4.2 Parametric confidence intervals

In some cases, such as the one- and two-sample $t$-tests discussed in the previous Chapter, the parametric form of the sampling distribution for the estimator is known – either exactly or asymptotically. Exactly means that the distribution is known regardless of the sample size $n$. Asymptotically means that the distribution is known only as $n\to \mathrm{\infty }$. In practice, we would treat this asymptotic distribution as the exact sampling distribution provided that the sample size $n$ is sufficiently large.

If this is the case, then we can construct a confidence interval exactly from the known sampling distribution. We demonstrate this for the population mean of the one-sample $t$-test and the difference in population means of the two-sample $t$-tests discussed in the previous chapter. Note that confidence intervals constructed in this way will be referred to simply as confidence intervals, as opposed to bootstrap confidence intervals, although the two intervals have the same interpretation despite the differences in their construction.