# Confidence Interval Calculator - Means, Proportions, t-test and z-test Online

> Free online statistical inference: confidence intervals for means and proportions, one- and two-sample tests with the statistic, df, p-value and decision.

- URL: https://qilingtools.com/en/math/hypothesis-test
- Category: Math Tools (https://qilingtools.com/en/math)
- Tags: [Math](https://qilingtools.com/en/tags/math), [Statistics](https://qilingtools.com/en/tags/statistics)
- Last updated: 2026-10-26
- Price: Free, no sign-up
- Privacy: Runs locally in the browser; files are never uploaded

Enter the sample mean, standard deviation and size to get a confidence interval or a test result straight away. Means use the t distribution when the population standard deviation is unknown, proportions come with both the Wald and the sturdier Wilson interval, and every test reports the statistic, degrees of freedom, p-value, critical value and decision.

## How to use

1. Choose what you need: interval for a mean, interval for a proportion, one-sample test or two-sample test
2. Enter the sample statistics (mean, standard deviation, size — or successes and trials)
3. Pick the confidence level or significance level α, and the tail for a test
4. Read the interval, or the p-value and the conclusion

## Features

- Interval for a mean uses t when σ is unknown and switches to z when it is known
- Interval for a proportion gives the Wald interval and the small-sample-friendly Wilson interval
- One-sample test: t or z statistic, degrees of freedom, one- or two-tailed p-value, critical value and decision
- Two-sample test uses Welch’s method, so equal variances are not assumed
- t quantiles come from the incomplete beta function and match printed tables to six decimals
- The standard error and margin of error are shown so the work matches the textbook formulas

## FAQ

### When do I use t instead of z?

Use z when the population standard deviation σ is known (or the sample is very large), and t with n−1 degrees of freedom when σ has to be estimated by the sample standard deviation s. The two converge as n grows and differ little past about n = 30.

### What does a p-value actually mean?

The probability of seeing a result at least as extreme as this one if the null hypothesis were true. A small p means the data sit badly with the null, and p < α is the rule for rejecting it — but p is not the probability that the null is true.

### What does “95% confidence” mean?

If you repeated the sampling many times, about 95% of the intervals produced this way would contain the true parameter. For one particular interval the parameter is either in it or not, so it is wrong to say the parameter has a 95% chance of being inside.

### Why is the Wilson interval better than Wald?

The Wald interval can run past 0 or 1 and under-covers badly when the sample is small or the proportion is near an endpoint. The Wilson interval always stays inside [0, 1] and its coverage stays close to the nominal level, so it is the one to trust.

### Why does the two-sample test default to Welch?

The pooled-variance t test assumes both groups have the same variance, which often is not true. Welch’s test drops that assumption and performs virtually identically when the variances are equal, so it is the safer default.

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