An approximate method is given to estimate the mean of a dependent variable after a linear model is fitted to the Box—Cox power transformation of this variable. The estimate is accurate except when the transformation power parameter is near zero. The properties of the estimate in the one-sample and regression cases are considered, by both asymptotic calculations and Monte Carlo simulations, and comparisons are made with the smearing estimate (Duan 1983). It is shown that there can be some cost due to estimating the power transformation, as opposed to assuming it is known; however, this cost is not severe.