A confidence interval for a binomial probability is calculated using the following formula:

Confidence Interval = p  +/-  z*(√p(1-p) / n)

where:

• p: proportion of “successes”
• z: the chosen z-value
• n: sample size

The z-value that you will use is dependent on the confidence level that you choose. The following table shows the z-value that corresponds to popular confidence level choices:

For example, suppose we want to estimate the proportion of residents in a county that are in favor of a certain law. We select a random sample of 100 residents and find that 56 of them are in favor of the law.

This tutorial explains three different ways to calculate a confidence interval for the true proportion of residents in the entire county that support the law.

Method 1: Use the prop.test() function

One way to calculate the 95% binomial confidence interval is to use the prop.test() function in base R:

#calculate 95% confidence interval
prop.test(x=56, n=100, conf.level=.95, correct=FALSE)


	1-sample proportions test without continuity correction

data:  56 out of 100, null probability 0.5
X-squared = 1.44, df = 1, p-value = 0.2301
alternative hypothesis: true p is not equal to 0.5
95 percent confidence interval:
 0.4622810 0.6532797
sample estimates:
   p 
0.56 

The 95% C.I. for the true proportion of residents in the county that support the law is [.46228, .65328].

Method 2: Use the binconf() function

Another way to calculate the confidence interval is to use the binconf() function from the Hmisc package:

library(Hmisc)

#calculate 95% confidence interval
binconf(x=56, n=100, alpha=.05)

 PointEst    Lower     Upper
     0.56 0.462281 0.6532797

Notice that this confidence interval matches the one calculated in the previous example.

Method 3: Calculate the Confidence Interval Manually

#define proportion
p <- 56/100

#define significance level
a <- .05

#calculate 95% confidence interval
p + c(-qnorm(1-a/2), qnorm(1-a/2))*sqrt((1/100)*p*(1-p))

[1] 0.4627099 0.6572901

Learn more about the qnorm() function here: A Guide to dnorm, pnorm, qnorm, and rnorm in R