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Let be a random sample from a population with mean and variance ${\sigma }_1^2$, and be a random sample from another population with mean and variance ${\sigma }_2^2$ independent of {}. Consider the two sample t-statistic $T=\frac{\overline{X}-\overline{Y}-({\mu }_1-{\mu }_2)}{\sqrt{s_1^2/n_1+s_2^2/n_2}}$. This paper shows that ln ²/2 for any satisfying → ∞, as provided 0 < < ∞. If, in addition, , , then $\frac{P(T\ge x)}{1-\Phi \left(x\right)}\to 1$ holds uniformly in