One Proportion Z-Test: Definition, Formula, and Example

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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:

• H₀: p = 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 0 (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

where:

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

• H₀: p = 0.60 (population proportion is equal to 0.60)
• H₁: p ≠ 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 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