Calculator
Example Data Table
| Scenario | Successes | Sample Size | p0 | Alpha | Approximate Z | Approximate P Value |
|---|---|---|---|---|---|---|
| Survey support check | 62 | 100 | 0.50 | 0.05 | 2.400 | 0.016 |
| Quality pass rate | 78 | 120 | 0.60 | 0.05 | 1.118 | 0.264 |
| Defect reduction review | 24 | 80 | 0.35 | 0.05 | -0.938 | 0.348 |
Formula Used
Sample proportion: p̂ = x / n
Null standard error: SE0 = sqrt[p0(1 - p0) / n]
Z score: z = (p̂ - p0) / SE0
Two tailed p value: p value = 2 × [1 - Φ(|z|)]
Decision rule: reject H0 when p value ≤ alpha.
Continuity correction: the absolute difference is reduced by 0.5 / n before the z score is calculated.
How to Use This Calculator
Enter the number of observed successes. Enter the total sample size. Add the claimed population proportion as a decimal between 0 and 1.
Choose alpha, confidence level, interval method, and rounding. Select continuity correction only when you want a more conservative normal approximation.
Press calculate to show the z score, p value, confidence interval, expected counts, effect size, and final decision.
Use the CSV button for spreadsheet work. Use the PDF button for a simple saved report.
About the One Sample Proportion Test
Why This Test Matters
A one sample proportion test checks one population rate. It compares observed successes with a claimed value. The two tailed form asks whether the rate is different. It does not assume a direction. That makes it useful for audits, surveys, trials, and quality checks.
What The Calculator Measures
The calculator starts with successes, sample size, and the null proportion. It finds the sample proportion. It then builds a standard error from the null value. The z score shows how far the sample sits from the claim. A large absolute z score means stronger evidence against the claim.
Interpreting The P Value
The two tailed p value measures both extremes. It includes results far below the claim. It also includes results far above the claim. When the p value is less than alpha, the result is statistically significant. In that case, you reject the null proportion. When it is larger, you do not have enough evidence.
Confidence Interval Use
A confidence interval gives a range for the true population proportion. It supports the test result, but it answers a slightly different question. The interval estimates plausible values. The test judges one target value. The calculator offers common interval styles for practical reporting.
Assumptions And Limits
This z method works best with random data. Observations should be independent. The sample should be small compared with the population, unless sampling uses replacement. Expected successes and failures should usually be at least five. Very small samples may need an exact binomial test.
Good Reporting Practice
Report the sample proportion, z score, p value, alpha, and conclusion. Also include the confidence interval and assumption notes. Avoid saying the null is proven true. A nonsignificant result only means evidence was not strong enough. Use subject knowledge with the numeric output.
Common Use Cases
Analysts use this test for conversion rates, defect rates, support pass rates, election polls, and medical response rates. Teachers can use it in statistics lessons because each input has a clear meaning. Business teams can compare an observed rate with a promised target. Researchers can document evidence without overstating certainty. The same structure also helps learners understand hypothesis testing step by step. It keeps reports clear and easy later.
FAQs
What is a one sample proportion test?
It tests whether one population proportion differs from a claimed proportion. It uses sample successes, sample size, and a null proportion.
Why is this calculator two tailed?
A two tailed test checks for any difference. The sample proportion may be higher or lower than the claimed value.
What does p value mean here?
The p value shows how unusual the sample result is if the null proportion is true. Smaller values give stronger evidence against H0.
When should I reject H0?
Reject H0 when the two tailed p value is less than or equal to alpha. Otherwise, fail to reject H0.
What is the sample proportion?
The sample proportion is successes divided by sample size. It estimates the unknown population proportion from observed data.
What alpha should I use?
Many reports use 0.05. Use a smaller alpha when false positives are costly. Follow your study plan or field standard.
What if expected counts are small?
The normal z method may be weak when expected successes or failures are below five. Consider an exact binomial test then.
Which confidence interval method is best?
Wilson is often stable for proportions. Wald is simple but weaker near zero or one. Agresti-Coull is also practical.