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Unformatted text preview: Tutorial 5 1. Suppose we have n = 10 observations (Xi , Yi ) and ﬁt the data with model Yi = β0 + β1 Xi + εi with εi , i = 1, ..., 10 are IID N (0, σ 2 ). We have the following calculations.
n ¯ X = 0.5669,
n i=1 ¯ Y = 0.9624,
i=1 n Yi2 = 10.2695, Xi2 = 4.0169, Xi Yi = 6.2841.
i=1 (a) Write down the estimated model (b) Test H0 : β1 = 1 with α = 0.05 (c) For a new X = 1, ﬁnd the 95% CI for its expected response (d) For a new X = 0.5, ﬁnd the 95% prediction interval for its possible response 2. For the least square estimator b0 , b1 of simple linear regression model, ﬁnd Cov (b0 , b1 ) 3. Suppose A : m × n is a constant matrix and Y : n × 1, is a random vector. Then Var(AY ) = AVar(Y )A Please give your proof for m = 2, n = 3. 4. For multiple linear regression, the normal equations are
n ei = 0
i=1 n ei Xi1 = 0
i=1 . . . n ei Xip = 0
i=1 Prove that n ˆ Yi ei = 0
i=1 1 5. Consider the simple regression model Y = β0 + β1 X + ε. We have n observations (Xi , Yi ) with sample correlation coeﬃcient rXY . Standardize ˜ Xi = ¯ Xi − X , n ¯ (Xj − X )2 /(n − 1) j =1 ˜ Yi = ¯ Yi − Y n ¯2 j =1 (Yj − Y ) /(n − 1) ˜ ˜˜ ˜ What is the estimated model for Yi = β0 + β1 Xi + εi ? 2 ...
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This note was uploaded on 10/04/2010 for the course STAT ST3131 taught by Professor Xiayingcun during the Fall '09 term at National University of Singapore.
 Fall '09
 XIAYingcun
 Regression Analysis

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