Z Test for One Proportion Calculator

Calculator Input

Study name

Number of successes

Sample size

Null proportion p0

Alpha level

Alternative hypothesis

Confidence level percent

Assumed true proportion for power

Decimal precision

Continuity correction

Use conservative correction

Example Data Table

Scenario	Successes	Sample Size	Null p0	Alternative	Alpha
Website sign ups	56	100	0.50	Two tailed	0.05
Product defect check	8	200	0.06	Less than	0.05
Voter support survey	410	800	0.48	Greater than	0.01
Quality pass rate	188	220	0.80	Greater than	0.05

Formula Used

Sample proportion: p̂ = x / n

Standard error under the null: SE0 = sqrt[p0 × (1 − p0) / n]

Z statistic: z = (p̂ − p0) / SE0

Right tailed p value: P(Z ≥ z)

Left tailed p value: P(Z ≤ z)

Two tailed p value: 2 × P(Z ≥ |z|)

Normal interval: p̂ ± z critical × sqrt[p̂ × (1 − p̂) / n]

Decision rule: reject H0 when p value is less than alpha.

How to Use This Calculator

Enter the number of successes from your sample.
Enter the total sample size.
Add the claimed population proportion as p0.
Choose the alpha level for the test.
Select a left, right, or two tailed alternative.
Enter a confidence level for interval estimates.
Add an assumed true proportion for optional power output.
Press calculate and review the result above the form.
Use CSV or PDF export for records.

Understanding the One Proportion Z Test

A one proportion z test checks one sample share. It compares observed successes with a claimed population proportion. The method is useful for surveys, quality checks, landing page tests, and compliance reviews. It works best when the sample is random. It also needs enough expected successes and failures. This calculator shows each step, so the decision is easy to audit.

What the Result Means

The sample proportion is the observed rate. The null proportion is the benchmark you want to test. The z score measures how far the sample rate is from that benchmark. It uses standard error under the null claim. A large positive z score supports a greater than claim. A large negative z score supports a less than claim. For a two tailed test, both directions count as evidence.

P Value and Decision

The p value converts the z score into probability. It estimates how unusual the sample result is, assuming the null claim is true. When the p value is less than alpha, reject the null hypothesis. When it is not less than alpha, do not reject it. This wording is important. The test does not prove the null claim. It only shows whether the sample gives enough evidence against it.

Confidence Interval View

The calculator also reports confidence intervals. A normal interval is quick and familiar. A Wilson interval is often more stable, especially near zero or one. Use the interval as an estimation tool. Use the hypothesis test for the formal decision. If the claimed proportion sits far outside the interval, the test result will usually agree.

Best Practice Notes

Check inputs before trusting output. Successes must not exceed trials. The claimed proportion must be between zero and one. Review the expected counts warning. Small samples may need an exact binomial test instead. Use the continuity correction only when you want a conservative normal approximation. For reports, export the result with the assumptions and selected alternative. Good records make the conclusion easier to review later.

Common Uses

Teams use this test for pass rates, defect rates, vote shares, and sign up rates. It helps compare one measured rate with target. Always explain the population, sample, and success definition.

FAQs

1. What is a one proportion z test?

It is a hypothesis test for one sample proportion. It compares the observed sample rate with a claimed population rate.

2. What does p0 mean?

p0 is the claimed population proportion. It is the value stated in the null hypothesis.

3. What is the sample proportion?

The sample proportion is successes divided by total trials. It is written as p̂ in formulas.

4. When should I use a two tailed test?

Use it when you want to check whether the true proportion is different from p0 in either direction.

5. What does the p value show?

It shows how unusual the sample result is if the null proportion is true. Smaller values give stronger evidence against H0.

6. What alpha level should I choose?

Many reports use 0.05. Stricter work may use 0.01. Choose alpha before viewing the final result.

7. Why does the calculator show warnings?

The z test needs enough expected successes and failures. Low expected counts can make the normal approximation weak.

8. What is the continuity correction?

It adjusts the normal approximation for count data. It can make the test more conservative in some cases.