Suppose you construct a 95 confidence interval for and get

Suppose you construct a 95% confidence interval for , and get the interval (a, b). You later consider the probability distribution of the population of primary interest (intuitively, that of the Xis), and you notice that Pr[a < Xi b] is much less than 95%. Explain this

Solution

The 95% confidence is valid only for the sampling distribution of the mean. It is not valid for the distirbution of the individual observations themselves.

Hence, we observe a much less probability for the second case.