  A reasonable confidence interval should have a confidence coefficient no less
than the given nominal level and a small expected length to reliably and
accurately estimate the parameter of interest, and the bootstrap interval is
considered to be an efficient interval estimation technique. In this paper, we
offer a first attempt at computing the coverage probability and expected length
of a parametric or percentile bootstrap interval by exact probabilistic
calculation for any fixed sample size. This method is applied to the basic
bootstrap intervals for functions of binomial proportions and a normal mean.
None of these intervals, however, are found to have a correct confidence
coefficient, which leads to illogical conclusions including that the bootstrap
interval is narrower than the z-interval when estimating a normal mean. This
raises a general question of how to utilize bootstrap intervals appropriately
in practice since the sample size is typically fixed.
