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

This tutorial explains the following:

- The motivation for performing a one proportion z-test.
- The formula to perform a one proportion z-test.
- An example of how to perform a one proportion z-test.

**One Proportion Z-Test: Motivation**

Suppose we want to know if the proportion of people in a certain county that are in favor of a certain law is equal to 60%. Since there are thousands of residents in the county, it would be too costly and time-consuming to go around and ask each resident about their stance on the law.

Instead, we might select a simple random sample of residents and ask each one whether or not they support the law:

However, it’s virtually guaranteed that the proportion of residents in the sample who support the law will be at least a little different from the proportion of residents in the entire population who support the law. **The question is whether or not this difference is statistically significant**. Fortunately, a one proportion z-test allows us to answer this question.

**One Proportion Z-Test:**** Formula**

A one proportion z-test always uses the following null hypothesis:

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

We use the following formula to calculate the test statistic z:

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****: Example**

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, will perform a one proportion z-test at significance level α = 0.05 using the following steps:

**Step 1: Gather the sample data.**

Suppose we survey a random sample of residents and end up with the following information:

**p:**observed sample proportion = 0.64**p**hypothesized population proportion = 0.60_{0}:**n:**sample size = 100

**Step 2: Define the hypotheses.**

We will perform the one sample t-test with the following hypotheses:

**H**p = 0.60 (population proportion is equal to 0.60)_{0}:**H**p ≠ 0.60 (population proportion is not equal to 0.60)_{1}:

**Step 3: Calculate the test statistic z.**

**z **= (p-p_{0}) / √p_{0}(1-p_{0})/n = (.64-.6) / √.6(1-.6)/100 = **0.816**

**Step 4: Calculate the p-value of the test statistic z.**

According to the Z Score to P Value Calculator, the two-tailed p-value associated with z = 0.816 is **0.4145**.

**Step 5: Draw a conclusion.**

Since this p-value is not less than our significance level 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.

**Note: **You can also perform this entire one proportion z-test by simply using the One Proportion Z-Test Calculator.

**Additional Resources**

How to Perform a One Proportion Z-Test in Excel

One Proportion Z-Test Calculator