Initially motivated by electron microscopy experiments, we develop an approximate prediction interval on the univariate response variable Y, where it is assumed that a normal-theory linear model is fit using a transformed version of Y, and the transformation type is contained in the Box-Cox family. Further motivated by A-10 single-engine climb experiments, we then develop an approximate prediction interval on the univariate response Y, in which a linear model is fit using a transformed version of Y, contained in the Manly exponential family. For each case, we derive a closed-form approximation to the kth moment of the original response variable Y, which is then used to estimate the mean and variance of Y, given parameter estimates obtained from fitting the model in the transformed domain. Chebychev’s inequality is then used to construct a 100(1 − α)% prediction interval estimator on Y based on these mean and variance estimators. Extended data obtained from the A-10 single-engine climb experiments motivates the development of prediction regions in the original domain of a q-variate response vector Y through the use of multivariate extensions of both the Box-Cox power transformation and the Manly exponential transformation. For each transformation, we derive closed-form approximations to the kth moment of each original response Y, as well as a closed-form approximation to E(Yi Yi'), which are used to estimate the mean and variance of each Y and the covariance between them, given parameter estimates obtained from fitting the model in the transformed domain. Exploiting two multivariate analogs of Chebyshev’s inequality, we construct an approximate 100(1 − α)% prediction sphere and ellipsoid on the original response vector Y.