Use the residual variance to estimate the variance of the estimated common mean (ANOVA)

Question

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.

1. Calculation of the mean:

$$m = 10\% \times 1.5 + 40\% \times 2 + 50\% \times 2.5 = 2.2$$

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!

Answer 1

The answer is:

$$var(m) = var( 10\% \times m_1 + 40\% \times m_2 + 50\% \times m_3) \\ = 0.1^2 var(m_1) + 0.4^2 var(m_2) + 0.5^2 var(m_3)$$

With $var(X_1) = var(X_2) = var(X_3) = s^2_R$, the residual variance.

We can deduce that $var(m_1) = s^2_R / n_1, var(m_2) = s^2_R / n_2, var(m_3) = s^2_R / n_3$. From the information given in the exercise $n_1=n_2=n_3 = 200$.