1. (50pts, Completely Randomized Design) Consider comparing mean of I groups under the following model. indep Yij N(μ₁,0

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1. (50pts, Completely Randomized Design) Consider comparing mean of I groups under the following model. indep Yij N(μ₁,0²), i, I; j = 1,2,..., J. (1) i = 1, with n = I x J. Our goal is to test the following hypotheses: Ho : μι = μ2 = · · = μι VS Not Ho. (2) (a) (7pts) Compute the maximum likelihood estimator (MLE) of μi, i = 1,..., I. (b) (8pts) Note that (1) can be equivalently rewritten as the following linear model: Yij = pi + €ij, ¤ij ~ N(0,0²), i = 1, · · · I; j = 1, 2, · · ·, J. (3) or equivalently y = Χμ + ε (4)
where y = (y11, , Y1J, Y21, ‚YI1, ···‚YIJ)¹ € R” € = (€11, ···, €1J, €21, ‚€I1, ···‚¤ÏJ)¹ € R" 2 9 μ = (μι, μ2, · ‚μI)T ERI Provide a proper design matrix X and compute the ordinary least square (OLS) estimator of Hi, i=1,2,.·, I. (c) (5pts) Compute an orthogonal projection matrix on col{X} noted by Px under (4). (d) (5pts) Compute both fitted-value vector ŷ and residual vector ê under (4). (e) (10 pts) It is well-known that the F-test statistic for testing (2) is 221( - 5)/(I−1) i=1 MStrt MSE F= F(I – 1, I(J — 1)) under Ho (5) - - E-1 Ej-1(Yij-i) ²/(I(J − 1)) j=1 where yi = -1 yij/J and ÿ = Σi-1 Σj=1 Yij/n. To justify (5), it is essential that MStrt and MSE (or equivalently their numerators) are independent. Justify the independence using the geometry of regression based on Px under (4). = 2 Y2J, €2J,