Example Data Table
| Successes | Sample Size | Null Proportion | Alternative | Alpha | Expected Result |
|---|---|---|---|---|---|
| 62 | 100 | 0.50 | Two sided | 0.05 | Sample rate is tested against 50%. |
| 85 | 120 | 0.65 | Greater | 0.05 | Checks whether the true rate is higher. |
| 18 | 80 | 0.30 | Less | 0.01 | Checks whether the true rate is lower. |
Formula Used
The sample proportion is:
p̂ = x / n
The normal test statistic is:
z = (p̂ - p0) / √(p0(1 - p0) / n)
For the exact method, the binomial probability is:
P(X = k) = C(n, k)p0k(1 - p0)n-k
The p-value depends on the selected alternative hypothesis. A two sided test uses both tails. Greater and less tests use one tail.
How to Use This Calculator
- Enter the number of observed successes.
- Enter the total sample size.
- Add the claimed null proportion.
- Select the alternative hypothesis before testing.
- Choose alpha, confidence level, test method, and interval type.
- Press calculate to view the result above the form.
- Use the CSV or PDF button to save the report.
Article
Understanding the One Sample Proportion Test
A one sample proportion test checks one population rate. It compares sample evidence with a claimed proportion. The claim may be a pass rate, defect rate, click rate, or support rate. The method is useful when every observation has two outcomes. Examples include yes or no, success or failure, and correct or incorrect.
When the Test Helps
Use this test when you know the sample count and total size. You also need a null proportion. The calculator turns these values into a sample proportion. It then estimates how far the sample sits from the claim. A large distance gives stronger evidence against the null statement.
Choosing the Alternative
The alternative hypothesis controls the tail of the test. Choose two sided when any difference matters. Choose greater when the sample rate should be higher. Choose less when the sample rate should be lower. This choice must be made before viewing results. Changing it after testing can mislead decisions.
Interpreting Results
The z score measures distance in standard error units. The p value shows how unusual the sample is under the null claim. A small p value means the observed result is unlikely under that claim. Compare it with the chosen significance level. When p is smaller than alpha, reject the null hypothesis. Otherwise, do not reject it.
Confidence Interval Use
The confidence interval estimates the likely range for the true proportion. A narrow interval means the estimate is more precise. A wider interval often means the sample is small or mixed. Wilson and Agresti-Coull intervals usually behave better near zero or one. The Wald interval is simple, but can be weak for small samples.
Good Practice
Check that expected successes and failures are large enough. If either value is small, the exact binomial result is safer. Report the sample size, successes, p value, confidence interval, and conclusion. Use plain language after the numbers. This makes the result easier for readers to trust. The calculator supports downloads for records, audits, and shared reports.
Record Keeping
Saved files help teams compare repeated studies. They also make classroom checking easier. Keep inputs with the conclusion, so another person can reproduce the same test without confusion later and doubt.
FAQs
What is a one sample proportion test?
It tests whether one population proportion equals a claimed value. It uses sample successes, total sample size, and a null proportion to measure evidence.
When should I use the exact binomial test?
Use it when sample size is small or expected successes or failures are low. It avoids weak normal approximation assumptions.
What does the p-value mean?
The p-value shows how unusual your sample result is if the null proportion is true. Smaller values give stronger evidence against the null hypothesis.
What is the null proportion?
It is the claimed population rate being tested. For example, 0.50 means the claimed rate is fifty percent.
Which alternative hypothesis should I choose?
Choose two sided for any difference. Choose greater for a higher rate. Choose less for a lower rate. Decide before calculating.
What is continuity correction?
It adjusts the normal test for count data. It can improve approximation because binomial counts are discrete while the normal curve is continuous.
Which confidence interval is best?
Wilson is often a strong default. Agresti-Coull is also useful. Wald is simple but may perform poorly with small samples.
Can I download my result?
Yes. Enter the values, then use the CSV or PDF button. The downloaded file includes inputs, test results, and the decision.