2 ,S XY = X ( X i- ¯ X )( Y i- ¯ Y ) . With these notations, estimates of β 1 and β are b 1 = S XY S XX ,b = ¯ Y- b 1 ¯ X. The following fact is very useful. Fact a) E ( b ) = β , b) E ( b 1 ) = β 1 , c) V ar ( b ) = h 1 n + ¯ X 2 S XX i σ 2 , d) V ar ( b 1 ) = σ 2 S XX . Notations in the text : V ar ( b ) is denoted by σ 2 ( b ), V ar ( b 1 ) is denoted by σ 2 ( b 1 ). Since σ 2 is estimated by MSE we can obtain estimates of V ar ( b ) and V ar ( b 1 ). Estimates of V ar ( b ) and V ar ( b 1 )are given by s 2 ( b ) = 1 n + ¯ X 2 S XX MSE , s 2 ( b 1 ) = MSE S XX . The following fact will be used to construct confidence intervals for β and β 1 . Fact a) The random variable b- β s ( b ) has a t-distribution with n- 2 degrees of freedom (df). b) The random variable b 1- β 1 s ( b 1 ) has a t-distribution with n- 2 df. Confidence intervals: A (1- α )100% confidence interval for β 1 is given by b 1 ± t (1- α/ 2; n- 2) s ( b 1 ), where area to the left of t (1- α/ 2; n- 2) under the t-curve with n- 2 df is 1- α/ 2 . Similary, (1- α )100% confidence interval for β is given by b ± t (1- α/ 2; n- 2) s ( b 1 ) . 1 For the Housing data, we have n = 19 , ¯ X = 15 . 719 , ¯ Y = 75 . 211 ,S XX = 40 . 805 ,S Y Y = 556 . 078 ,S XY = 120 . 001 , Fitted regression line: ˆ Y = 28 . 981 + 2 . 941 X,...
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## This note was uploaded on 09/28/2011 for the course STA STA 108 taught by Professor Jiang during the Summer '09 term at UC Davis.

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sta108_handout2 - Handout 2 The observations are X 1,Y 1 X...

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