In testing a hypothesis concerning the correlation coefficient in a bivariate normal distribution where all the parameters are unknown, the Pearson product moment statistic is appropriate. It may happen, however, that there are relations among the parameters in the distribution, in which case the Pearson statistic would not utilize this information. In particular if the variances of the two marginal distributions are equal, it is possible to test the correlation coefficient by means of a simpler statistic which makes use of this information. In this paper we explain how this statistic arises and present some properties of its distribution. This statistic as well as its properties developed here are utilized in the latter part of this paper where we consider the problem of estimating the difference of the means when some of the observations corresponding to one of the variables are missing.