One Proportion Z Test Calculator

Calculator

Number of Successes Example: 58 favorable responses.

Sample Size Total observations in the sample.

Hypothesized Proportion Use 0.50 or 50%.

Alternative Hypothesis

Significance Level Use 0.05 or 5%.

Confidence Level Use 0.95 or 95%.

Continuity Correction

Apply 0.5 continuity adjustment

Useful when approximating discrete counts.

Formula Used

Sample proportion: p̂ = x / n

Null standard error: SE₀ = √[p₀(1 − p₀) / n]

Z statistic: z = (p̂ − p₀) / SE₀

Two tailed p value: 2 × [1 − Φ(|z|)]

Right tailed p value: 1 − Φ(z)

Left tailed p value: Φ(z)

Confidence interval: p̂ ± z* × √[p̂(1 − p̂) / n]

How to Use This Calculator

Enter the number of successes in your sample.
Enter the total sample size.
Enter the hypothesized population proportion.
Choose the alternative hypothesis.
Add alpha and confidence level values.
Select continuity correction when your course requires it.
Press Calculate to view the result below the header.
Use CSV or PDF buttons to save the output.

Example Data Table

Scenario	Successes	Sample Size	p₀	Alpha	Alternative	Expected Use
Customer approval	58	100	0.50	0.05	Two tailed	Check if approval differs from 50%.
Defect reduction	12	300	0.06	0.05	Left tailed	Check if defect rate is lower.
Campaign response	84	200	0.35	0.01	Right tailed	Check if response rate improved.

Understanding the One Proportion Z Test

A one proportion z test checks whether one sample proportion differs from a claimed population proportion. It is useful when data has only two possible outcomes. Common examples include pass or fail, yes or no, and defective or acceptable. The method compares the observed sample rate with the null value. It then converts the difference into a z score.

Why the Test Matters

This test helps users make decisions with sample evidence. A business may test whether the complaint rate is below a target. A teacher may test whether the pass rate has improved. A researcher may test whether support for an option differs from a stated benchmark. The calculator keeps these checks structured and repeatable.

Key Inputs

The main inputs are successes, sample size, hypothesized proportion, significance level, confidence level, and alternative hypothesis. Successes must be between zero and the sample size. The hypothesized proportion can be entered as a decimal or percent. The alpha value controls the rejection rule. Lower alpha values require stronger evidence against the null hypothesis.

Reading the Output

The z score shows how many standard errors the sample proportion is from the hypothesized value. The p value measures how unusual the sample result is under the null hypothesis. When the p value is less than or equal to alpha, the calculator rejects the null hypothesis. Otherwise, it does not reject it.

Practical Notes

The normal approximation works best when the expected successes and failures are large enough. Many introductory courses use five as a simple minimum. If expected counts are too small, an exact binomial test may be better. The confidence interval gives a helpful range for the true proportion. It should be reviewed with the test result, not used alone.

Good Interpretation

A rejected null does not prove the alternative with complete certainty. It means the sample gives enough statistical evidence at the chosen alpha level. A non rejected result does not prove equality. It means the sample evidence was not strong enough. Always consider data quality, sampling method, and practical importance before making final decisions.

Use results as evidence, not automatic truth. Clear reporting makes the conclusion easier for readers to audit and compare confidently later.

FAQs

What is a one proportion z test?

It is a hypothesis test for one sample proportion. It checks whether the observed sample proportion differs from a claimed population proportion.

What does p₀ mean?

p₀ is the hypothesized population proportion. It is the benchmark value used in the null hypothesis.

What does the z score show?

The z score shows how far the sample proportion is from p₀ in standard error units.

When should I reject the null hypothesis?

Reject the null hypothesis when the p value is less than or equal to your selected alpha level.

Can I enter percentages?

Yes. You can enter values such as 50% or 0.50. The calculator converts percentage style inputs into proportions.

What is alpha?

Alpha is the significance level. It sets the evidence cutoff used to reject or not reject the null hypothesis.

What is continuity correction?

Continuity correction adjusts the z statistic when a discrete count is approximated by a continuous normal curve.

When is this test not ideal?

It is not ideal when expected successes or failures are very small. In that case, an exact binomial test may be better.