We consider here the relative merits of different statistics available for testing two means or two regression coefficients in relation to one-sided (asymmetric) and two-sided (symmetric) alternatives in case of unequal population variances. In so far as the Behrens-Fisher statistic is concerned we confine ourselves to the consideration of the behavior of its probability of Type I error in repeated sampling from populations with a fixed value of the unknown ratio of variances. In connection with the tests between two means, the present study takes its point of departure from the existing tests and investigates the question of utilizing an approximately determinate knowledge about the unknown ratio of variances. In connection with the comparison of two regression coefficients and also of two linear regression functions, we consider the effect of two concomitant sources of variation, viz., the unknown ratio of residual variances and the ratio of the sums of squares of the fixed variates, on the probability of Type I and Type II errors of certain well known statistics.