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Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_univar_ci_binomial (g07aa)

## Purpose

nag_univar_ci_binomial (g07aa) computes a confidence interval for the parameter p$p$ (the probability of a success) of a binomial distribution.

## Syntax

[pl, pu, ifail] = g07aa(n, k, clevel)
[pl, pu, ifail] = nag_univar_ci_binomial(n, k, clevel)

## Description

Given the number of trials, n$n$, and the number of successes, k$k$, this function computes a 100(1α)%$100\left(1-\alpha \right)%$ confidence interval for p$p$, the probability parameter of a binomial distribution with probability function,
f(x) =
 ( n ) x
px(1p)nx,  x = 0,1,,n,
$f(x)= n x px(1-p)n-x, x=0,1,…,n,$
where α$\alpha$ is in the interval (0,1)$\left(0,1\right)$.
Let the confidence interval be denoted by [pl,pu${p}_{l},{p}_{u}$].
The point estimate for p$p$ is = k / n$\stackrel{^}{p}=k/n$.
The lower and upper confidence limits pl${p}_{l}$ and pu${p}_{u}$ are estimated by the solutions to the equations;
n
 ( n ) x
plx(1pl)nx = α / 2,
x = k
$∑x=kn n x plx (1-pl) n-x =α/2 ,$
k
 ( n ) x
pux(1pu)nx = α / 2.
x = 0
$∑x= 0k n x pux (1-pu) n-x =α /2 .$
Three different methods are used depending on the number of trials, n$n$, and the number of successes, k$k$.
1. If max (k,nk) < 106$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(k,n-k\right)<{10}^{6}$.
The relationship between the beta and binomial distributions (see page 38 of Hastings and Peacock (1975)) is used to derive the equivalent equations,
 pl = βk,n − k + 1,α / 2, pu = βk + 1,n − k,1 − α / 2,
$pl = βk,n-k+1,α/2, pu = βk+1,n-k,1-α/2,$
where βa,b,δ${\beta }_{a,b,\delta }$ is the deviate associated with the lower tail probability, δ$\delta$, of the beta distribution with parameters a$a$ and b$b$. These beta deviates are computed using nag_stat_inv_cdf_beta (g01fe).
2. If max (k,nk)106$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(k,n-k\right)\ge {10}^{6}$ and min (k,nk) 1000 $\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(k,n-k\right)\le 1000$.
The binomial variate with parameters n$n$ and p$p$ is approximated by a Poisson variate with mean np$np$, see page 38 of Hastings and Peacock (1975).
The relationship between the Poisson and χ2${\chi }^{2}$-distributions (see page 112 of Hastings and Peacock (1975)) is used to derive the following equations;
 pl = 1/(2n)χ2k,α / 22, pu = 1/(2n)χ2k + 2,1 − α / 22,
$pl = 12n χ2k,α/22, pu = 12n χ2k+2,1-α/22,$
where χδ,ν2${\chi }_{\delta ,\nu }^{2}$ is the deviate associated with the lower tail probability, δ$\delta$, of the χ2${\chi }^{2}$-distribution with ν$\nu$ degrees of freedom.
In turn the relationship between the χ2${\chi }^{2}$-distribution and the gamma distribution (see page 70 of Hastings and Peacock (1975)) yields the following equivalent equations;
 pl = 1/(2n)γk,2;α / 2, pu = 1/(2n)γk + 1,2;1 − α / 2,
$pl = 12n γk,2;α/2, pu = 12n γk+1,2;1-α/2,$
where γα,β;δ${\gamma }_{\alpha ,\beta \text{;}\delta }$ is the deviate associated with the lower tail probability, δ$\delta$, of the gamma distribution with shape parameter α$\alpha$ and scale parameter β$\beta$. These deviates are computed using nag_stat_inv_cdf_gamma (g01ff).
3. If max (k,nk) > 106$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(k,n-k\right)>{10}^{6}$ and min (k,nk) > 1000 $\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(k,n-k\right)>1000$.
The binomial variate with parameters n$n$ and p$p$ is approximated by a Normal variate with mean np$np$ and variance np(1p)$np\left(1-p\right)$, see page 38 of Hastings and Peacock (1975).
The approximate lower and upper confidence limits pl${p}_{l}$ and pu${p}_{u}$ are the solutions to the equations;
 (k − npl)/(sqrt(npl(1 − pl))) = z1 − α / 2, (k − npu)/(sqrt(npu(1 − pu))) = zα / 2,
$k-npl npl(1-pl) = z1-α/2, k-npu npu(1-pu) = zα/2,$
where zδ${z}_{\delta }$ is the deviate associated with the lower tail probability, δ$\delta$, of the standard Normal distribution. These equations are solved using a quadratic equation solver (nag_zeros_quadratic_real (c02aj)).

