Please show steps.

The oneway random factor ANOVA model with n, observations for each level of the random factor has the form yij2+03i+ 221,2,m,k; j=1121...,ni_ where a is a constant, but the a, and 6,-3- are independent random variables distributed as (1,: ~ N(0,o:) and 6,, ~ N(0,02). The total number of observations is n = m + n; + ' - - + in. a) Show that the model assumptions stated above imply E(y,j) = p for all i and 3'. Answer: b) In general, if U1 and U2 are independent (and therefore uncorrelated) random variables, then we know that VeT(U1 + U2) = Var(U1) + Var(U2). Using this fact, obtain a simplied expression for Ve'r(y,3-). Answer: c) For any 4 random variables U1, U2, U3, U4, we know that COU(U1+ U2, U3 + U4) = CWU1,U3) + COU(U1, U4) + COU(U2, U3) + COU(U2, U4). Using these facts, show that if i * i', then Cov(yij, yay) = 0, i.e., observations from different groups are uncorrelated. Answer: d) Now, we show that observations from the same group have nonzero covariance. Show that if j # j', then Cov(yij, yiji) = 2. (Useful fact: for any random variable U, Cov(U, U) = Var(U)) Answer: e) The final step is to obtain the intra-class correlation coefficient for two observations from the same group. Remembering the definition of correlation, Corr(U, V) = Cov(U, V) Var(U) Var(V) show that if j # j' then 02 Corr(yij, yij') = 02 + 02