Validity of the bootstrap method. Why extrapolate to the general population?

Question

Algorithms

Suppose we have a sample that we consider representative. Using the boostrapping method, we make several thousand resamples and each time we find, for example, the average value of the received sample. We build the probability distribution function and calculate the 2.5 and 97.5 percentiles. Why, based on this, can we be sure that this interval will cover the population mean with a probability of 95%, not knowing the distribution, but based only on samples?

Answer 1

Something is a little off. After resampling, we get the empirical distribution function of (for example) the mean value. For this distribution (and not for the distribution of the original sample), we can build a confidence interval, i.e. such limits, in which (conditionally) in 95 cases out of a hundred the average of our sample will fall.
Those. the real mean of the real sample or the mean of the population may well not fall into this confidence interval, but the probability of this is less than 5%. Moreover, we made this conclusion solely on the basis of the available data. If suddenly we have additional data from the same general population, then it is quite possible that our conclusion will have to be corrected.
The main thing to understand is that statistics are not about certainty. Never! Statistics is really aboutthe likelihood of being mistaken in your confidence.
PS All the same, take a look at the book that I recommended to you elsewhere.