Given the regression eqn: $y_0= \beta_0 +\beta_1 x_i + \epsilon_i$
I am having difficulty in calculating the variance of $\beta_0$
Here is how I proceeded:-
$\operatorname{Var}(b_0)= \operatorname{Var}(\bar Y -b_1\bar X)$, where $b_0,b_1 \text{are parameters estimator} $;
\begin{align} \operatorname{Var}(b_0)& =\operatorname{Var}(\bar Y)+\operatorname{Var}(b_1\bar X) -2\operatorname{Cov}(\bar Y,b_1\bar X)\\[10pt] &= \operatorname{Var}(\bar Y)+(\bar X)^2\operatorname{Var}(b_1) -2\bar X\operatorname{Cov}(\bar Y,b_1)\\ \end{align}
I have already got the value of $Var(b_1)$,but i cannot prove $\operatorname{Cov}(\bar Y,b_1)=0$.
Thanks!