# Suggestions for estimating variance in finite population with bootstrap?

I have a set of 990 samples with size=3 (n=3). All come from a different finite population (size between 10 and 20). Each population has its own mean, which is known, but it is reasonable to assume that they have equal variance. Therefore I substracted the known means from the sampled values. That gives 2970 values ($$990*n$$), lets name them $$Y_i$$ with $$i=1,\dots,2970$$. My reasoning was that the sample variance of these values would be an unbiased estimate of the variance in each finite population.

1. Is that correct so far?

Then I would want to estimate the variance of the sample mean for different sample sizes. I thought of bootstrapping by resampling the desired sample size from $$Y_i$$ with replacement giving each time a value of $$\bar{Y}^*_j$$ for bootstrap sample $$j$$. I would then estimate $$var(\bar{Y})$$ as is traditionally done in bootstrapping and multiply with a Finite Population Correction (FPC) so the estimator becomes:

$$Var(\bar{Y}) * \frac{N-n}{N-1}$$

1. Is it correct to use FPC in combination with a bootstrap variance estimate, or do you have other suggestions?

Thanks for your help!