What Is Hypothesis Statistics?
A hypothesis test checks whether sample data are consistent with a claim about a population. You work out a test statistic, convert it to a p-value, and compare the p-value with your significance level.
This calculator covers four common tests: a one-sample z test, a one-sample t test, a one-proportion z test and a two-sample Welch t test. Choose one, enter the summary numbers and the alternative hypothesis.
It shows the hypotheses, the test statistic, the p-value, the critical value and the decision, along with a graph of the shaded p-value area.
How to Use the Hypothesis Statistics Calculator
- Choose the test and the alternative hypothesis.
- Enter the summary statistics.
- Read the test statistic, p-value and decision.
Hypothesis Statistics Formula
Mean test
Proportion test
Welch two-sample
Hypothesis Statistics Variables Explained
| Symbol | Meaning | Notes |
|---|---|---|
| x̄, s, n | Sample mean, standard deviation and size | |
| μ₀, p₀ | Value of the mean or proportion under the null hypothesis | |
| t or z | Test statistic | (estimate − null) / SE |
| p | p-value | Reject H₀ if p ≤ α |
Hypothesis Statistics Worked Example
With the example values loaded in the calculator above, the steps are:
- One-sample t test for a mean. H₀: μ = 50 against H₁: μ ≠ 50.
- Standard error = 1.22.
- t = (52.3 − 50) ÷ 1.22 = 1.8852 with 24 degrees of freedom.
- p-value (two-tailed) = 0.07156.
- Compare with α = 0.05: p > α, fail to reject H₀.
Result: Test statistic t = 1.8852; p-value = 0.07156; Decision = Fail to reject H₀ at α = 0.05.
Worked example: one-sample t test
Suppose x̄ = 52.3, μ₀ = 50, s = 6.1 and n = 25. The standard error is 6.1 ÷ 5 = 1.22, so t = 2.3 ÷ 1.22 = 1.885 with 24 degrees of freedom.
The two-tailed p-value is about 0.0715. At α = 0.05 that is above the cut-off, so you fail to reject H₀.
Choosing the test
Use a z test for a mean when σ is known. Use t when you estimate it from the sample. Use the proportion test for yes or no data. Use Welch for comparing two independent means without assuming equal variances.
Where Hypothesis Statistics Is Used
Statistics homework
Run a test from summary data and get the decision.
Checking a claim
Compare a sample mean or proportion with a target.
Comparing two groups
Test a difference in means without equal-variance assumptions.
Common Hypothesis Statistics Mistakes to Avoid
- Choose the alternative before you see the data.
- Report the p-value and an effect size, not only the decision.
Hypothesis Statistics Terms and Definitions
- Null hypothesis
- The default claim, usually no effect or no difference.
- p-value
- The probability of a result at least this extreme if H₀ is true.
Hypothesis Statistics FAQs
What does a hypothesis test tell me?
Whether the sample gives enough evidence against the null hypothesis at your significance level.
When do I use z and when t?
Use z when the population standard deviation is known, t when you estimate it from the sample.
What is a Welch test?
A two-sample t test that does not assume the two groups have equal variances.
What does it mean to fail to reject?
The data did not provide enough evidence against H₀. It does not show H₀ is true.