2. Consider the following model for the responses measured in a randomized block design containing b blocks and & treatments: Yij = #+ Tit Bjteij: where: Yay = response to treatment i in block j; " = overall mean; 71 = nonrandom effect of treatment i, where _ ri = 0; i=1 3, = random effect of block j, where Bys are independent, normally distributed random variables with E [Bj] = 0 and V (8;) = of for j = 1, 2, .., b; By = random error terms where es are independent, normally distributed random variables with E[ejl = 0 and V (e;) = of for i = 1, 2, .... k and j = 1, 2, .., b. Further assume that the Bis and eys are also independent. This model differs from that presented in Section 13.8 of the textbook in that the block effects are assumed to be fixed but unknown constants. (a) Assuming the model just described is accurate, show that observations taken from different blocks are independent of one another. That is, show that Y and You are independent if j # j', as are Vij and Yoy if i f i' and j # j'. (b) Derive the covariance of two observations from the same block. That is, find Cov (Vij, Y;) if ifi. (c) Two random variables that have a joint normal distribution are independent if and only if their covariance is 0. Use the result from part (b) to determine the conditions under which two observations from the same block are independent of one another. (d) Find the expected value and variance of Vij. (e) Let Yi, denote the average of all responses to treatment i. Use the model to derive E [Y;] and V (Yi ). (f) Calculate the bias of Yo. Is it an unbiased estimator of the mean response to treatment