Identifying the number of components in a finite mixture is hard problem. Generally, the likelihood ratio test provides a robust method for statistical inference for this problem. However, the classical theorem for the asymptotic null distribution of the LRT statistic cannot be applied to finite mixture alternatives. So other inferential methods have been proposed to assess the statistical significance of an observed LRT value. Two such methods are the bootstrap and posterior predictive check (PPC). In this dissertation we conducted simulation studies to compare the power of the bootstrap method to the PPC method as it applies to identify the number of components in a Poisson mixture. We considered two simple hypothesis tests where we test a single Poisson distribution against a mixture of two Poisson distributions and a zero inflated Poisson (ZIP) distribution. For the two-component Poisson mixture alternative, we compared the power of the PPC method to the Bootstrap method. In the case of the zero inflated Poisson (ZIP) alternative, we compared the PPC method to the bootstrap method and two asymptotic tests proposed by Rao and Chakravarti  and van den Broek  for detecting zero inflation in a Poisson. Simulated data sets were used to compare the performance of the methods for each test. A wide range of cases under these alternative hypotheses were considered with the objective of seeing whether one method is uniformly more powerful than the others for each of these alternatives.