Formula Used
| z test statistic | z = (x̄ − μ₀) / (σ / √n) |
|---|---|
| t test statistic | t = (x̄ − μ₀) / (s / √n) |
| Standard error | SE = standard deviation / √n |
| Confidence interval | x̄ ± critical value × SE |
| Effect size | Cohen’s d = (x̄ − μ₀) / standard deviation |
How to Use This Calculator
- Choose auto, t test, or z test.
- Select the correct alternative hypothesis.
- Enter the hypothesized mean and sample details.
- Use raw data when individual observations are available.
- Set alpha and the confidence level.
- Press Calculate to view the result above the form.
- Use CSV or PDF buttons to save the report.
Example Data Table
| Scenario | μ₀ | x̄ | n | s or σ | Alpha | Tail |
|---|---|---|---|---|---|---|
| Exam score check | 50 | 52.4 | 36 | 7.8 | 0.05 | Two tailed |
| Package fill audit | 16 | 16.3 | 64 | 1.2 | 0.01 | Right tailed |
| Cycle time reduction | 25 | 23.9 | 28 | 3.4 | 0.05 | Left tailed |
Understanding a Mean Hypothesis Test
A mean hypothesis test checks a claim about one population average. It compares a sample mean with a claimed value. The calculator supports both common cases. Use a z test when the population standard deviation is known. Use a t test when it is unknown and the sample standard deviation is used.
Why the Test Matters
Many decisions need more than a simple average. A school may test whether scores changed. A factory may test whether fill weight meets a target. A researcher may test whether a treatment shifts a measured response. The test turns those questions into a statistic, a p value, and a decision.
Key Inputs
You need the hypothesized mean, sample size, sample mean, and variation. You also choose alpha. Alpha is the risk level for rejecting a true claim. Common values are 0.05, 0.01, and 0.10. The alternative hypothesis sets the direction. Use two tailed for any difference. Use right tailed for a higher mean. Use left tailed for a lower mean.
Reading the Result
The test statistic shows how many standard errors separate the sample mean from the claim. A larger absolute value gives stronger evidence. The p value shows how unusual the sample is if the null claim is true. If the p value is less than alpha, reject the null hypothesis. If it is not, fail to reject it.
Confidence Interval and Effect
The confidence interval gives a range of plausible population means. It should be read with the test result. When a two tailed test rejects the claim, the claimed mean usually falls outside the matching confidence interval. Effect size helps judge practical importance. Cohen’s d divides the mean difference by standard deviation. A small p value may still have a small effect.
Good Practice
Check that sampling is reasonable before using the result. The sample should be random or representative. Very small samples need data that is roughly normal. Large samples are more forgiving. Always report the test type, tail choice, alpha, statistic, p value, and conclusion. Keep units with every mean and interval.
Save each report when comparing many scenarios. Consistent records make peer review easier and reduce later calculation mistakes too.
FAQs
What is a mean hypothesis test?
It is a statistical test for a claim about a population mean. It compares the sample mean with the hypothesized mean and reports a p value.
When should I use a t test?
Use a t test when the population standard deviation is unknown. This is common because most real samples only provide a sample standard deviation.
When should I use a z test?
Use a z test when the population standard deviation is known. It is also common with large samples and trusted process variation.
What does alpha mean?
Alpha is the chosen significance level. It is the risk of rejecting the null hypothesis when the null hypothesis is actually true.
What does the p value show?
The p value shows how likely the sample result is under the null hypothesis. Smaller values give stronger evidence against the null claim.
What is a two tailed test?
A two tailed test checks whether the population mean is different from the claimed mean. It tests both higher and lower directions.
Can I enter raw data?
Yes. Enter values separated by commas, spaces, or new lines. The calculator will compute the sample mean, size, and standard deviation.
What is Cohen’s d?
Cohen’s d is an effect size. It divides the mean difference by standard deviation and helps judge practical importance.