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Confidence Intervals and Tests using the t-Distribution

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Confidence Intervals & Tests using the t-Distribution

Big idea: when you don't actually know how spread out a population is (its true variance), and you've only got a small sample to go on, you can't trust the normal distribution anymore — you need the t-distribution instead, because it's built to be a bit more "forgiving" about that uncertainty.

Why the t-distribution? Reading t-tables Confidence intervals Hypothesis test (one mean) Paired t-test Two means — CI Two means — test What to memorise Checklist Exam tips

Summary

The whole chapter in seven bite-sized ideas

Why not just use z? With a small sample and unknown variance, the sample variance isn't a reliable enough stand-in for the true variance — so we swap the normal distribution for the "wider" t-distribution.

Reading the tables t-tables are read using degrees of freedom (rows) and tail probability (columns). Always sketch the curve first.

CI for one mean x̄ ± t(n-1)(α/2) × s/√n gives a range of plausible values for the true population mean.

Hypothesis test for one mean Same 5-step process as z-tests, just swap in the t-statistic and t-critical-values.

Paired t-test For before/after data — test the mean of the differences, not the two samples separately.

Two independent means Pool the two sample variances into one shared estimate, then build a CI or run a test on the difference of the means.

1. Why do we need the t-distribution?

The problem it solves

Start with something you already know: if X is normally distributed with mean μ and (crucially) a known variance σ², then the sample mean X̄ is also normal: X̄ ~ N(μ, σ²/n). That lets you standardise using (X̄ − μ)/(σ/√n), which follows the standard normal N(0,1) — the familiar z-score.

But here's the catch: in real life, you almost never actually know σ². You only have your sample, from which you can calculate the sample variance, s². Two situations then arise:

Large n: s² is a good enough approximation for σ², so you can basically carry on as before and treat it like a z-problem.
Small n: s² is not reliably close to σ² — a small sample can easily give you a wonky estimate of spread just by bad luck. This extra uncertainty means (X̄ − μ)/(S/√n) is more spread out and has fatter tails than N(0,1). We give it a new name, t, and a new distribution to match: the t-distribution.

Core definition

If X₁, X₂, …, Xₙ is a random sample from N(μ, σ²) with unknown variance, then
t = (X̄ − μ) / (S/√n) follows a t-distribution with v = n − 1 degrees of freedom,
where S² = (1/n)(Σx² − nx̄²) is an unbiased estimator of σ².

In plain English: "how many standard errors away from μ is my sample mean" — but measured on a ruler that gets a bit more generous (wider) the smaller your sample is, to account for the fact that s is a shakier estimate of σ when n is small.

Think of it like this

What happens as n gets bigger?

only used for small samples

Quick Check

2. Reading t-distribution tables

The single most important practical skill in this chapter

t

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Also in the full note

3. Confidence interval for the mean (unknown variance)
4. Hypothesis test for the mean (unknown variance)
5. The paired t-test
6. Confidence interval for the difference between two means
7. Hypothesis test for the difference between two means
What to Memorise
Concepts Checklist
Exam Tips

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