What Is Hypothesis Mean?
A hypothesis test for a mean checks whether a sample mean is far enough from a claimed value μ₀ to cast doubt on the claim. Distance is measured in standard errors, giving a z or t statistic.
Choose a t test when the population standard deviation is unknown, which is the usual case, or a z test when it is given. Enter x̄, μ₀, the standard deviation and n, pick the direction of the alternative hypothesis, and set alpha.
The result lists H₀ and H₁, the test statistic, the p-value, the critical value, the decision and a confidence interval for the mean. Failing to reject H₀ means the data give too little evidence against it, not that it is proven.
How to Use the Hypothesis Mean Calculator
- Choose z if the population standard deviation is known, otherwise t.
- Enter x̄, μ₀, the standard deviation and n.
- Pick the direction of the alternative hypothesis and read the decision.
Hypothesis Mean Formula
Test statistic
Standard error
Degrees of freedom (t)
Hypothesis Mean Variables Explained
| Symbol | Meaning | Notes |
|---|---|---|
| x̄ | Sample mean | |
| μ₀ | Hypothesised population mean | |
| s or σ | Sample or population standard deviation | |
| n | Sample size | |
| α | Significance level | Commonly 0.05 |
Hypothesis Mean Worked Example
With the example values loaded in the calculator above, the steps are:
- H₀: μ = 50 versus H₁: μ ≠ 50.
- Standard error = 6 ÷ √25 = 1.2.
- t = (52 − 50) ÷ 1.2 = 1.6667.
- p-value = 0.10858, compared with α = 0.05.
- p > α, so fail to reject H₀.
Result: Test statistic t = 1.6667; p-value = 0.10858; Decision at α = 0.05 = Fail to reject H₀.
Worked example
A sample of n = 25 has mean 52 and s = 6. Test H₀: μ = 50 against H₁: μ ≠ 50 at α = 0.05.
SE = 6 ÷ 5 = 1.2 and t = (52 − 50) ÷ 1.2 = 1.667 with 24 degrees of freedom. The two-tailed p-value is about 0.109, above 0.05, so you fail to reject H₀.
How to word the conclusion
Say reject H₀ or fail to reject H₀. Failing to reject does not prove μ equals μ₀. It means the sample gives too little evidence against it.
Where Hypothesis Mean Is Used
Claims about an average
Test whether a mean weight, time or score differs from a stated value.
Quality control
Check that a process mean still matches its target.
Statistics exams
Work a hypothesis test from summary data and confirm the p-value.
Common Hypothesis Mean Mistakes to Avoid
- Decide the tail before looking at the data.
- If the confidence interval excludes μ₀, a two-tailed test rejects at the same alpha.
Hypothesis Mean Terms and Definitions
- Null hypothesis
- The claim being tested, usually that the mean equals a given value.
- p-value
- The probability, if H₀ is true, of a result at least as extreme as the one observed.
Hypothesis Mean FAQs
How do you test a hypothesis about a mean?
State H₀ and H₁, compute (x̄ − μ₀) ÷ (s ÷ √n), find the p-value and compare it with alpha.
When do I use a z test instead of a t test?
Use z when the population standard deviation is known. In practice t is used far more often because σ is usually unknown.
What does fail to reject H₀ mean?
The data do not give enough evidence against the claimed mean. It is not proof that the claim is true.
What are the assumptions?
Independent observations and roughly normal data, or a large enough sample for the mean to be approximately normal.
How is the confidence interval related to the test?
A 95% interval that excludes μ₀ matches a two-tailed rejection at α = 0.05.
Can I run this from raw data?
Use the t-test calculator for raw data. This page works from the summary numbers.