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Tut1-2 - NATIONAL UNIVERSITY OF SINGAPORE Department of...

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1 NATIONAL UNIVERSITY OF SINGAPORE Department of Statistics and Applied Probability 2012/13 Semester 2 ST3131 Regression Analysis Tutorial 1 1. The data given below were obtained from an experiment with 10 cars. The amount of additive x and the amount of reduction in nitrogen oxide y are measured in some suitable units. x 1 1 2 3 4 4 5 6 6 7 y 2.1 2.5 3.1 3.0 3.8 3.2 4.3 3.9 4.4 4.6 The following simple linear regression model is expected to be appropriate y i = β 0 + β 1 x i + ε i , i = 1, ..., 10, where ε i ’s are independent and distributed as N (0, σ 2 ). (i) Restate the model using the matrix notation. (ii) Obtain the matrices X y , X X and ( X X ) 1 . (iii) Let . Obtain the least squares estimate of β . (iv) Find an estimate of . Hence obtain the standard errors of and . (v) Explain the meaning of β 1 . Does β 0 have any specific meaning here? (vi) Test for the significance of the simple linear relationship by using (a) a t -test for the slope; (b) the ANOVA approach.

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