What is the definition of one proportion z-test?

The one proportion z-test is a hypothesis test that is used to compare a sample proportion to a hypothesized proportion. It is used to test the difference between the sample proportion and the hypothesized proportion, and is based on the normal distribution. It is commonly used to test differences in proportions between two populations.

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 of residents and ask each one whether or not they support the law:

Population proportion estimation example

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:

  • H0p = p0 (population proportion is equal to some hypothesized population proportion p0)

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

  • H1 (two-tailed): p ≠ p0 (population proportion is not equal to some hypothesized value p0)
  • H1 (left-tailed): p < p0 (population proportion is less than some hypothesized value p0)
  • H1 (right-tailed): p > p0 (population proportion is greater than some hypothesized value p0)

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

z = (p-p0) / √p0(1-p0)/n


  • p: observed sample proportion
  • p0: hypothesized population proportion
  • n: sample size

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
  • p0: hypothesized population proportion = 0.60
  • n: sample size = 100

Step 2: Define the hypotheses.

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

  • H0p = 0.60 (population proportion is equal to 0.60)
  • H1p ≠ 0.60 (population proportion is not equal to 0.60)

Step 3: Calculate the test statistic z.

z = (p-p0) / √p0(1-p0)/n = (.64-.6) / √.6(1-.6)/100 = 0.816

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

According to the , 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 .