sta108_handout2

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

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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 = + 1 X i + i ,i = 1 ,...,n, where 1 ,..., n are independent N (0 , 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 1 and are b 1 = S XY S XX ,b = Y- b 1 X. The following fact is very useful. Fact a) E ( b ) = , b) E ( b 1 ) = 1 , c) V ar ( b ) = h 1 n + X 2 S XX i 2 , d) V ar ( b 1 ) = 2 S XX . Notations in the text : V ar ( b ) is denoted by 2 ( b ), V ar ( b 1 ) is denoted by 2 ( b 1 ). Since 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 and 1 . Fact a) The random variable b- s ( b ) has a t-distribution with n- 2 degrees of freedom (df). b) The random variable b 1- 1 s ( b 1 ) has a t-distribution with n- 2 df. Confidence intervals: A (1- )100% confidence interval for 1 is given by b 1 t (1- / 2; n- 2) s ( b 1 ), where area to the left of t (1- / 2; n- 2) under the t-curve with n- 2 df is 1- / 2 . Similary, (1- )100% confidence interval for is given by b t (1- / 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: Y = 28 . 981 + 2 . 941 X,...
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sta108_handout2 - Handout 2 The observations are ( X 1 ,Y 1...

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