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A problem I've found and been thinking about for a while but not sure I can articulate properly. Any help appreciated.

A survey is being planned with the goal of interviewing $n$ people in some number $J$ of clusters. For simplicity, assume simple random sampling of clusters and a simple random sample of size $n/J$ (appropriately rounded) within each sampled cluster. Consider inferences for the proportion of Yes responses in the population for some question of interest. The estimate will be simply the average response for then people in the sample. Suppose that the true proportion of Yes responses is not too far from 0.5 and that the standard deviation among the mean responses of custers is $0.1$.

Suppose the cost of the survey is \$50 per interview, plus \$500 per cluster. Further suppose that the goal is to estimate the proportion of Yes responses in the population with a standard error of no more than 2%. What values of $n$ and $J$ will achieve this at the lowest cost?

The problem I'm having is determining how to use the information given about the standard deviation among the mean responses of clusters being $0.1$. I'm interpreting that to mean that for each cluster mean $\theta_j$ we have $\text{sd}(\theta_j)=0.1$ but that seems to imply the overall standard error is free of $n$, which doesn't seem to make sense.

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The answer I was looking for (and the original statement of the problem) can be found here. http://www.stat.columbia.edu/~gelman/stuff_for_blog/chap20.pdf

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