Confidence Limits for the Fraction of a Normal Population which Lies Between Two Given Limits

Abstract

Let $\mu$ and $\sigma^2$ be the unknown mean and variance, respectively, of a normally distributed population on which $N$ independent observations $x_1, \cdots, x_N$ have been made. Let $L_1$ and $L_2, L_1 < L_2$, and $\alpha, 0 < \alpha < 1$, be given constants. We define the following symbols:\begin{equation*}\tag{(a)} \gamma = (\sqrt{2\pi}\sigma)^{-1} \int^{L_2}_{L_1} \exp \big\{-\frac{1}{2} \frac{(y - \mu)^2}{\sigma^2}\big\} dy\end{equation*} \begin{equation*}\tag{(b)} \bar{x} = N^{-1}\sigma\ x_i\end{equation*} \begin{equation*}\tag{(c)} s^2 = (N - 1)^{-1}\sigma (x_i - \bar{x})^2\end{equation*} (d) $\chi^2_{1-\alpha}$ as that number for which $P\{\chi^2 < \chi^2_{1-\alpha}\} = 1 - \alpha$ where $\chi^2$ has $N - 1$ degrees of freedom. \begin{equation*}\tag{(e)} w = \sqrt{N - 1}\frac{s}{\chi 1 - \alpha}\end{equation*} \begin{equation*}\tag{(f)} D = (2\pi)^{-1/2} \int^{(L_2 - {\bar{x})/w}_{(L_1-{\bar{x})/w} \exp \big\{ - \frac{1}{2} y^2\big\} dy\end{equation*} It is proved that, under restrictions stated precisely below, and before the observations are made, the probability that $D \leq \gamma$ differs from $\alpha$ by a number which can be made arbitrarily small by making $N$ sufficiently large. Thus an approximate (large sample) lower confidence limit for $\gamma$ is obtained. Similar methods can be applied to obtain upper and two-sided confidence limits. A problem raised by the present paper (but not attacked here) is to investigate the rapidity of approach to $\alpha$ of $P\{D \leq \gamma\}$. It would perhaps be useful to obtain a series for the latter in powers of $N^{-1\frac{1}{2}$; the first term of such an expansion is obtained here.