X2 and its components as tests of normality for grouped data

We consider testing for an unobservable normal distribution with unspecified mean and variance. It is only possible to observe the counts in groups with boundaries specified before sighting the data. On the basis of a small power study, we recommend the usual X2 test be used as an omnibus test, augmented by informal examination of the first two non-zero components of X2. We also recommend use of maximum likelihood and method of moments estimation.