Hypothesis Tests About μ, σ Not Known, the Critical 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 a Mean: σ Not Known — Critical-Value Approach

Perform a one-sample t test when the population standard deviation σ is unknown. Choose the tail (two/left/right), set α, compute the t statistic, find critical t value(s), and decide: reject or do not reject H₀.

Test setup

Alternative direction (tail)

α (significance level)

How will you provide sample values?

Use a t distribution with df = n − 1. With small n, results are most reliable when the population is approximately normal.

Null value and sample information

Null mean μ₀

H₀ style (wording)

Optional note

Sample mean x̄

Sample standard deviation s

Sample size n

Quick reference: tails, df, and rejection rules

Tail	H₁ sign	df	Critical value(s)	Reject H₀ if…
Two-tailed	≠	n − 1	± t_{α/2, df}	\|t\| ≥ t_{α/2, df}
Left-tailed	<	n − 1	t_{α, df} (negative)	t ≤ t_{α, df}
Right-tailed	>	n − 1	t_{1−α, df} (positive)	t ≥ t_{1−α, df}

Decisions are stated as reject H₀ or do not reject H₀.

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One-Sample t Test for a Mean (σ Unknown): Critical-Value Approach

When the population standard deviation σ is unknown, we replace it with the sample standard deviation s. The standardized test statistic follows a t distribution with df = n − 1. The critical-value approach compares the observed test statistic to one or two critical t values determined by the significance level α and the tail direction of the alternative hypothesis.

When to use the t distribution

σ is not known and you have a random sample of size n.
For small samples (often n < 30), the method is most reliable when the population distribution is approximately normal (and there are no strong outliers).
For larger samples, the procedure is generally robust to mild non-normality.
If n is small and the population is strongly non-normal or unknown with outliers, consider a nonparametric method.

Hypotheses and tails

Let μ be the population mean and let μ₀ be the hypothesized value. The alternative hypothesis determines the tail(s):

Two-tailed: H₁: μ ≠ μ₀
Left-tailed: H₁: μ < μ₀
Right-tailed: H₁: μ > μ₀

Test statistic

With sample mean x̄, sample standard deviation s, and sample size n, define the standard error:

\[ \text{SE} = \frac{s}{\sqrt{n}} \qquad\text{and}\qquad t = \frac{\bar{x} - \mu_0}{s/\sqrt{n}} \]

Critical values and rejection regions

Choose a significance level α (Type I error rate). Let df = n − 1. Then:

Two-tailed: critical values are ±t_{α/2, df}; reject if |t| ≥ t_{α/2, df}.
Left-tailed: critical value is t_{α, df} (negative); reject if t ≤ t_{α, df}.
Right-tailed: critical value is t_{1−α, df} (positive); reject if t ≥ t_{1−α, df}.

Decision statement

The critical-value approach concludes either:

Reject H₀ (the sample result is in the rejection region), or
Do not reject H₀ (the sample result is in the nonrejection region).

Practical note: The t distribution has heavier tails than the normal distribution for small df. As df increases, the t distribution becomes closer to the normal distribution.

Frequently Asked Questions

How do you run a hypothesis test for a mean when sigma is not known using the critical value approach?

Compute t = (xbar - mu0) / (s / sqrt(n)) and df = n - 1, then find the critical t cutoff(s) from alpha and the tail type. Reject H0 if the observed t is in the rejection region defined by the critical value(s).

What is the difference between the t test and the z test for a mean?

A z test assumes the population standard deviation sigma is known, while a t test uses the sample standard deviation s when sigma is unknown. The t distribution has heavier tails, especially for small df, reflecting additional uncertainty.

How do critical t values work for two-tailed versus one-tailed tests?

For a two-tailed test, the rejection region is split into alpha/2 in each tail, producing ±t(1 - alpha/2, df). For a one-tailed test, all alpha is placed in one tail, producing a single cutoff t(1 - alpha, df) or t(alpha, df) depending on direction.

What assumptions are needed for a one-sample t test?

The test assumes the sample is random and independent and that the population distribution is approximately normal for small samples. For larger n, the sampling distribution of xbar becomes more stable due to the central limit theorem.

Why is degrees of freedom equal to n minus 1 in this calculator?

In a one-sample t procedure, df = n - 1 because one parameter (the sample mean) is estimated from the data when computing the sample standard deviation. This affects the shape of the t distribution used for critical values.

Hypothesis Tests About a Mean: σ Not Known — Critical-Value Approach

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