STAT2004J – Linear Modelling
Tutorial 3

Question 1. For the simple linear regression model Yi = β0  + β1Xi  + Ei, i = 1, . . . , n with

i = Yi - Y(ˆ)i ,

show that

(a)Σ i = 0

(b)Σ iXi = 0

(c)  Hence or otherwise showΣ Y(ˆ)i i = 0

(d)  Use (a) and (b) to show that the sample correlation between i  and Xi  is zero.

Question 2. In a simple linear regression model with

E(Y) = α0 + α1 (X - X(-))

under the standard model assumptions, show that the expectation of the residual sum of squares (RSS) is (n - 2)σ2 , where

RSS = (Yi -Y(ˆ)i)2

andY(ˆ)i  =ˆ(α)0 + ˆ(α)1 (Xi - X(-)) is the ith  itted value. Deduce that ˆ(σ)2 =  is an unbiased estimator of σ2 .

Question 3. Observations Y in an experiment have constant variance σ2 and linear regression on a predetermined variable X.  The experiment is divided into two groups.  In the irst group the regression equation is

E(Y) = α1 + βX,

whereas in the second group the regression coefficient is the same but the intercept is diferent, i.e.

E(Y) = α2 + βX.

A sample of size n is taken from the irst group, and independently, a sample of size n is taken from the second group, so in total there are 2n independent pairs of observations (X, Y).  Obtain the least square estimators of α1 , α2 , β .

Question 4.

For any particular vehicle tyre run under given conditions of load,inlation pressure and ambient temperature, the equilibrium temperature T(。C)generated in the shoulder of the tyre may be assumed to vary with the vehicle speed, S (in mph), according to an equation of the form T = α + βS. As part of an investigation into tyre performance,two tyres of the same size were run under the same load, pressure and ambient conditions at a number of diferent speeds,and the following shoulder temperatures were recorded: