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you in advance.
1. (50pts, Completely Randomized Design) Consider comparing mean of I groups under the following model. indep Yij N(μi, 0²), i = 1, · I; j = 1,2,..., J. (1) with n = I x J. Our goal is to test the following hypotheses: Ho 12. = μI 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, tij ~ N(0,0²), i = 1,... I; j = 1,2,..., J. (3) or equivalently y = Χμ + ε (4) where y = (y11,,Y1J, Y21, Y2J, YI1, YIJ)¹ € Rn 2 " € = (€11,, €1J, €21,, €2J,, €11, ... , €1J)¹ € R μ = (μ₁₂ μ₂,₂ μ1)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} denoted by Px under (4). (d)) (5pts) Compute both fitted-value vector y and residual vector ê under (4).