  Normal mixture distributions are arguably the most important mixture models,
and also the most technically challenging. The likelihood function of the
normal mixture model is unbounded based on a set of random samples, unless an
artificial bound is placed on its component variance parameter. Moreover, the
model is not strongly identifiable so it is hard to differentiate between over
dispersion caused by the presence of a mixture and that caused by a large
variance, and it has infinite Fisher information with respect to mixing
proportions. There has been extensive research on finite normal mixture models,
but much of it addresses merely consistency of the point estimation or useful
practical procedures, and many results require undesirable restrictions on the
parameter space. We show that an EM-test for homogeneity is effective at
overcoming many challenges in the context of finite normal mixtures. We find
that the limiting distribution of the EM-test is a simple function of the
$0.5\chi^2_0+0.5\chi^2_1$ and $\chi^2_1$ distributions when the mixing
variances are equal but unknown and the $\chi^2_2$ when variances are unequal
and unknown. Simulations show that the limiting distributions approximate the
finite sample distribution satisfactorily. Two genetic examples are used to
illustrate the application of the EM-test.
