In an experiment, with t=4 populations with means, consider the four linear combinations of those means.

l1 = mu1 -3mu2 + mu4

L2 = mu1 + mu2 - 2mu4

L3 = mu1 + mu2 + mu3 + mu4

L4 = mu1 + mu2 - 3mu3 +mu4

a. Which of the four linear combinations are in contrasts?

b. Which pairs of contrasts are orthogonal?

c. Suppose we have two contrasts:

L1 = mu1 + mu2 + mu3 - 3mu4 and L2 = (1/3)mu1 + (1/3)mu2 + (1/3) mu3 - mu4

Is testing H0: L0 = 0 equivalent to testing HO: L2 = 0? Justify your answer.