Calculator Inputs
Example Data Table
| Case | Successes | Sample Size | Null Proportion | Alpha | Alternative | Use Case |
|---|---|---|---|---|---|---|
| Survey approval | 56 | 120 | 0.50 | 0.05 | Two tailed | Check if approval differs from half. |
| Defect reduction | 14 | 220 | 0.10 | 0.05 | Left tailed | Check if the defect rate fell. |
| Conversion lift | 88 | 400 | 0.18 | 0.01 | Right tailed | Check if conversion improved. |
Formula Used
The calculator uses the one sample z test for a population proportion.
Sample proportion: p̂ = x / n
Standard error under null: SE0 = sqrt(p0 × (1 - p0) / n)
Z statistic: z = (p̂ - p0) / SE0
Two tailed p value: 2 × P(Z ≥ |z|)
Right tailed p value: P(Z ≥ z)
Left tailed p value: P(Z ≤ z)
The decision rejects the null hypothesis when the p value is less than alpha.
How to Use This Calculator
- Enter the number of successes from your sample.
- Enter the total sample size.
- Enter the claimed null proportion.
- Choose alpha, such as 0.05 or 0.01.
- Select the correct alternative hypothesis direction.
- Enter a confidence level for interval estimates.
- Add a planning proportion if you want power.
- Press submit, then review the result above the form.
- Use CSV or PDF export for records.
Article
Why this calculator matters
A proportion test checks a claim about a population share. It is useful when the data has only two outcomes. Examples include yes or no, pass or fail, and defect or clean. The calculator turns those counts into a formal test. It gives a z statistic, a p value, and a decision. This helps users avoid guesses.
What the result means
The null proportion is the claim being tested. The sample proportion is the observed rate. The z statistic measures how far the sample rate is from the claim. It uses the standard error under the null claim. A larger distance gives stronger evidence. The p value translates that distance into probability. A small p value suggests the sample is unusual under the claim.
Choosing the right tail
Use a two tailed test when the sample may differ in either direction. Use a right tailed test when you expect the true proportion to be higher. Use a left tailed test when you expect it to be lower. The calculator changes the p value and critical rule for each choice. This makes the decision match the research question.
Confidence and power
The confidence interval estimates a practical range for the true proportion. The normal interval is simple. The Wilson interval is often steadier near zero or one. The power estimate uses an optional true proportion. It shows the chance of rejecting the null when that chosen value is real. This is helpful for planning studies.
Best practice notes
Check the expected successes and failures before trusting the z test. Many guides prefer at least ten in each expected group. Small samples may need an exact binomial test instead. Also enter independent observations. Duplicate or biased records can distort the result. Report the test side, alpha level, p value, and confidence interval together. These details make the conclusion clear.
When the report is exported, keep the input values with the output. This keeps the calculation traceable. It also helps reviewers repeat the same test later. For teaching, compare several alpha levels. Students can see how the rejection rule changes. For business, save the table before decisions are shared. That habit improves audit quality and team confidence over time.
FAQs
What is a proportion hypothesis test?
It tests a claim about one population proportion. The method compares the observed sample proportion with a claimed null value.
When should I use a two tailed test?
Use it when you only want to know whether the true proportion is different. The direction can be higher or lower.
When should I use a right tailed test?
Use it when your research claim says the true proportion is greater than the null proportion.
When should I use a left tailed test?
Use it when your research claim says the true proportion is less than the null proportion.
What does the p value mean?
It shows how unusual the sample result is under the null claim. Smaller values give stronger evidence against the null.
What alpha level should I enter?
Common choices are 0.05, 0.01, and 0.10. Use the level required by your study, class, or reporting rule.
Why is the expected count check important?
The z test works best when expected successes and failures are large enough. Low expected counts may need exact binomial testing.
What is the Wilson interval?
It is a confidence interval for a proportion. It is often more stable than the simple normal interval for difficult samples.