  Two-sample inference for the difference of population means typically relies
upon a Central Limit Theorem approximation. When data are drawn from a Negative
Binomial distribution, previous work of Shilane et al. (2010) showed that a
Normal approximation is often unreliable in one-sample inference and proposed
alternative techniques. We seek to extend these methods to the problem of
two-sample inference on the difference of sample means in highly dispersed
Negative Binomial models. We demonstrate that the Normal approximation is
considerably more robust in the two-sample setting and may often be applied
even when it is not appropriate for either sample individually. We also provide
an intuitive extension of Bernstein's Inequality to the two-sample case. A
simple mixture of these two methods may improve coverage in borderline and
small sample settings. We subsequently investigate the coverage quality of
confidence intervals and sample size considerations in a wide variety of
simulation studies. Overall, we demonstrate that the Normal approximation is
considerably more robust in approximating the distribution of the mean
difference than the corresponding one-sample theory would suggest.
