Hypothesis Test Population Mean Calculator

Calculator

Test method

Alternative hypothesis

Null mean μ₀

Sample mean x̄

Sample size n

Sample standard deviation s

Population standard deviation σ

Significance level α

Confidence level percent

Display note

Measurement units

Claim label

Optional raw data

Formula used

Z test statistic: z = (x̄ − μ₀) / (σ / √n)

T test statistic: t = (x̄ − μ₀) / (s / √n)

Degrees of freedom: 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

Select automatic, z test, or t test.
Choose the alternative hypothesis before viewing results.
Enter the null mean, sample mean, sample size, and deviation.
Paste raw data when you want the calculator to compute summary values.
Enter alpha and confidence level.
Press calculate to view the result above the form.
Use CSV or PDF buttons to save the report.

Example data table

Scenario	Method	Tail	μ₀	x̄	n	s or σ	α
Quality target check	T	Two tailed	100	104.2	36	s = 12.5	0.05
Minimum service score	Z	Right tailed	80	83.6	64	σ = 10	0.01
Cost reduction claim	T	Left tailed	55	52.4	25	s = 6.1	0.05

Why Population Mean Testing Matters

A population mean test checks whether sample evidence supports a claim about an unknown average. It is useful when a business, school, lab, or survey team must compare a measured average with a target value. The method turns sample information into a test statistic. That statistic is then matched with a probability. The probability is the p value. A small p value suggests the sample result would be unusual if the null claim were true.

Z And T Method Selection

This calculator supports two common methods. 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. The automatic option follows the same rule. It selects z when sigma is supplied. It selects t when sigma is missing. The t method also uses degrees of freedom. That value is n minus one.

Tail Choices And Claims

A two tailed test checks whether the mean is different from the null mean. A right tailed test checks whether the mean is greater. A left tailed test checks whether the mean is lower. Your tail choice should match the research question before any result is viewed. Changing tails after seeing results can lead to biased decisions.

Interpreting The Output

The result panel shows the test statistic, standard error, p value, critical value, confidence interval, and decision. Reject the null hypothesis when the p value is less than or equal to alpha. Otherwise, fail to reject it. This wording matters. A non rejected null is not proven true. It only means the sample does not give enough evidence at the selected alpha.

Practical Use

The calculator also accepts raw data. Paste values separated by commas, spaces, or new lines. It will compute the sample size, mean, and sample standard deviation. This helps prevent typing mistakes. The example table shows realistic settings for one sample mean tests. The CSV and PDF buttons make reports easier to save. Always check units, sample design, and assumptions before using the final conclusion. Reliable testing needs random sampling, independent observations, and measurements that match the claim being tested. Document each assumption, because later audits depend on clear records too.

FAQs

What is a population mean hypothesis test?

It is a statistical test that compares a sample mean with a claimed population mean. It helps decide whether the sample provides enough evidence against the null hypothesis.

When should I use a z test?

Use a z test when the population standard deviation is known and the sample observations are suitable for mean testing. Enter sigma to use this method.

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 the p value mean?

The p value estimates how unusual the sample result is if the null hypothesis is true. Smaller values give stronger evidence against the null claim.

What does alpha mean?

Alpha is the chosen significance level. It is the cutoff used for the decision. Common alpha values are 0.05, 0.01, and 0.10.

Can I paste raw data?

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

Why does the calculator show a confidence interval?

The interval gives a useful estimate range for the population mean. It supports interpretation beyond the reject or fail to reject decision.

Does fail to reject mean the null is true?

No. It means the sample does not provide enough evidence against the null hypothesis at the selected alpha level.