Contents · 0% complete

Learning objectives
Core idea
Key formulae
Worked examples
Explore the graph
Common mistakes
Exam tip
Guided practice
Practice questions
Challenge
Summary
Quick quiz
Same topic, different board
Past paper questions

Learn7367Paper 3 · Statistics optionInferencet-tests and confidence intervals

Inference · Subtopic 3 of 3

Inference with the t-Distribution

For a small sample from a normal population with unknown variance, $\frac{\bar X-\mu}{S/\sqrt n}$ follows a $t$ -distribution with $n-1$ degrees of freedom. It has heavier tails than $N(0,1)$ , so intervals are wider: $\bar x\pm t_{n-1}\frac s{\sqrt n}$ .

Cambridge 9231 Paper 4Edexcel 9FM0 OptionAQA 7367 Paper 3≈ 45 minRevision 3 · updated 2026-11-04

Practise this topicTimed test Ask Salam TutorSwitch board

Section 1 of 14

Learning objectives

Know when to use $t$ rather than $z$ .
Find confidence intervals using $t_{n-1}$ .
Carry out one-sample and paired $t$ -tests.

Pause and think

With only 5 measurements, why is the sample standard deviation an unreliable stand-in for $\sigma$ ?

Section 2 of 14

Core idea

Start hereBeer and small samples

At the Guinness brewery in 1908, William Gosset had only a handful of barley samples. Using the normal distribution with his estimated standard deviation gave intervals that were too narrow. His fix, the t-distribution, is now used everywhere small samples are.

The big idea

Replacing $\sigma$ by the estimate $s$ adds uncertainty. For a normal population, $\frac{\bar X-\mu}{S/\sqrt n}\sim t_{n-1}$ , which has heavier tails than $N(0,1)$ , giving wider intervals and stricter tests.

1. Confidence intervals

Building the interval

$P\left(-t<\frac{\bar X-\mu}{S/\sqrt n}<t\right)=0.95$
$t$ is the 97.5th percentile of $t_{n-1}$ .
$P\left(\bar X-t\tfrac S{\sqrt n}<\mu<\bar X+t\tfrac S{\sqrt n}\right)=0.95$
Rearrange for $\mu$ .

Interval:

\bar x\pm t_{n-1}\frac{s}{\sqrt n}

Worked example95% interval

$n=10$ , $\bar x=50$ , $s=4$ . Find a 95% CI for $\mu$ .

Goal: Interval.
Strategy: $t_9=2.262$ .

$\frac{s}{\sqrt n}=\frac4{\sqrt{10}}=1.265$
Standard error.
$2.262\times1.265=2.861$
Half-width.
$50\pm2.861$
Centre on $\bar x$ .

Answer

(47.1,\ 52.9)

, assuming a normal population.

✓ CheckWith

z=1.96

it would be narrower,

(47.5,52.5)

✓.

2. t-tests

Worked exampleOne-sample test

Same data. Test $H_0:\mu=48$ against $H_1:\mu>48$ at 5%.

Goal: Decision.
Strategy: Compute $t$ , compare with $t_9(95\%)$ .

$t=\frac{50-48}{1.265}=1.581$
Test statistic.
$t_9(5\%\text{ one-tail})=1.833$
Critical value.
$1.581<1.833$
Not in the critical region.

AnswerInsufficient evidence that the mean exceeds 48.

✓ Check

48

lies inside the 95% interval, consistent with not rejecting ✓.

Choosing the distribution

Drawing diagram…

z or t? Depends on what you know.

Key takeawayPaired samples Same subjects measured twice: work with the differences

d

, and test

\mu_d=0

with

t=\frac{\bar d}{s_d/\sqrt n}

Common misconceptionUnbiased s

s^2

divides by

n-1

. Calculators show both

\sigma_n

and

s_{n-1}

; choose the latter.

Pause and think

Why does the $t$ critical value shrink as $n$ grows?

Summary

Unknown $\sigma$ , small $n$ , normal population: $t_{n-1}$ .
CI $\bar x\pm t\frac s{\sqrt n}$ .
Paired: test differences.

t-distribution: Symmetric distribution with heavier tails than normal.
Standard error: $\frac s{\sqrt n}$ : spread of the sample mean.

Also on your syllabus

Two independent small samples: if both populations are normal with a common variance, use the pooled estimate $s_p^2=\dfrac{(n_1-1)s_1^2+(n_2-1)s_2^2}{n_1+n_2-2}$ .
Testing a difference of means: $t=\dfrac{\bar x_1-\bar x_2}{s_p\sqrt{\frac1{n_1}+\frac1{n_2}}}$ with $n_1+n_2-2$ degrees of freedom. For large samples use $z$ with $\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}$ instead; for paired data, test the mean of the differences.
Confidence interval for a difference of means: $(\bar x_1-\bar x_2)\pm t\,s_p\sqrt{\frac1{n_1}+\frac1{n_2}}$ .

