For statistical process control, a number of single charts that jointly monitor both process mean and variability recently have been developed. For quality control-related hypothesis testing, however, there has been little analogous development of joint mean-variance tests: only one two-sample statistic that is not computationally intensive has been designed specifically for the one-sided test of Ho: Mean2<=Mean1 and StDev2<=StDev1 vs. Ha: Mean2>Mean1 OR StDev2>StDev1 (see Opdyke, 2006). Many have proposed other commonly used tests, such as tests of stochastic dominance, exceedance tests, or permutation tests for this joint hypothesis, but the first can exhibit prohibitively poor type I error control, and the latter two can have virtually no power under many conditions. This paper further develops and generalizes the maximum test proposed in Opdyke (2006) and demonstrates via extensive simulations that, for comparing two independent samples under typical quality control conditions, it a) always maintains good type I error control; b) has good power under symmetry and modest power under notable asymmetry; and c) often has dramatically more power and much better type I error control than the only other widely endorsed competitor. The statistic (OBMax2) is not computationally intensive, and although initially designed for quality control testing in regulatory telecommunications, its range of application is as broad as the number of quality control settings requiring a one-sided, joint test of both the mean and the variance.