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 $p \leqq \frac{1}{4}$ and $k \geqq np$ or if $p \leqq \frac{1}{2}$ and $np \leqq k \leqq n(1 - p)$. A local refinement is given for $np \leqq k \leqq n(1 - p), k \geqq 2$, and for $p \leqq \frac{1}{4}, k \geqq n(1 - p)$. 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.