STAT2004J – Linear Modelling Tutorial 3
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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:
Tyre 1 |
Tyre 2 |
S T |
S T |
15 53 |
15 57 |
20 55 |
20 65 |
25 63 |
25 78 |
30 65 |
30 77 |
35 78 |
35 91 |
40 83 |
40 95 |
Assuming that the tyres have a common value of β but possibly diferent values for α , estimate β and the two α’s.
2024-01-02