Given the number of trials,
$n$, and the number of successes,
$k$, this function computes a
$100\left(1\alpha \right)\%$ confidence interval for
$p$, the probability parameter of a binomial distribution with probability function,
where
$\alpha $ is in the interval
$\left(0,1\right)$.
The lower and upper confidence limits
${p}_{l}$ and
${p}_{u}$ are estimated by the solutions to the equations;
Three different methods are used depending on the number of trials,
$n$, and the number of successes,
$k$.

1.If $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(k,nk\right)<{10}^{6}$.
The relationship between the beta and binomial distributions (see page 38 of
Hastings and Peacock (1975)) is used to derive the equivalent equations,
where
${\beta}_{a,b,\delta}$ is the deviate associated with the lower tail probability,
$\delta $, of the beta distribution with parameters
$a$ and
$b$. These beta deviates are computed using
g01fec.

2.If $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(k,nk\right)\ge {10}^{6}$ and $\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(k,nk\right)\le 1000$.
The binomial variate with parameters
$n$ and
$p$ is approximated by a Poisson variate with mean
$np$, see page 38 of
Hastings and Peacock (1975).
The relationship between the Poisson and
${\chi}^{2}$distributions (see page 112 of
Hastings and Peacock (1975)) is used to derive the following equations;
where
${\chi}_{\delta ,\nu}^{2}$ is the deviate associated with the lower tail probability,
$\delta $, of the
${\chi}^{2}$distribution with
$\nu $ degrees of freedom.
In turn the relationship between the
${\chi}^{2}$distribution and the gamma distribution (see page 70 of
Hastings and Peacock (1975)) yields the following equivalent equations;
where
${\gamma}_{\alpha ,\beta \text{;}\delta}$ is the deviate associated with the lower tail probability,
$\delta $, of the gamma distribution with shape parameter
$\alpha $ and scale parameter
$\beta $. These deviates are computed using
g01ffc.

3.If $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(k,nk\right)>{10}^{6}$ and $\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(k,nk\right)>1000$.
The binomial variate with parameters
$n$ and
$p$ is approximated by a Normal variate with mean
$np$ and variance
$np\left(1p\right)$, see page 38 of
Hastings and Peacock (1975).
The approximate lower and upper confidence limits
${p}_{l}$ and
${p}_{u}$ are the solutions to the equations;
where
${z}_{\delta}$ is the deviate associated with the lower tail probability,
$\delta $, of the standard Normal distribution. These equations are solved using a quadratic equation solver
.

1:
$\mathbf{n}$ – Integer
Input

On entry: $n$, the number of trials.
Constraint:
${\mathbf{n}}\ge 1$.

2:
$\mathbf{k}$ – Integer
Input

On entry: $k$, the number of successes.
Constraint:
$0\le {\mathbf{k}}\le {\mathbf{n}}$.

3:
$\mathbf{clevel}$ – double
Input

On entry: the confidence level, $\left(1\alpha \right)$, for twosided interval estimate. For example ${\mathbf{clevel}}=0.95$ will give a $95\%$ confidence interval.
Constraint:
$0.0<{\mathbf{clevel}}<1.0$.

4:
$\mathbf{pl}$ – double *
Output

On exit: the lower limit, ${p}_{l}$, of the confidence interval.

5:
$\mathbf{pu}$ – double *
Output

On exit: the upper limit, ${p}_{u}$, of the confidence interval.

6:
$\mathbf{fail}$ – NagError *
Input/Output

The NAG error argument (see
Section 7 in the Introduction to the NAG Library CL Interface).
For most cases using the beta deviates the results should have a relative accuracy of
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(\text{0.5e\u221212},50.0\times \epsilon \right)$ where
$\epsilon $ is the
machine precision (see
X02AJC). Thus on machines with sufficiently high precision the results should be accurate to
$12$ significant figures. Some accuracy may be lost when
$\alpha /2$ or
$1\alpha /2$ is very close to
$0.0$, which will occur if
clevel is very close to
$1.0$. This should not affect the usual confidence levels used.
None.
The following example program reads in the number of deaths recorded among male recipients of war pensions in a six year period following an initial questionnaire in 1956. We consider two classes, nonsmokers and those who reported that they smoked pipes only. The total number of males in each class is also read in. The data is taken from page 216 of
Snedecor and Cochran (1967). An estimate of the probability of a death in the six year period in each class is computed together with 95% confidence intervals for these estimates.