Hypothesis Tests About μ, σ Known, the P Value Approach

Statistics • Hypothesis Tests About the Mean and Proportion

Written by STEM Calculators Team Published December 28, 2025 Updated February 24, 2026

Hypothesis Tests About μ (σ Known): the p-value approach

Enter μ₀, known σ, a tail type, and either raw sample data (paste from CSV) or a summary (n and x̄). The calculator computes z, the p-value, and the decision at significance level α.

Inputs

Test type (tail)

The sign in H₁ determines where “extreme” results live.

Significance level α

Null mean μ₀

H₀ includes equality.

Known population σ

This is the z-test case where σ is treated as known.

Data input method

Raw mode is best when you paste a column from a CSV file.

Sample data (paste from CSV)

Tip: copy a column from Excel/Sheets/CSV and paste here. Separators: comma, semicolon, tab, space, new line.

Visualization

Run Calculate to see z, p-value, and shading.

Exploration controls activate after calculation.

Shading

Switch what the filled area represents.

Explore z

Manual z (clamped to −4 … 4).

z slider

Drag to watch p-value / critical regions change.

Simple motion across tails.

Shaded = p-value (mode: p-value) Shaded = critical region(s) (mode: critical) Dashed = critical value(s) for α Solid = z

The curve is standard normal (mean 0, standard deviation 1). The observed test statistic z and the shaded area visualize the p-value (or the α rejection region), depending on the selected shading mode.

Results & p-value steps

Quantity	Value	Meaning

p-value rule: reject H₀ if p-value ≤ α; otherwise, do not reject H₀.

Definition (p-value): assuming H₀ is true, the p-value is the probability of a sample result at least as extreme as the observed one in the direction of H₁.

Help: assumptions & what “σ known” means

This calculator applies the z-test for a population mean when the population standard deviation σ is treated as known. A normal distribution procedure is commonly used when either (a) the population is approximately normal and n is small, or (b) n is large (so the sampling distribution of x̄ is approximately normal).

Situation	Typical condition	What to do
Small sample + population approx normal	n < 30 and distribution is close to normal (no extreme outliers)	Use the normal z procedure
Large sample	n ≥ 30 (sampling distribution of x̄ is usually approx normal)	Use the normal z procedure
Small sample + not normal / unknown	n < 30 and distribution is clearly non-normal or unknown	Consider a nonparametric approach (outside this calculator)

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Hypothesis Tests About μ (σ Known) — p-value approach

This page explains the core ideas used by the calculator: hypotheses, tails, the z test statistic, the p-value definition, and the reject / do-not-reject rule using a chosen significance level α.

Two hypotheses

A statistical hypothesis test compares a claim about a population parameter to evidence from a sample. We set up two competing statements:

Null hypothesis H₀: a claim assumed true unless the data provide strong evidence against it.
Alternative hypothesis H₁: the claim supported when H₀ is rejected.

In hypotheses we use the population parameter (here, μ), not the sample statistic (x̄).

Rejection and nonrejection

A test has two possible decisions: reject H₀ (evidence supports H₁) or do not reject H₀ (not enough evidence to support H₁).

“Do not reject” does not mean H₀ is proven true—it means the sample did not provide sufficiently strong evidence against H₀ at the chosen α.

Two types of errors

Because decisions are made from samples, errors are possible:

Type I error: reject a true H₀. Its probability is α.
Type II error: do not reject a false H₀. Its probability is β.

The power of a test is 1 − β (probability of correctly rejecting a false H₀). For a fixed sample size, decreasing α generally increases β (and vice versa).

Tails of a test

The sign in the alternative hypothesis determines where “extreme” sample results live:

Test type	Alternative hypothesis	Where the rejection region is
Two-tailed	H₁: μ ≠ μ₀	Both tails
Left-tailed	H₁: μ < μ₀	Left tail
Right-tailed	H₁: μ > μ₀	Right tail

H₀ always includes equality ( = ), while H₁ uses ( ≠ ), ( < ), or ( > ).

When σ is known: the z test statistic

When the population standard deviation σ is treated as known, a common test for μ uses the standard normal model. The standard error of the sample mean is:

\[ \sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} \]

The observed z value (test statistic) is:

\[ z = \frac{\bar{x} - \mu_0}{\sigma/\sqrt{n}} \]

The calculator uses these formulas and then computes the tail probability under the standard normal curve.

p-value definition and decision rule

Assuming H₀ is true, the p-value is the probability of getting a sample result at least as extreme as the observed one in the direction of H₁.

Right-tailed: p-value is the area to the right of the observed z.
Left-tailed: p-value is the area to the left of the observed z.
Two-tailed: p-value is twice the tail area beyond |z|.

Decision rule (p-value approach):
Reject H₀ if p-value ≤ α; otherwise, do not reject H₀.

Steps used by the calculator

State H₀ and H₁ (choose the correct tail).
Confirm the normal z procedure is reasonable (often n ≥ 30, or population approximately normal when n < 30).
Compute z using \(\, z = \dfrac{\bar{x} - \mu_0}{\sigma/\sqrt{n}} \,\).
Compute the p-value from the standard normal curve (tail area based on H₁).
Compare p-value to α and report the decision.

Frequently Asked Questions

What does the p-value mean in a hypothesis test for a mean?

Assuming H0 is true, the p-value is the probability of getting a sample result at least as extreme as the observed one in the direction of H1. Smaller p-values indicate stronger evidence against H0.

How is the z test statistic computed when sigma is known?

The calculator uses z = (x-bar - mu0) / (sigma / sqrt(n)). This standardizes the difference between the sample mean and the null mean using the known standard error.

How is the two-tailed p-value calculated?

For a two-tailed test, the p-value is twice the tail area beyond the magnitude of the observed statistic, so it is 2 x P(Z >= |z|) under the standard normal model.

When is it reasonable to use the z test for a mean with sigma known?

It is commonly used when sigma is treated as known and either the sample size is large (often n >= 30) or the population distribution is approximately normal for smaller samples. If the population is clearly non-normal and n is small, the normal z procedure may be unreliable.

What decision rule does the calculator use with alpha?

Using the p-value approach, reject H0 if p-value <= alpha; otherwise, do not reject H0. “Do not reject” means there is not enough evidence at the chosen alpha, not that H0 is proven true.