Consider the data shown in Problem 3.26.

(a) Write out the least squares normal equations for this problem and solve them for ????̂ and ????̂i, using the usual constraint

(Σ3 i=1 ̂ ????i = 0

)

. Estimate ????1 − ????2.

(b) Solve the equations in

(a) using the constraint ̂ ????3 = 0.

Are the estimators ̂ ????i and ????̂ the same as you found in (a)? Why? Now estimate ????1 − ????2 and compare your answer with that for (a). What statement can you make about estimating contrasts in the ????i?

(c) Estimate ???? + ????1, 2????1 − ????2 − ????3, and ???? + ????1 + ????2 using the two solutions to the normal equations. Compare the results obtained in each case.