Question

Could you please explain the step 1 and step 2 for me? I know the notes want to show the formula for E(SSTR), but I don't know what are these two steps doing.

1.2 Expectations of sums of squares Expectation of the sums of squares be derived from the following fact about sample variance. If X1, . . . , Xn are iid, with mean μ and variance σ2, then The sample variance is an unbiased estimator of the variance σ2, i.e., E(s2)-σ2. Equivalently, If we further assume Xi's to be normal random variables, then . Ση 1 (Xi X)2 follows a X2 distribution with n 1 degrees of freedom, which is usually denoted by X(n-1) Why "n 1"Iis the degrees of freedom of s2 . The sarnple mean -n-1 Ση 1° i is independent with the residuals (Xi , . . . , xn X), simce Cov(X, X X)-0, fori1,... ,n, riables, covariance b and for normal randoin va eing zero Imeans ndependence For each i 1, , r, {Y'y}n-i are 1.1.d. N(μί, σ2), thus and ience Note also that SSE has the χ(nr 1) distribution, in particular, dr(SSE) = nT r. Y t N(0, σ2/m); and nr Στ i ni(Yi 4)-. Y.. .. μ., so that Y.. i. _ μί are independen has N (0, σ2/nr) distribution μ e Thus に! . By simple algebra,

Answer 1