What is the error of the global mean given known equal errors for each stratum? This was asked to me by a colleague. It seems to be an easy one but I wasn't able to find the derivation online. I've added my own attempt below.
 A: We have $H$ strata, each with sampled mean $\bar Y_{h}$, taken from population of size $N_h$ and error $\epsilon_h$. $\bar Y, N, \epsilon$ are the corresponding quantities for the whole sample.
\begin{align}
\bar Y &= \sum_{h=1}^H {N_h \over N}\bar Y_h\\
\Rightarrow V(\bar Y)&= \sum_{h=1}^H {N_h^2 \over N^2} V(\bar Y_h)\\
\end{align}
The error for any confidence interval $\%$ is proportional to the standard error, $\epsilon = k\sqrt{V(\bar Y)}$. Also, we said all $\epsilon_h$ are the same. It is reasonable that the confidence level is also fixed:
\begin{align}
V(\bar Y) &= \sum_{h=1}^H {N_h^2 \over N^2} {\epsilon_h^2 \over k_h^2}\\
V(\bar Y) &= {\epsilon_H^2 \over k_H^2N^2} \sum_{h=1}^H N_h^2\\
\Rightarrow \epsilon &= k{\epsilon_H \over k_H N} \sqrt{\sum_{h=1}^H N_h^2}
\end{align}
If we again assume the same confidence level is desired for the global mean, we end up with
$$\epsilon = {\epsilon_H \over N} \sqrt{\sum_{h=1}^H N_h^2}$$
Since the sum of squares is equal or less than the square of the sum, the error for the global mean $\bar Y$ is guaranteed not to exceed the error at each stratum, and it diminishes with the number of strata used.
