A number of transformations of the form of have been given in the literature for particular parametric families. The hypothesis testing problem of is often not a very useful model in practice. The conditional probability integral transformations are given for a number of parametric families of probability distributions. Some important classes of distributions which are not covered by the transformation theory are the truncation parameter families considered by Quesenberry. The hypothesis testing problem is often not a very useful model in practice. It is more meaningful in many instances to test that the distribution function F has some specified functional form without assuming that the values of all parameters are known. Any of the multitude of goodness-of-fit tests for the completely specified null hypothesis testing problem of may be used to test normality of the normal uniform residuals.