Given the number of trials,
n, and the number of successes,
k, this routine computes a
1001-α% confidence interval for
p, the probability parameter of a binomial distribution with probability function,
where
α is in the interval
0,1.
The lower and upper confidence limits
pl and
pu 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.
- If maxk,n-k<106.
The relationship between the beta and binomial distributions (see page 38 of
Hastings and Peacock (1975)) is used to derive the equivalent equations,
where
βa,b,δ is the deviate associated with the lower tail probability,
δ, of the beta distribution with parameters
a and
b. These beta deviates are computed using
G01FEF.
- If maxk,n-k≥106 and
mink,n-k
≤
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
χ2-distributions (see page 112 of
Hastings and Peacock (1975)) is used to derive the following equations;
where
χδ,ν2 is the deviate associated with the lower tail probability,
δ, of the
χ2-distribution with
ν degrees of freedom.
In turn the relationship between the
χ2-distribution and the gamma distribution (see page 70 of
Hastings and Peacock (1975)) yields the following equivalent equations;
where
γα,β;δ is the deviate associated with the lower tail probability,
δ, of the gamma distribution with shape parameter
α and scale parameter
β. These deviates are computed using
G01FFF.
- If maxk,n-k>106 and
mink,n-k
>
1000
.
The binomial variate with parameters
n and
p is approximated by a Normal variate with mean
np and variance
np1-p, see page 38 of
Hastings and Peacock (1975).
The approximate lower and upper confidence limits
pl and
pu are the solutions to the equations;
where
zδ is the deviate associated with the lower tail probability,
δ, of the standard Normal distribution. These equations are solved using a quadratic equation solver
(
C02AJF).
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
If on entry
IFAIL=0 or
-1, explanatory error messages are output on the current error message unit (as defined by
X04AAF).
For most cases using the beta deviates the results should have a relative accuracy of
max0.5E-12,50.0×ε where
ε is the
machine precision (see
X02AJF). Thus on machines with sufficiently high precision the results should be accurate to
12 significant figures. Some accuracy may be lost when
α/2 or
1-α/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, non-smokers 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.