One Proportion Z Test Calculator

Select null and alternative hypotheses:

Hypothesized Population Proportion

The claimed or assumed proportion, between 0 and 1.

Sample Proportion

What you actually observed in your sample, between 0 and 1.

Sample Size

Total number of people or items in your sample.

Significance Level

Usually 0.05 — your risk tolerance for a false positive.

About This Test

What Is the One-Proportion Z-Test?

The one-proportion Z-test checks whether your sample's observed proportion is significantly different from a claimed or hypothesized population proportion — useful whenever you want to test a specific claimed rate against what you actually measured.

When to Use It

Testing whether a company's claimed conversion rate matches reality
Testing whether a defect rate matches a stated specification
Testing whether a survey response rate matches an expected value
Any single sample with a yes/no outcome compared against a claimed proportion

The Formula

\[ Z = \frac{\hat{p} - p_0}{\sqrt{\dfrac{p_0(1-p_0)}{n}}} \]

Worked Example

A company claims 30% of customers use their loyalty program. A survey of 200 customers finds 72 use it. At α = 0.05, is there evidence the true rate differs from 30%?

\( p_0 = 0.30, \ \hat{p} = 72/200 = 0.36, \ n = 200 \)
\( Z \approx 1.85 \), two-tailed p-value ≈ 0.0641
Since 0.0641 > 0.05, we fail to reject H₀ — not quite enough evidence to say the true rate differs from 30% at this significance level.

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