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

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:

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

1. 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!

$$var(m) = var( 10\% \times m_1 + 40\% \times m_2 + 50\% \times m_3) \\ = 0.1^2 var(m_1) + 0.4^2 var(m_2) + 0.5^2 var(m_3)$$
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 $$n_1=n_2=n_3 = 200$$.