Hypothesis Tests About μ, σ Not Known, the P Value Approach

Statistics • Hypothesis Tests About the Mean and Proportion

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

Hypothesis Tests About μ: σ Not Known (t Test — p-value approach)

Enter a hypothesized mean μ₀, choose the tail and α, and provide sample data (summary or raw/CSV). The calculator computes the t statistic, df, and the p-value, then gives the decision.

Test setup

Alternative direction (tail)

α (significance level)

Null (hypothesized) mean μ₀

Because σ is unknown, the test statistic uses s and follows a t distribution with df = n − 1 (under standard conditions).

Sample information

How will you provide data?

Visualization shading

Explore t (optional)

Sample mean x̄

Sample standard deviation s

Sample size n

Quick reference: signs and tails

Tail	H₀	H₁	p-value idea
Two-tailed	μ = μ₀	μ ≠ μ₀	2 · area beyond \|t\| in both tails
Left-tailed	μ = μ₀ (or μ ≥ μ₀)	μ < μ₀	area to the left of t
Right-tailed	μ = μ₀ (or μ ≤ μ₀)	μ > μ₀	area to the right of t

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

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Hypothesis Tests About a Mean (σ Unknown): the t Test (p-value approach)

When the population standard deviation σ is not known, we replace it with the sample standard deviation s. Under common conditions, the resulting test statistic follows a t distribution with df = n − 1.

When is the t procedure appropriate?

The t test is used for testing a population mean when σ is unknown. It works best when the sample comes from a population that is approximately normal, and it is typically robust for larger samples. For very small samples from clearly non-normal populations, a nonparametric alternative may be more appropriate.

Situation	Typical approach
σ unknown, n < 30, population approximately normal	Use the t distribution (one-sample t test)
σ unknown, n ≥ 30	Use the t distribution (often robust in practice)
σ unknown, n < 30, population clearly not normal / shape uncertain	Consider a nonparametric method (depends on context)

Test statistic (one-sample t)

Let x̄ be the sample mean, s be the sample standard deviation, and n be the sample size. The standard error is:

\[ s_{\bar{x}}=\frac{s}{\sqrt{n}} \]

The t test statistic is:

\[ t=\frac{\bar{x}-\mu_0}{s/\sqrt{n}} \quad\text{with}\quad df=n-1 \]

The p-value approach

The p-value measures how extreme the observed test statistic is if the null hypothesis were true. The tail type is determined by the alternative hypothesis:

State hypotheses. H₀ includes equality (μ = μ₀), and H₁ sets the direction (≠, <, or >).
Compute the test statistic. Find t using x̄, s, and n; set df = n − 1.
Find the p-value. Use the t distribution with df degrees of freedom to get the tail area(s) beyond the observed t.
Decide. If p-value ≤ α, reject H₀. If p-value > α, do not reject H₀.

How this calculator supports the method

Summary mode: enter n, x̄, and s.
Raw/CSV mode: paste values or upload a CSV; the tool computes n, x̄, and s automatically.
Outputs: t, df, p-value, and the decision using the p-value rule.
Visualization: a t-curve with optional shading for the p-value region and/or α rejection region, plus a slider and animation.

Mini example (concept)

Suppose you test whether a product’s mean lifetime is less than 65 months. You sample n values, compute x̄ and s, then use the left-tailed t test. If the p-value is below your chosen α, you reject H₀ and conclude the mean lifetime is lower than 65.

Frequently Asked Questions

What is a one-sample t test and when is it used?

A one-sample t test checks whether a population mean differs from a hypothesized value mu0 when the population standard deviation sigma is unknown. It uses the sample standard deviation s and a t distribution with df = n - 1 under standard conditions.

How is the t test statistic calculated when sigma is not known?

The calculator uses t = (xbar - mu0) / (s / sqrt(n)). The denominator s / sqrt(n) is the estimated standard error of the sample mean.

How do I choose between two-tailed, left-tailed, and right-tailed tests?

Use two-tailed when you are testing for a difference (mu != mu0), left-tailed when you suspect the mean is smaller (mu < mu0), and right-tailed when you suspect the mean is larger (mu > mu0). The tail choice determines which t tail area is used for the p-value.

What is the decision rule in the p-value approach?

Reject H0 if p-value <= alpha; otherwise, do not reject H0. A smaller alpha makes rejection harder because it requires stronger evidence (a smaller p-value).

How does entering raw sample data change the calculation?

When you paste or upload sample values, the calculator first computes n, xbar, and s from your data and then runs the t test using those computed statistics. This helps prevent manual summary mistakes when working from a dataset.

Hypothesis Tests About μ: σ Not Known (t Test — p-value approach)

Calculation steps (p-value approach)

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