More Damned Lies and Statistics: How Numbers Confuse Public Issues and Chance: A Guide to Gambling,Love, the Stock Market, & Just About Everything Else

In this article we review two historical approximations to the Poisson and binomial cumulative distribution functions (CDFs); that is, the Wilson–Hilferty and Camp–Paulson approximations. Both of these approximations reduce to standard normal formulas that produce very accurate estimates of the Poisson and binomial CDFs, and are thus quite simple to implement. Additionally, in an upper-division undergraduate or master’s level probability and inference course, the derivation of these approximations presents a nice opportunity to introduce and study the distributional relationships between the gamma and Poisson CDFs, and the binomial, beta, and F CDFs. This article presents the basic theorems and lemmas needed to derive each approximation, along with some relevant examples that compare and contrast the precision of these approximations with their large-sample, limiting normal counterparts.     

Some Properties of the Exact and Score Methods for Binomial Proportion and Sample Size Calculation

In this article, we point out some interesting relations between the exact test and the score test for a binomial proportion p. Based on the properties of the tests, we propose some approximate as well as exact methods of computing sample sizes required for the tests to attain a specified power. Sample sizes required for the tests are tabulated for various values of p to attain a power of 0.80 at level 0.05. We also propose approximate and exact methods of computing sample sizes needed to construct confidence intervals with a given precision. Using the proposed exact methods, sample sizes required to construct 95% confidence intervals with various precisions are tabulated for p = .05(.05).5. The approximate methods for computing sample sizes for score confidence intervals are very satisfactory and the results coincide with those of the exact methods for many cases.