An approximate solution to the problem of testing equality of means of normal populations, when the variances may differ, is developed. The proposed solution utilizes the F-test with a random critical function. A series expansion based on results of Gideon and Gurland (1976) yields the true level of significance of the test. For a small number of populations in the test the technique is viable, but for a larger number a simulation study is made to compare the behavior of this test with that of some other well-known tests. The proposed test appears to provide better control of the true significance level, especially for small sample sizes. Futhermore, since it utilizes the ordinary F-statistic, it is simpler and more easily computed than some others.