Calculator
Example Data Table
| Case | n | Sample mean | Sample SD | μ₀ | Tail | α | Expected result |
|---|---|---|---|---|---|---|---|
| Student score check | 36 | 82.4 | 8.7 | 80 | Two tailed | 0.05 | t test decision |
| Factory fill amount | 50 | 502.1 | 6.4 | 500 | Right tailed | 0.01 | Upper mean check |
| Delivery time review | 25 | 18.2 | 3.1 | 20 | Left tailed | 0.05 | Lower mean check |
Formula Used
For a known population standard deviation, the calculator uses a z statistic:
z = (x̄ − μ₀) / (σ / √n)
For an unknown population standard deviation, it uses a t statistic:
t = (x̄ − μ₀) / (s / √n)
The standard error is σ / √n for a z test. It is s / √n for a t test. The degrees of freedom for a t test are n − 1.
The confidence interval is:
x̄ ± critical value × standard error
How to Use This Calculator
Select summary statistics or raw sample data. Enter the sample size, sample mean, and sample deviation when using summary mode. Paste raw values when using raw mode. Add the hypothesized mean. Choose a test method, tail direction, alpha level, and confidence level. Press submit to view the test statistic, p value, confidence interval, and decision.
Population Mean Test Guide
What This Calculator Does
A population mean test checks whether sample evidence supports a claim about a larger population mean. It is useful when a study, quality check, survey, or experiment gives one sample. The calculator compares the sample mean with the hypothesized mean. It then reports a test statistic, p value, critical value, and decision.
Choosing the Correct Test
Use a z test when the population standard deviation is known. Use a t test when the population standard deviation is unknown and the sample standard deviation estimates spread. The t test is common in real work because the true population deviation is often unavailable. Auto mode selects a z test when population deviation is entered.
Understanding Tail Direction
A two tailed test checks whether the mean is different from the claim. A right tailed test checks whether the mean is greater. A left tailed test checks whether the mean is lower. Choose the tail before looking at results. This keeps the test fair and clear.
Reading the P Value
The p value shows how unusual the sample result is under the null hypothesis. A small p value gives stronger evidence against the null claim. If the p value is less than or equal to alpha, the calculator rejects the null hypothesis. Otherwise, it fails to reject it.
Confidence Interval Meaning
The confidence interval gives a practical range for the population mean. It is based on the sample mean, standard error, and selected confidence level. A narrow interval shows more precision. Larger samples usually reduce standard error and make the interval narrower.
Practical Notes
Always check data quality before testing. Extreme outliers can change the mean and deviation. Small samples should come from a roughly normal population, especially for a t test. Large samples are more stable. The final decision should be combined with subject knowledge, effect size, and data context.
FAQs
What is a population mean test?
It is a hypothesis test that compares a sample mean with a claimed population mean. It tells whether the sample gives enough evidence to reject the claimed mean.
When should I use a z test?
Use a z test when the population standard deviation is known. This is less common in practice, but it is useful in controlled processes or established studies.
When should I use a t test?
Use a t test when the population standard deviation is unknown. The calculator then uses the sample standard deviation and degrees of freedom.
What does alpha mean?
Alpha is the significance level. It is the cutoff for rejecting the null hypothesis. Common values are 0.05, 0.01, and 0.10.
What does the p value show?
The p value measures evidence against the null hypothesis. Smaller values mean the sample result is less likely if the null claim is true.
What is a two tailed test?
A two tailed test checks for any difference from the hypothesized mean. It does not assume the sample mean should be higher or lower.
Can I use raw data?
Yes. Choose raw sample data and paste numbers separated by commas, spaces, lines, or semicolons. The calculator computes the mean and sample deviation.
Why is the confidence interval useful?
It gives a range of likely values for the population mean. It helps interpret the size and practical meaning of the test result.