Suppose that I have a population, each represented by a bit $b_i$ for $i \in \{1,\ldots, n\}$. I would like to compute an estimate $\hat{B}$ to the statistic $B = \sum_{i=1}^nb_i$ so that with high probability, the error $|\hat{B}-B| \leq k$ for some fixed $k$. However, I have to pay a cost $c_i$ to sample bit $b_i$, and this cost may be different for each $i$. I want to find the minimum cost sample that satisfies my accuracy constraint. Clearly, uniform sampling is not necessarily optimal.

Has this been studied? Is there a known optimal solution specifying the probability $p_i$ that I should sample each bit $b_i$ to compute $\hat{B}$, as a function of the costs $c_i$?