Test For One Proportion Calculator

Calculator Form

Number of successes

Sample size

Null proportion p₀

Alternative hypothesis

Significance level

Confidence level

Planning proportion for power

Continuity correction

Apply correction to z test

Exact binomial test

Calculate exact p value

Example Data Table

Scenario	Successes	Sample Size	Claimed Proportion	Alternative	Alpha
Customer approval check	142	250	0.50	Greater	0.05
Defect rate audit	9	180	0.08	Less	0.05
Survey support rate	312	600	0.50	Two sided	0.01

Formula Used

Sample proportion: p̂ = x / n

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

Z statistic: z = (p̂ - p₀) / SE₀

Continuity corrected difference: |p̂ - p₀| is reduced by 0.5 / n before the z statistic is calculated.

Two sided normal p value: p = 2 × P(Z ≥ |z|)

Exact binomial p value: probabilities are summed from the binomial distribution using n and p₀.

Wald interval: p̂ ± z*√[p̂(1 - p̂) / n]

Cohen h: h = 2asin(√p̂) - 2asin(√p₀)

How To Use This Calculator

Enter the number of observed successes.
Enter the total sample size.
Enter the claimed population proportion as a decimal.
Select the alternative hypothesis.
Set the significance level and confidence level.
Add a planning proportion to estimate power.
Choose exact testing or continuity correction when needed.
Press calculate and review the result above the form.
Download the result as CSV or PDF for reporting.

Understanding The One Proportion Test

A one proportion test checks whether one sample rate supports a claimed population proportion. It is useful when an outcome has two categories, such as pass or fail, yes or no, defect or no defect. The calculator compares the observed sample proportion with the hypothesized proportion.

When This Test Fits

Use this method when observations are independent and each trial has the same outcome definition. The sample should be random or reasonably representative. The normal z test works best when expected successes and expected failures under the claim are both large enough. When counts are small, the exact binomial option gives safer evidence.

What The Result Means

The z statistic measures distance from the claim in standard error units. A large positive z supports a greater alternative. A large negative z supports a less alternative. A two sided test looks for evidence in either direction. The p value reports how unusual the sample is when the claim is treated as true.

Confidence Intervals

Confidence intervals estimate the likely range for the true population proportion. The Wald interval is simple, but it may behave poorly near zero or one. The Wilson interval is usually more stable. The Agresti adjusted interval is also helpful for practical reporting.

Advanced Options

The continuity correction makes the normal test more conservative by adjusting the gap between the sample count and the claimed count. Exact probability sums use the binomial distribution directly. Power estimates show the chance of rejecting the claim when a chosen planning proportion is true.

Good Reporting Practice

Report the sample size, successes, observed proportion, null proportion, alternative, p value, confidence interval, and decision rule. Also mention whether the exact test or continuity correction was used. These details help readers understand the strength and limits of the conclusion.

Common Interpretation Errors

Do not read a p value as the probability that the claim is true. It measures sample extremeness under the claim. Do not ignore practical size either. A tiny difference can become significant with a very large sample. A clear report should connect statistical evidence with the real question, cost, risk, or quality target being studied. Always explain the sampling method before applying the final decision rule.

FAQs

What is a one proportion test?

It tests whether a sample proportion gives enough evidence against a claimed population proportion. The outcome must have two categories, such as success or failure.

When should I use the exact binomial option?

Use it when the sample is small, expected counts are low, or the sample proportion is near zero or one. It avoids weak normal approximation behavior.

What does the p value mean?

The p value measures how unusual the sample result is if the null proportion is true. Smaller values give stronger evidence against the claim.

What is the null proportion?

The null proportion is the claimed population rate being tested. It may come from a standard, target, past rate, policy, or research hypothesis.

Which confidence interval is best?

The Wilson interval is often preferred for proportions. It usually behaves better than the Wald interval, especially with smaller samples or extreme rates.

What does continuity correction do?

It adjusts the normal z test for the discrete nature of counts. This usually makes the normal test slightly more conservative.

What is Cohen h?

Cohen h is an effect size for proportions. It helps describe the practical size of the difference, not only statistical significance.

Why is power included?

Power estimates the chance of rejecting the null claim when a selected true proportion is assumed. It helps with planning and interpretation.