---
title: "Hypothesis Tests for a Mean with the t-Distribution"
description: "Test a claim about a mean when the population spread is unknown: a t statistic, degrees of freedom, and a p-value read as the probability of data at least this extreme if the claim held."
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source: https://lightmysky.com/learn/mathematics/hypothesis-tests-for-a-mean-with-the-t-distribution-mt_ig2o39hAwh.md
retrieved: 2026-09-12
---

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# Hypothesis Tests for a Mean with the t-Distribution

Test a claim about a mean when the population spread is unknown: a t statistic, degrees of freedom, and a p-value read as the probability of data at least this extreme if the claim held.

Subject: Mathematics · Area: Data & Statistics · Ages 20 to 21
Page: https://lightmysky.com/learn/mathematics/hypothesis-tests-for-a-mean-with-the-t-distribution-mt_ig2o39hAwh

## Ready when they can

- Carry out a one-sample t-test and state the conclusion in context
- Choose between one-tailed and two-tailed alternatives before seeing the data
- State what a p-value is and what it is not

## Lesson: Testing a claim about a mean

To test a claim about a mean with sigma unknown, scale the gap between the sample mean and the null value by the estimated standard error. The t statistic is the sample mean minus the null value, over s divided by the square root of n. It runs on the t distribution with n minus 1 degrees of freedom. With 4 observations, a mean of 12, spread 4, and null 10, the statistic is 1.

**Example.** Fix the alternative before examining the data. A predicted direction, like a new drug lowering blood pressure, calls for a one-tailed test. A plain difference with no predicted direction calls for two tails. Choosing the direction after peeking rigs the test, so lock it in early.

The p-value is the chance of data this extreme if the null were true. Values below the significance level, often 0.05, reject the null; larger ones simply fail to reject it. With t equal to 3.1 and p equal to 0.004 at the 5 percent level, rejecting is the right call. State the conclusion in context, naming the variable and direction.

**Tip.** The p-value is not the chance that the null is true, and failing to reject never proves the null. It only says the data do not contradict it. Keep that modesty in every write-up and never overclaim.

**Recap.** Scale the gap into a t statistic, fix tails early, read the p-value against the level, and phrase conclusions with restraint.

## Practice

14 questions on this page, each with its working shown.

## Needs first

- [Setting Up a Hypothesis Test](https://lightmysky.com/learn/mathematics/setting-up-a-hypothesis-test-mt_JmpLNHK0Nf)
- [Confidence Intervals for Means and Proportions](https://lightmysky.com/learn/mathematics/confidence-intervals-for-means-and-proportions-mt_Pg2fPDW1sn)

## Opens up

- [Errors, Power and the Design of a Test](https://lightmysky.com/learn/mathematics/errors-power-and-the-design-of-a-test-mt_D7ZXbvSD0l)