A one proportion z-test is used to compare an observed proportion to a theoretical one.

This test uses the following null hypotheses:

**H**p = p_{0}:_{0}(population proportion is equal to hypothesized proportion p_{0})

The alternative hypothesis can be either two-tailed, left-tailed, or right-tailed:

**H**p ≠ p_{1}(two-tailed):_{0}(population proportion is not equal to some hypothesized value p_{0})**H**p < p_{1}(left-tailed):_{0}(population proportion is less than some hypothesized value p_{0})**H**p > p_{1}(right-tailed):_{0}(population proportion is greater than some hypothesized value p_{0})

The test statistic is calculated as:

z = (p-p_{0}) / √p_{0}(1-p_{0})/n

where:

**p:**observed sample proportion**p**hypothesized population proportion_{0}:**n:**sample size

If the p-value that corresponds to the test statistic z is less than your chosen significance level (common choices are 0.10, 0.05, and 0.01) then you can reject the null hypothesis.

**One Proportion Z-Test in Python**

To perform a one proportion z-test in Python, we can use the proportions_ztest() function from the **statsmodels** library, which uses the following syntax:

**proportions_ztest(count, nobs, value=None, alternative=’two-sided’) **

where:

**count:**The number of successes**nobs:**The number of trials**value:**The hypothesized population proportion**alternative:**The alternative hypothesis

This function returns a z test-statistic and a corresponding p-value.

The following example shows how to use this function to perform a one proportion z-test in Python.

**Example: One Proportion Z-Test in Python**

Suppose we want to know whether or not the proportion of residents in a certain county who support a certain law is equal to 60%. To test this, we collect the following data on a random sample:

**p**hypothesized population proportion = 0.60_{0}:**x:**residents who support law: 64**n:**sample size = 100

The following code shows how to use the **proportions_ztest** function to perform a one sample z-test:

#import proportions_ztest function from statsmodels.stats.proportion import proportions_ztest #perform one proportion z-test proportions_ztest(count=60, nobs=100, value=0.64) (-0.8164965809277268, 0.41421617824252466)

From the output we can see that the z test-statistic is **-0.8165 **and the corresponding p-value is **0.4142**. Since this value is not less than α = 0.05, we fail to reject the null hypothesis. We do not have sufficient evidence to say that the proportion of residents who support the law is different from 0.60.

**Additional Resources**

An Introduction to the One Proportion Z-Test

One Proportion Z-Test Calculator

How to Perform a One Proportion Z-Test in Excel

How to Perform a One Proportion Z-Test in R