In this derivation the variance of y-bar given x is shown to be N * sigma^2
However when I derive it ybar = beta_0 + beta_1Xbar + Ubar which boils down to the variance of Ubar because beta_0 beta_1 and Xbar are constants once we have drawn our sample.
Then we have: Var(Ubar) = (1/n)^2 Var(U_i) = (n times sigma^2)/n^2
which is sigma^2/n
and that is what I know the var(ybar) to be, how does this gel with the above proof?