On the number of observations requir ed for testing hypotheses of the binomial distribution parameter

The testing procedure for the binomial scheme with constant success probability $p$ being considered, the hypotheses to be tested are $H_0\colon p=p_0$ and $H_1\colon p=p_1$, $p_1>p_0$. Two methods are presented to estimate the number of observations required for testing $H_0$ against $H_1$ by means of the most powerful nonrandomized test at given first $(\varepsilon)$ and second kind $(\omega)$ error probabilities. The first method gives an interval estimate, i.e. a closed interval with integer end points, which, for fixed $p_0$, $\varepsilon$ and $\omega$ and all $p_1$'s sufficiently close to $p_0$, will contain the exact number of observations required. The second method provides a point estimate of the number of observations required and the corresponding critical constant. As the calculations show, these estimates are extiremely accurate for $\varepsilon$ and $\omega$ close to 0.05–0.1 and $p_0\le0.08$. Yet, the author can draw no final conclusion about the accuracy of the estimates.