The main theorem is the following inequality between binomial and their approximating normal tail probabilities: if \(p\leq1/4\) and \(k\geq np\), or if \(p\leq1/2\) and \(np\leq k\leq n(1-p)\), then the binomial probability \(\sum\limits_{j=k}^nb(j,n,p)\geq1-\Phi((k-np)/(npq)^{1/2})\). The tools of proof are elementary. The point of view taken is that such distribution inequalities reveal systematic errors in the common approximations used in tests of significance. The results are applied directly to construct significance tests concervative with respect to type II errors.