Hypothesis Test for a Population Proportion Calculator

Calculator Inputs

Number of successes

Sample size

Claimed proportion p0

Alpha level

Alternative hypothesis

Confidence level

Interval method

Continuity correction

True proportion for power

Reset

Example Data Table

Scenario	Successes	Sample Size	Claimed p0	Alpha	Alternative
Product approval survey	48	120	0.35	0.05	p ≠ p0
Defect rate audit	9	200	0.07	0.01	p < p0
Conversion lift test	86	300	0.24	0.05	p > p0

Formula Used

Sample proportion:

p̂ = x / n

Standard error under the null claim:

SE0 = √[p0(1 - p0) / n]

Z test statistic:

z = (p̂ - p0) / SE0

Two tailed p value:

p value = 2 × min[P(Z ≤ z), P(Z ≥ z)]

Right tailed p value:

p value = P(Z ≥ z)

Left tailed p value:

p value = P(Z ≤ z)

Wald confidence interval:

p̂ ± z* × √[p̂(1 - p̂) / n]

Wilson confidence interval:

[(p̂ + z*² / 2n) ± z*√{[p̂(1 - p̂) + z*² / 4n] / n}] / [1 + z*² / n]

How to Use This Calculator

Enter the number of successes found in your sample.
Enter the total sample size.
Enter the claimed population proportion as a decimal.
Choose the alpha level for the decision rule.
Select the alternative hypothesis before calculating.
Choose a confidence level and interval method.
Add a possible true proportion when power is needed.
Press Calculate to show the result above the form.
Use CSV or PDF export for reporting.

Understanding One Proportion Testing

A population proportion test checks a claim about one percentage. It uses sample results to judge a stated target. The target may be a pass rate, defect rate, voter share, conversion rate, or recovery rate. The method compares the sample proportion with the hypothesized proportion. Then it measures the gap in standard error units.

When This Test Helps

Use this test when each observation has two outcomes. Examples include yes or no, success or failure, and defective or acceptable. The sample should be random. Observations should be independent. The null model also needs enough expected successes and failures. Many courses use at least five in each group. Some analysts prefer ten for safer normal results.

What The Output Means

The z statistic shows how far the sample result sits from the claim. A large positive value means the sample proportion is above the claim. A large negative value means it is below the claim. The p value measures how unusual the result is under the null claim. A small p value gives evidence against the null hypothesis.

Choosing The Alternative

The alternative hypothesis sets the direction of the test. Use two tailed when any difference matters. Use right tailed when you test for a higher proportion. Use left tailed when you test for a lower proportion. This choice must be made before viewing results. Changing it later can weaken the conclusion.

Intervals And Decisions

The confidence interval gives a practical range for the unknown proportion. It is not the same as a p value. It helps show the size of the possible effect. Wilson intervals often behave better than basic Wald intervals, especially with smaller samples. The calculator reports both testing and interval information, so the result is easier to explain.

Good Reporting Practice

Report the sample size, successes, sample proportion, null proportion, test direction, z statistic, p value, alpha level, and conclusion. Avoid saying the null hypothesis is proven. Say whether the evidence is strong enough to reject it. Also mention assumptions and any limits. This keeps the statistical conclusion clear, honest, and useful for decisions.

For serious studies, combine this output with study design notes and subject knowledge before action, review, or public reporting.

FAQs

What is a population proportion test?

It is a hypothesis test for one percentage or probability. It compares a sample proportion with a claimed population proportion and reports whether the sample gives enough evidence against the claim.

When should I use a two tailed test?

Use a two tailed test when you want to detect any difference from the claimed proportion. It checks both higher and lower sample results against the null hypothesis.

When should I use a right tailed test?

Use a right tailed test when your research question asks whether the true proportion is greater than the claimed proportion. Choose it before reviewing the sample result.

When should I use a left tailed test?

Use a left tailed test when your research question asks whether the true proportion is less than the claimed proportion. It places the rejection region on the lower side.

What does the p value mean?

The p value is the probability of getting evidence this extreme, assuming the null claim is true. Smaller values show stronger evidence against the null hypothesis.

What alpha level should I choose?

Common alpha levels are 0.05, 0.01, and 0.10. Lower alpha values require stronger evidence before rejecting the null hypothesis.

Why is the Wilson interval included?

The Wilson interval often performs better than the simple Wald interval, especially for small samples or proportions near zero or one. It gives a more stable interval.

Can this calculator prove the null hypothesis?

No. A failed rejection does not prove the null hypothesis. It only means the sample did not provide enough evidence against the claim at the selected alpha level.