--- title: "Hypothesis Tests for a Mean with the t-Distribution" description: "Test a population mean using a one-sample t statistic when the population standard deviation is unknown. Check the sampling model, use the planned tails and degrees of freedom, and interpret the p-val" canonical: https://lightmysky.com/learn/mathematics/hypothesis-tests-for-a-mean-with-the-t-distribution-mt_ig2o39hAwh source: https://lightmysky.com/learn/mathematics/hypothesis-tests-for-a-mean-with-the-t-distribution-mt_ig2o39hAwh.md retrieved: 2026-11-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Hypothesis Tests for a Mean with the t-Distribution Test a population mean using a one-sample t statistic when the population standard deviation is unknown. Check the sampling model, use the planned tails and degrees of freedom, and interpret the p-value assuming the null hypothesis. 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 Calculate t as the sample mean minus null mean μ0, divided by SE = s / sqrt(n). Here s is the sample standard deviation, n the sample size and sqrt(n) its square root. Use df = n - 1, so n = df + 1. Under H0, the t reference is exact for independent observations from one normal population with positive variance. Large independent samples can allow a non-normal approximation; skewness and outliers matter. Calculating t alone does not validate this reference. *(drawing: Calculate the estimated standard error before dividing the gap from the null mean.)* State the population mean, null value, alternative and level α before examining results. An alternative μ < μ0 uses the lower tail; μ > μ0 uses the upper tail; μ ≠ μ0 uses both tails. Choose this from the planned question, not the observed sign. A prediction alone does not require a one-tailed test if the planned question concerns departures in either direction. Under H0 and the assumed model, the p-value is the probability of a statistic at least as extreme as observed. The planned alternative defines extremeness. Let T have the null t distribution with n - 1 degrees of freedom. Use its lower tail for μ < μ0 and upper tail for μ > μ0. For μ ≠ μ0, use both tails: by symmetry, double the upper-tail probability beyond |t|, the size of t without its sign. This is not the probability that H0 is true. *(drawing: Software gives the rounded upper-tail p-value. At planned α = 0.05, reject H0. This supports a mean-fill difference, not a cause.)* Calculate SE = s / sqrt(n), then t = (sample mean - μ0) / SE. Keep the gap's sign. For n = 36, sample mean 18, s = 12 and μ0 = 24, sqrt(36) = 6, SE = 12 / 6 = 2 and t = (18 - 24) / 2 = -3. Under a symmetric null t model, negative t has upper-tail p > 0.5. Recover the sample mean as μ0 + t × SE. Use supplied square-root approximations and round only the final result. At the level α chosen in advance, reject when p < α; otherwise fail to reject. Failure to reject means the evidence did not cross that threshold, not that H0 is true or that no evidence opposes it. Report the population, variable and planned direction. For a matching two-sided t-test at α = 0.05, null values within the closed 95% t interval, including its endpoints, are not rejected; values outside are rejected. **Recap.** Check the model and plan the alternative and level. Calculate t, df and the matching tail probability. State the population conclusion; failing to reject does not prove H0. *Original lesson, questions and flashcards about one-sample t-tests, authored for this learning dataset.* ## 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)