Four objects, Ou 0 2, 0 3, and 0 4, whose individual weights are to be determined are weighed in a common balance according to the following design:

In the design above, 1 indicates that the object was placed in the right pan, and - 1 indicates that the object was placed in the left pan of the balance. Assuming that errors of measurements are independent and normally distributed with zero mean and common unknown variance a 2:

(a) Find the maximum-likelihood estimates of the four objects.

(b) Find the covariance matrix of these estimates.

(c) Find the estimate of a 2.

0 0 O. Weight 1 1 1 1 20.2 1 -1 1 -1 8.1 1 1 -1 9.7 1 -1 -1 1 1.9 1 1 1 I 18.9 1 -1 1 -1 8.8 1 1 -1 1 10.4 1 -1 -1 12.0