To obtain test probabilities based on empirical approximations to
the distribution of a Studentized function of a mean, we need the
approximations to be accurate with sufficiently high probability. In
particular, when these test probabilities are small it is best to consider
relative errors. Here we show that in the case of univariate standardized means
and in the general case of tests based on smooth functions of means, the
empirical approximations have asymptotically small relative errors on sets with
probability differing from 1 by an exponentially small quantity and that these
error rates hold for moderately large deviations. In particular, for
standardized deviations of order $n^{1/6}$, the probabilities approximated are
exponentially small with exponents of order $n^{1/3}$ and the corresponding
relative errors tend to zero on sets whose complements have probabilities of
the order of the probabilities being approximated.