Hypothesis Test for Population Mean Calculator

Calculator

Claimed mean, μ₀

Sample mean, x̄

Sample size, n

Sample standard deviation, s

Population standard deviation, σ

Significance level, α

Confidence level (%)

Alternative hypothesis

Test type

Displayed decimals

Main assumption

Report note

Optional raw sample data

Raw data overrides the manual sample mean, sample size, and sample standard deviation.

Formula Used

Null hypothesis: H₀: μ = μ₀

Z statistic: z = (x̄ − μ₀) / (σ / √n)

T statistic: t = (x̄ − μ₀) / (s / √n), with df = n − 1

Standard error: SE = standard deviation / √n

Confidence interval: x̄ ± critical value × SE

Decision rule: reject H₀ when p value ≤ α.

How to Use This Calculator

Enter the claimed population mean.
Add the sample mean, sample size, and sample standard deviation.
Enter population standard deviation only when it is truly known.
Choose the alternative hypothesis direction.
Set alpha and the confidence level.
Press Calculate to see the result above the form.
Use CSV or PDF buttons to save the report.

Example Data Table

Case	μ₀	x̄	n	s or σ	Tail	α	Suggested Test
Class score check	50	52.4	36	9.1	Two tailed	0.05	T test
Known process spread	100	103.2	64	12	Right tailed	0.01	Z test
Fill weight audit	250	246.8	25	7.4	Left tailed	0.05	T test

About This Population Mean Test

A hypothesis test for a population mean checks a claim about an unknown average. It compares a sample result with a stated mean. The calculator handles left, right, and two tailed tests. It supports z tests when population standard deviation is known. It supports t tests when only sample standard deviation is available.

Why This Test Matters

Mean testing is common in quality control, surveys, education, finance, and lab work. A factory may test whether fill weight equals a label value. A teacher may test whether a class average changed after a program. A researcher may test whether a process mean is above a target. The method gives a structured way to judge evidence.

Inputs Used By The Calculator

Enter claimed mean, sample mean, sample size, standard deviation, alpha level, and tail direction. You may also paste raw sample data. When raw data is supplied, the tool computes size, mean, and sample standard deviation automatically. This helps reduce errors.

Understanding The Results

The test statistic measures distance from the claimed mean. It uses standard error as the unit. A large statistic usually gives stronger evidence against the null claim. The p value shows how unusual the sample would be if the null claim were true. The critical value shows the cutoff set by alpha.

Decision And Interpretation

If the p value is less than or equal to alpha, reject the null hypothesis. This means the sample gives enough evidence for the chosen alternative. If the p value is greater than alpha, fail to reject the null hypothesis. This does not prove the null claim. It only means the evidence is not strong enough.

Confidence Interval View

The calculator also reports a confidence interval for the population mean. For two tailed tests, this interval matches the selected confidence level. For one tailed tests, it still provides a useful estimate range. Use the interval with the decision statement for a clearer report.

Best Practice Notes

Check assumptions before using the result. Samples should be random and independent. Use a z test only when population standard deviation is known. Use a t test for small samples with unknown population standard deviation. Larger samples make the method more stable.

FAQs

What is a population mean hypothesis test?

It is a statistical test that checks whether sample evidence supports or challenges a claimed population average. It compares the sample mean with the claimed mean using standard error.

When should I use a z test?

Use a z test when the population standard deviation is known. This is common in controlled processes with established long-term variation.

When should I use a t test?

Use a t test when the population standard deviation is unknown and you rely on the sample standard deviation. This is common in most sample studies.

What does the p value mean?

The p value measures how unusual the sample result is under the null hypothesis. Smaller values give stronger evidence against the null claim.

What does alpha mean?

Alpha is the selected significance level. It is the cutoff used to decide whether the p value is small enough to reject the null hypothesis.

What is a two tailed test?

A two tailed test checks for any difference from the claimed mean. It can detect sample evidence above or below the claimed value.

Can I paste raw data?

Yes. Paste numbers separated by commas, spaces, or lines. The calculator will compute the sample size, mean, and sample standard deviation.

Does failing to reject prove the claim?

No. Failing to reject means the sample evidence is not strong enough against the claim. It does not prove the population mean equals the claim.