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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, o²), 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 pi, 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, Y2J, YIL, •, YIJ)¹ € Rn 3 € = (€11,, €1J, €21,, €2J,, €11,..., €1J)¹ € R μ = (μ₁, μ₂,₂ μ₁) ERI Provide a proper design matrix X and compute the ordinary least square (OLS) estimator of Hi, i=1,2,. , I. (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 è under (4).