How do you know if your confidence interval is exact or approximate?

Question

I would like to know the rules between the two, and know how you know the differences.

Answer 1

If you know the exact distribution of the test statistic, and use that distribution's quantiles to make confidence intervals, that interval is exact.

If you approximate the distribution of the test statistic, then the interval is approximate. You often fail to know the exact distribution of the test statistic when the assumptions involved in the setup are not met.

Example:

Consider the problem of constructing confidence intervals for the mean of a Normal distribution with unknown variance from $n$ independent samples. A confidence interval constructed using a $t_{n-1}$ distribution quantiles is exact, and a confidence interval constructed using a $z$ distribution is approximate, where the approximation is better when $n$ is large.

If the assumption of independence of the sample is not met, then you have $n$ dependent realization from a Normal distribution, and the confidence interval obtained from using $t_{n-1}$ distribution quantiles is no longer exact.