Consider a simple linear model with normal error: Y = 0 + 1x + , N(0, ^2 ) Suppose that 10 pairs of observed values from this model given in the following table are obtained:

i xi yi

1 1.9 0.7

2 1.1 -0.2

3 5.5 3.7

4 3.4 2.0

5 -0.1 0.0

6 4.6 -0.1

7 1.6 0.8

8 0.8 -1.0

9 4.4 3.4

10 0.1 -1.2

(1) Calculate the values of the M.L.E.'s 0, 1, and ^ 2 .

(2) Determine an unbiased estimator of Var( 0) and calculate its value.

(3) Let = 20+c1, where c is a constant. Determine an unbiased estimator of . For what value of c will the M.S.E. of be smallest?