Consider the data shown in Problem 3-10. (a) Write out the least squares normal equations for this problem, and solve them for and

Consider the data shown in Problem 3-10.

(a) Write out the least squares normal equations for this problem, and solve them for µ̂ and τ̂_i, using the usual constraint (Σ³_{i =1} τ̂_i = 0). Estimate τ₁ – τ₂.

(b) Solve the equations in (a) using the constraint τ₃ = 0. Are the estimators τ̂_i, and µ̂ the same as you found in (a)? Why? Now estimate τ₁ - τ₂ and compare your answer with that for (a). What statement can you make about estimating contrasts in the τ_i?

(c) Estimate µ + τ_i, 2τ₁ – τ₂ – τ₃ and µ + τ₁ + τ₂ using the two solutions to the normal equations. Compare the results obtained in each case.

Problem 3-10.

The response time in milliseconds was determined for three different types of circuits that could be used in an automatic valve shutoff mechanism. The results are shown in the following table:

Circuit Type Response Time 1 9. 12 10 8 15 20 21 23 17 30 3 5 8 16 7