# Estimating Effective Sample Size in Unequal clustered design (ratio-type estimator)

I'm having trouble understanding how these experimenters calculated their effective sample size.

The free article is located in this link here: https://academic.oup.com/icesjms/article/60/2/297/625881. The study was done to evaluate the effect of a clustered design on sample sizes collected in a cod fishery due to intraclass correlation.

On page 299 of the paper. They used this equation to calculate the effective sample sizes:

How did they calculate var(mu-hat) and the variance sigma-hat (using what methods or equations)? They mentioned that they used "non-parametric bootstrapping [...] to estimate the variance". What is this "variance" referring to? There are so many variance equations in the paper on page 298 that I find myself confused about what they did.

For example, the also mentioned these variance equations:

If anyone understands how the effective sample size based on the first equation I posted was calculated, an explanation would be greatly appreciated! Thank you!