## References

Hastings N A J and Peacock J B (1975) Statistical Distributions Butterworth
Snedecor G W and Cochran W G (1967) Statistical Methods Iowa State University Press

## Parameters

### Compulsory Input Parameters

1:     n – int64int32nag_int scalar
n$n$, the number of trials.
Constraint: n1${\mathbf{n}}\ge 1$.
2:     k – int64int32nag_int scalar
k$k$, the number of successes.
Constraint: 0kn$0\le {\mathbf{k}}\le {\mathbf{n}}$.
3:     clevel – double scalar
The confidence level, (1α)$\left(1-\alpha \right)$, for two-sided interval estimate. For example clevel = 0.95${\mathbf{clevel}}=0.95$ will give a 95%$95%$ confidence interval.
Constraint: 0.0 < clevel < 1.0$0.0<{\mathbf{clevel}}<1.0$.

None.

None.

### Output Parameters

1:     pl – double scalar
The lower limit, pl${p}_{l}$, of the confidence interval.
2:     pu – double scalar
The upper limit, pu${p}_{u}$, of the confidence interval.
3:     ifail – int64int32nag_int scalar
${\mathrm{ifail}}={\mathbf{0}}$ unless the function detects an error (see [Error Indicators and Warnings]).

## Error Indicators and Warnings

Errors or warnings detected by the function:
ifail = 1${\mathbf{ifail}}=1$
 On entry, n < 1${\mathbf{n}}<1$, or k < 0${\mathbf{k}}<0$, or n < k${\mathbf{n}}<{\mathbf{k}}$, or clevel ≤ 0.0${\mathbf{clevel}}\le 0.0$, or clevel ≥ 1.0${\mathbf{clevel}}\ge 1.0$.
ifail = 2${\mathbf{ifail}}=2$
When using the relationship with the gamma distribution to calculate one of the confidence limits, the series to calculate the gamma probabilities has failed to converge. Both pl and pu are set to zero. This is a very unlikely error exit and if it occurs please contact NAG.

## Accuracy

For most cases using the beta deviates the results should have a relative accuracy of max (0.5e−12,50.0 × ε)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(\text{0.5e−12},50.0×\epsilon \right)$ where ε$\epsilon$ is the machine precision (see nag_machine_precision (x02aj)). Thus on machines with sufficiently high precision the results should be accurate to 12$12$ significant figures. Some accuracy may be lost when α / 2$\alpha /2$ or 1α / 2$1-\alpha /2$ is very close to 0.0$0.0$, which will occur if clevel is very close to 1.0$1.0$. This should not affect the usual confidence levels used.
The approximations used when n$n$ is large are accurate to at least 3$3$ significant digits but usually to more.

None.

## Example

```function nag_univar_ci_binomial_example
n = int64(1067);
k = int64(117);
clevel = 0.95;
[pl, pu, ifail] = nag_univar_ci_binomial(n, k, clevel)
```
```

pl =

0.0915

pu =

0.1300

ifail =

0

```
```function g07aa_example
n = int64(1067);
k = int64(117);
clevel = 0.95;
[pl, pu, ifail] = g07aa(n, k, clevel)
```
```

pl =

0.0915

pu =

0.1300

ifail =

0

```