Distribution Inequalities for the Binomial Law

We prove that the probability of at least k successes, in n Bernoulli trials with success-probability p, is larger than its normal approximant if p 1/4 and k np or if p 1/2 and np k n(1 - p). A local refinement is given for np k n(1 - p), k 2, and for p 1/4, k n(1 - p). Bounds below for individual binomial probabilities $b(k, n, p)$ are also given under various conditions. Finally, we discuss applications to significance tests in one-way layouts.