Evaluating the Normal Approximation to the Binomial Test

Abstract

The normal approximation to the binomial test with and without a continuity correction is evaluated in terms of control of Type I errors and power. The normal approximations are evaluated as robust for a given sample size, N, and at a given level α if the true Type I error rate never exceeds 1.5 α. The uncorrected normal test is found to be less robust than is implied by the currently applied guidelines. The most stringent currently used guideline of requiring σ2≥10 is adequate at α = .05 but must be increased to σ2 ≥35 at α = .01. The corrected test is shown to be robust but not conservative. Both tests are shown to have substantial power loss in comparison to the exact binomial test.