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 easiest way to calculate this type of confidence interval in Python is to use the **proportion_confint()** function from the **statsmodels** package:

proportion_confint(count, nobs, alpha=0.05, method='normal')

where:

**count**: Number of successes**nobs**: Total number of trials**alpha**: Significance level (default is 0.05)**method**: Method to use for confidence interval (default is “normal”)

The following example shows how to use this function in practice.

**Example: Calculate Binomial Confidence Interval in Python**

Suppose we want to estimate the proportion of residents in a county that are in favor of a certain law.

We decide to select a random sample of 100 residents and find that 56 of them are in favor of the law.

We can use the **proportion_confint()** function to calculate the 95% confidence interval for the true proportion of residents who suppose this law in the entire county:

from statsmodels.stats.proportion import proportion_confint #calculate 95% confidence interval with 56 successes in 100 trials proportion_confint(count=56, nobs=100) (0.4627099463758483, 0.6572900536241518)

The 95% confidence interval for the true proportion of residents in the county that support the law is **[.4627, .6573]**.

By default, this function uses the asymptotic normal approximation to calculate the confidence interval. However, we can use the **method** argument to use a different method.

For example, the default function used in the R programming language to calculate a binomial confidence interval is the Wilson Score Interval.

We can use the following syntax to specify this method when calculating the confidence interval in Python:

from statsmodels.stats.proportion import proportion_confint #calculate 95% confidence interval with 56 successes in 100 trials proportion_confint(count=56, nobs=100, method='wilson') (0.4622810465167698, 0.6532797336983921)

This tells us that the 95% confidence interval for the true proportion of residents in the county that support the law is **[.4623, .6533]**.

This confidence interval is just slightly different than the one calculated using the normal approximation.

Note that we can also adjust the **alpha** value to calculate a different confidence interval.

For example, we can set alpha to be 0.10 to calculate a 90% confidence interval:

from statsmodels.stats.proportion import proportion_confint #calculate 90% confidence interval with 56 successes in 100 trials proportion_confint(count=56, nobs=100, alpha=0.10, method='wilson') (0.47783814499647415, 0.6390007285095451)

This tells us that the 90% confidence interval for the true proportion of residents in the county that support the law is **[.4778, .6390]**.

**Note**: You can find the complete documentation for the **proportion_confint()** function here.

**Additional Resources**

The following tutorials explain how to perform other common operations in Python:

How to Plot a Confidence Interval in Python

How to Use the Binomial Distribution in Python

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