So, I have a sample dataset of size 500, and I've bootstrapped it 1000 times and took the mean of each bootstrap sample. So now, I have a list of means a.k.a listofmeans and I'm trying to calculate the confidence interval of the listofmeans. When I google the regular formula for confidence interval, the formula is z +/- std.dev/sqrt(n)

However, the reply in this stackoverflow post and answer by Bogdan Lalu in this post, the calculation of the bootstrap confidence interval seem to only take the standard deviation and not dividing it by square root.

So why don't we take the division of the square root for the bootstrap confidence interval?

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    $\begingroup$ Hint: what happens to your alternative equation when you have enough computational resources to run a huge number of bootstrap samples? $\endgroup$ – whuber Aug 10 at 21:15
  • $\begingroup$ @whuber I guess with a huge number of bootstrap samples, the alternative equation with sqrt(n) goes to zero.. and that means we will get the exact population mean..? $\endgroup$ – misheekoh Aug 10 at 21:26
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    $\begingroup$ Almost: it means you will get the exact sample mean. You don't know the population mean because all you have is one sample. $\endgroup$ – whuber Aug 11 at 13:51

The idea behind bootstrapping is to get the standard error of an estimator ($\approx$ standard deviation of the sampling distribution of the estimator) by looking at the variation (standard deviation) you see in the value of the estimator across boostrapped datasets.

Your SE should not be going to zero (and thus, confidence interval width going to zero and any p-values going to <0.0001) just because you do more bootstrap samples. More boostrap samples produce a better approximation to what the sampling distribution of your quantity of interest is, but they do not "magically" generate more data.

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