1. Consider the simple linear regression model Least square estimate of B = 8 - Ziel(Ti -)(y -y) _Et(T - x)yi y = Bo + Bix+E, where E ~ N(0, 62). Liz(Ti - x)2 Since y's are independent and Var(yi) = 02 Vi then Var(B1) = Lil(Ti -T)2Var(yi) Suppose that all the x's are between 0 and 1 (i.e., x; is in the interval [0,1]). Suppose (Etz(Ti - x)2)2 you are going to run n = 10 experiments and you are able to design the experiment Now Var(B1) is minimum when (I; - 1) is maximum. (i.e., you choose the choose the x's). What is the choice of (x1, X2, ..., X10) that will Now _(i - 1)' s _(ri -0.5)2 (since _ (ti - c)? is least when c = 1) minimize the variance of 1. (Hint: Write out the formula for the variance of the estimator of the slope, argue that minimizing the variance is equivalent to maximizing = _ (x;-0.5)2 + (24-0.5)2 s _ (0-0.5)2 + (1-0.5)2 Sri 50.5 0.5