I have read an exercise and its correction and I am not sure I understand it: The population of a country is divided into 3 housing zones in proportions: 10%, 40% and 50%. 200 subjects are chosen at random from each zone. The (oberved) means of a dummy variable $X$ in each area were: $m1 = 1.5$, $m2 = 2$ and $m3 = 2.5$. The estimate of the common (residual) variance is $s^2_R = 5$.
The question is to find the mean of the dummy variable in the country and to estimate its variance.
- Calculation of the mean:
$$m = 10\% \times 1.5 + 40\% \times 2 + 50\% \times 2.5 = 2.2$$
- Calculation of the variance of the mean:
$$ var(m) = (0.1^2 + 0.4^2 + 05^2)/200 \times s^2_R $$
I don't understand all the formula of the variance.
The begining is $var(aX) = a^2 var(X)$ but why should we divide it by $200$?
Thanks for any clarification!