Calculator
Example Data Table
| Successes | Sample Size | p₀ | Alpha | Alternative | Approximate Result |
|---|---|---|---|---|---|
| 64 | 100 | 0.50 | 0.05 | p > p₀ | Strong evidence above 0.50 |
| 47 | 80 | 0.60 | 0.05 | p ≠ p₀ | Weak evidence against 0.60 |
| 35 | 120 | 0.35 | 0.10 | p < p₀ | Check p value before rejecting |
Formula Used
The sample proportion is calculated as:
p̂ = x / n
The null standard error is:
SE₀ = √[p₀(1 - p₀) / n]
The one proportion z statistic is:
z = (p̂ - p₀) / SE₀
The p value depends on the alternative hypothesis. A right tailed test uses P(Z ≥ z). A left tailed test uses P(Z ≤ z). A two tailed test uses 2 × min[P(Z ≤ z), P(Z ≥ z)].
The Wald confidence interval is:
p̂ ± z* × √[p̂(1 - p̂) / n]
The Wilson interval uses center and spread adjustment. It performs better for small samples and extreme proportions.
How to Use This Calculator
- Enter the number of successes from your sample.
- Enter the total sample size.
- Enter the claimed population proportion as p₀.
- Choose alpha, such as 0.05 or 0.01.
- Select the correct alternative hypothesis.
- Add optional population size, margin, or power proportion.
- Press Calculate to view the result above the form.
- Use CSV or PDF download for saving the report.
Why This Calculator Matters
A population proportion test checks one claimed rate. It is useful when data has only two outcomes. Common examples include pass or fail, yes or no, and defect or no defect. The calculator turns sample counts into a clear statistical decision.
What The Test Measures
The null hypothesis states that the true proportion equals a chosen value. The sample proportion is compared with that value. The calculator uses the standard one proportion z test. It reports the z score, p value, critical value, and conclusion. It also shows confidence intervals, effect size, and approximation warnings.
Better Inputs Give Better Results
Enter the number of successes and the full sample size. Then enter the hypothesized proportion. Choose a left tailed, right tailed, or two tailed test. The selected direction should match the research claim before the data is examined. This keeps the test honest.
Advanced Options
Continuity correction can make the normal test more conservative. It is helpful when counts are close to the normal approximation limit. The calculator also includes Wilson and Wald confidence intervals. Wilson intervals are often steadier when proportions are near zero or one. Optional finite population correction adjusts the Wald interval when sampling without replacement from a limited population.
Reading The Output
A small p value means the sample result is unlikely under the null hypothesis. If the p value is less than alpha, reject the null hypothesis. If it is not less, do not reject it. This does not prove the null claim. It means the sample did not give enough evidence against it.
Use With Care
Check the expected successes and failures under the null value. Both should usually be at least five. Larger counts give a better z approximation. For very small samples, an exact binomial test may be better. Also remember that statistical significance is not the same as practical importance. Review the effect size, confidence interval, and study design together.
Practical Reporting
A good report states the sample count, sample size, null value, alternative, alpha, z score, p value, and decision. Include the confidence interval when possible. Download the CSV or PDF record to keep the result with your notes. It supports repeatable review and class assignments.
FAQs
What is a population proportion test?
It tests whether a true population rate matches a claimed value. The data must have two outcomes, such as success and failure.
When should I use a two tailed test?
Use it when the claim only says the population proportion is different from p₀. It checks both higher and lower directions.
When should I use a right tailed test?
Use it when the research claim says the population proportion is greater than the hypothesized value. Choose this before seeing results.
What does the p value mean?
The p value is the probability of getting a result this extreme, assuming the null hypothesis is true. Smaller values give stronger evidence.
What does alpha mean?
Alpha is the chosen significance level. Common values are 0.05 and 0.01. It sets the cutoff for rejecting the null hypothesis.
What is continuity correction?
Continuity correction adjusts a discrete count for use with a continuous normal curve. It often makes the z test more conservative.
Why show Wilson and Wald intervals?
Wald intervals are simple. Wilson intervals are often more stable, especially when the sample is small or the proportion is near zero or one.
Can this replace an exact binomial test?
No. For very small samples or weak expected counts, an exact binomial test may be better. This calculator shows a warning when needed.