Example Data Table
| Case | Input | Confidence | Expected Use |
|---|---|---|---|
| Mean | x̄ = 102.5, σ = 12, n = 64 | 95% | Estimate a population mean. |
| Proportion | x = 78, n = 120 | 95% | Estimate one population proportion. |
| Two Means | x̄1 = 55, x̄2 = 51, σ1 = 10, σ2 = 9 | 95% | Compare independent averages. |
| Two Proportions | x1 = 84 of 150, x2 = 70 of 140 | 95% | Compare independent rates. |
Formula Used
One mean: CI = x̄ ± z × σ / √n.
One proportion: CI = p̂ ± z × √(p̂(1 − p̂) / n).
Difference of means: CI = (x̄1 − x̄2) ± z × √(σ1² / n1 + σ2² / n2).
Difference of proportions: CI = (p̂1 − p̂2) ± z × √(p̂1(1 − p̂1) / n1 + p̂2(1 − p̂2) / n2).
The test statistic is z = (estimate − null value) / standard error.
How to Use This Calculator
Select the interval type first. Enter only the values required for that method. Add the confidence level as a percent. Leave custom z blank unless you already know the critical value. Choose the alternative hypothesis for the p value. Press Calculate. Review the interval, margin, z score, and p value. Use the export buttons for reports.
Z Test Confidence Interval Guide
Purpose
A z test confidence interval gives a practical range for an unknown population value. The range is built from a sample estimate. It also includes a margin of error. The method uses the standard normal curve. It is common in quality control, surveys, manufacturing, health studies, and business reporting.
When It Fits
Use a z interval for a mean when the population standard deviation is known. It can also be useful when a large sample supports normal behavior. Use a z interval for a proportion when expected successes and failures are large enough. Small samples can need exact or adjusted methods.
What the Output Means
The estimate is the center of the interval. The standard error measures sampling variation. The z critical value controls the confidence width. A higher confidence level gives a wider interval. The margin of error is added and subtracted from the estimate.
The lower and upper bounds are the final interval limits. If a null value lies inside the interval, the sample does not strongly contradict that value at the matching two sided level. If it lies outside, the result gives stronger evidence against it. The p value gives a separate test summary. It depends on the selected alternative hypothesis.
Good Practice
Check assumptions before trusting results. Samples should be independent. Data should be collected without strong bias. For means, the known standard deviation should be realistic. For proportions, success counts should not be too close to zero or the full sample size. Report the confidence level with every interval.
Do not read a confidence interval as a guarantee. It is a method based statement. In repeated sampling, the chosen method captures the true value at the stated long run rate. One interval either contains the true value or it does not. Clear data entry, correct units, and honest assumptions make the result more useful.
Reporting the Interval
Write the result with the estimate, interval bounds, and confidence level. Include sample size and known standard deviation when needed. For proportions, include success counts. State whether the interval supports the practical claim. Avoid saying the true value has a personal probability of sitting inside one computed interval during final reporting clearly.
FAQs
What is a z test confidence interval?
It is a range estimate built with a sample statistic, standard error, and z critical value. It estimates a population mean, proportion, or difference under normal approximation conditions.
When should I use a z interval for a mean?
Use it when the population standard deviation is known. It is also common with large samples when normal approximation is reasonable and data are independent.
Can this calculator compare two groups?
Yes. It supports independent mean differences and independent proportion differences. Enter both sample summaries, then review the estimate, standard error, and interval.
What does the margin of error mean?
The margin of error is the distance from the estimate to each interval bound. It increases when confidence rises or standard error becomes larger.
What is the custom z critical field?
It lets you override the automatic z value. Use it when your course, report, or table requires a specific rounded critical value.
Why is my proportion interval clipped at zero or one?
A simple normal proportion interval can extend outside valid probability limits. This tool limits one proportion results to the valid range from zero to one.
Does the p value match the confidence interval?
For two sided tests, it usually tells a related story. One sided p values use the chosen direction, so their interpretation can differ from a two sided interval.
Can I download the result?
Yes. Use Download CSV for spreadsheet work. Use Download PDF after calculating to save a simple report for records or sharing.