# Question about bootstrap and confidence intervals

Assume I have a sample of real data and from which I create a set of new samples with replacement. I next compute the means for each 'synthetic' sample I generated.

My first question is as follows. I presume the central limit theorem applies and the distribution of these means will be normal, or at least approach normal as the size of the number of samples generated gets bigger. Is that true?

If this is true then I assume the 2.5 and 97.5 percentiles of the distribution of the means will be symmetrical even though the original distribution from which the synthetic samples were generated was not normal? Is that true?

I'm a bit confused on these points as I thought the confidence interval would be asymmetric if the underlying distribution was also asymmetric. I've done simulations and I do seem to get asymmetric intervals but this could be because I'm not sampling enough (I generated 500,000 samples) and as suggested above the central limit theroem suggests the distribution should be normal.