I'm studying confidence intervals, and I'm curious about how one might generate a confidence interval for the confidence interval, if that even makes sense.
For example, let's say I draw simple random samples of n=100 from some population, calculate sample means and standard deviations, and construct 95% confidence intervals. I repeat this procedure 100 times. I know I expect about 95 of these intervals to capture the population mean, and about 5 of them not to. However, can I construct a confidence interval around this expectation? If I were to repeat this entire "100 samples of 100 samples" over and over again, what can I say about the distribution of how often the intervals captures?
Essentially, could I construct a confidence interval for the confidence interval? Would that even make any sense?