You have grades ($Y $) for men ($D = 0$) and women ($D = 1$). The mean grades (out of total possible score of 100) are 65 for men and 72 for women. Regression of $Y$ on $D$ yields:
$Y_i = b_0 + b_1D_i + e_i $.
(a) What values would you get for $b_0$ and $b_1$? Start from the expressions for the OLS estimator of the slope and intercept to defend your answer.
(b) Use the expression for $V (b_1)$ to prove that the effective $b_1$ is estimated most precisely (that is, with smallest variance) if you have equal numbers of men and women in your sample.
My thoughts:
a) Since there is no further information the best estimates are the means.
$b_1=Corr(X,Y)\frac{sy}{sx}$ $b_0= Y* - b_1 X*$
where $Y*$ is the mean of $Y$ and $X*$ is the mean of $X$. So can I say $Y=65 + 7D$ or tov does this work when I have no more information!?
b)
$Var(b_1) = \frac{\sigma^2}{\sum_{i=1}^n(X_i − X*)^2} No idea since I think a) is wrong and I still have no further information than the means.