Let $n_1, n_2, n_3$ denote sample sizes of $3$ samples of people, where $n_1 < n_2 < n_2$ and let $X_1, X_2, X_3$ denote the number of left handed people in each sample. Let $\hat{p_{1}} = \frac{(X_1)}{n_1}, \hat{p_{2}} = \frac{(X_2)}{n_2}, \hat{p_{3}} = \frac{(X_3)}{n_3}$
Show that $\hat{p_{12}} = \frac{(n_1)(\hat{p_1}) + (n_2)(\hat{p_2})}{n_1 +n_2}$ is an unbiased estimator, and find the variance. In order to show unbiasedness you prove that $E(\hat{p_{12}}) = \mu$, by allowing $\frac{X_1}{n_1}$ etc to = $\mu$ as they follow a normal distribution. In order to find theoretical variance, do i need to prove that $\frac{\sigma^2}{n} = 0$, or find it in terms of $X_1, n_1$ etc by showing that $\frac{\sigma}{n} = \frac{\sum(X_i - X_{bar})}{n}$.