In this paper, we study the application of a power transformation for the purpose of accelerating the rate of convergence of a statistics sampling distribution to that of its limiting normal. The power transformation is chosen so that the sampling distribution of the transformed test statistic is less skewed. Scale invariant sufficient conditions on the cummulants of the statistic are given which guarantee the reduction of skewness. Unfortunately, this power transformation depends on the first three moments of the test statistic for which exact expressions are not always available. We propose the estimation of these moments via a parametric bootstrap. The effectiveness of this data based power transformation in an application to goodness-of-fit testing is established through computer simulations. We demonstrate that the resulting normal approximation to the sampling distribution of the transformed goodness-offit test statistic is better than the approximation provided by the bootstrapped sampling distribution based on 100 bootstrap samples. This computational savings is important in applications for which each bootstrap realization is computationally intensive.