Variance of a Normal population: $\dfrac{(n-1)S^2}{\sigma^2}\sim\chi^2_{n-1}$ . To test $H_0:\sigma^2=\sigma_0^2$ compare $\frac{(n-1)s^2}{\sigma_0^2}$ with $\chi^2_{n-1}$ critical values. A confidence interval for $\sigma^2$ is $\left(\frac{(n-1)s^2}{b},\frac{(n-1)s^2}{a}\right)$ where $a,b$ cut off the lower and upper tails of $\chi^2_{n-1}$ .
F-test for equal variances: for independent samples from two Normal populations, $F=\dfrac{s_1^2}{s_2^2}$ (put the larger variance on top) follows the F-distribution $F_{n_1-1,\,n_2-1}$ under $H_0:\sigma_1^2=\sigma_2^2$ . Compare with the upper critical value, halving the significance level for a two-tailed test. Equal variances justify pooling before a two-sample $t$ -test.

History"Student". William Gosset developed the

t

-distribution in 1908 while testing barley for Guinness, publishing as "Student" because the brewery banned staff from publishing.

Fun factHeavier tails. With 1 degree of freedom the

t

-distribution is the Cauchy distribution, whose mean doesn't even exist.

Section 3 of 14

Key formulae

Statistic

T=\frac{\bar X-\mu}{S/\sqrt n}\sim t_{n-1}

Unbiased variance

s^2=\frac{1}{n-1}\left(\sum x^2-\frac{(\sum x)^2}n\right)

Interval

\bar x\pm t_{n-1}\frac{s}{\sqrt n}

TrickPaired data. For before-and-after data, take differences first and do a one-sample

t

-test on them with

\mu_d=0

Section 4 of 14

Worked examples

Example 1 · Interval0 / 3 steps

n=10

\bar x=50

s=4

. Find a 95% CI for

\mu

Example 2 · Test0 / 2 steps

Test

H_0:\mu=48

H_1:\mu>48

at 5% using the same data.

Section 5 of 14

Explore the graph

t against normal

Loading the interactive graph…

Try this

The flatter curve is $t_3$ ; the other is $N(0,1)$ .
$t_3$ has a lower peak and heavier tails; with more degrees of freedom it merges with the normal.

$t_\nu\to N(0,1)$ as $\nu\to\infty$ .

In the real worldDrug trials. Early clinical trials have few patients, so effects on blood pressure are analysed with paired

t

-tests.

Section 6 of 14

Common mistakes

IncorrectUsing

z

with small

n

, unknown

\sigma

CorrectUse

t_{n-1}

Extra uncertainty in

s

IncorrectDegrees of freedom

n

Correct

n-1

One lost estimating the mean.

IncorrectDividing by

n

for

s^2

CorrectDivide by

n-1

Unbiased estimate.

Sample of 6, σ unknown

Student's t: brewed for exactly this.

Section 7 of 14

Exam tip

Exam tipState the assumption. Always say "assuming the population is normally distributed": the

t

method relies on it.

Section 8 of 14

Guided practice

SA-ST-TD-G011 markL1 FoundationNon-calculatorIntervals

How many degrees of freedom are used for a

t

-interval from a sample of 12?

Salam Academy Original Practice · Proposed mark scheme

Section 9 of 14

Practice questions

SA-ST-TD-0013 marksL2 StandardCalculatorIntervals

n=10

\bar x=50

s=4

, and

t_9(97.5\%)=2.262

. Find the upper limit of the 95% confidence interval (3 s.f.).

Salam Academy Original Practice · Proposed mark scheme

SA-ST-TD-0022 marksL2 StandardCalculatort-tests

With the same data, find the test statistic for

H_0:\mu=48

(3 s.f.).

Salam Academy Original Practice · Proposed mark scheme

SA-ST-TD-0032 marksL2 StandardCalculatorIntervals

A sample of 5 has

\sum x=40

and

\sum x^2=330

. Find the unbiased estimate of the population variance.

Salam Academy Original Practice · Proposed mark scheme

More questions on every skill in this subtopic are in the question bank under Practise.

Section 10 of 14

Challenge

1 more challenge question with hints, worked solutions and instant marking
Included in Student Premium.

See Premium

Section 11 of 14

Summary

$t_{n-1}$ for unknown variance.
CI: $\bar x\pm t\frac s{\sqrt n}$ .
Paired: test the differences.
Assume normality.

Section 12 of 14

Quick quiz

1 more quiz question with hints, worked solutions and instant marking
Included in Student Premium.

See Premium

Section 13 of 14

Same topic, different board

Specification: 9231 AS & A Level Further Mathematics
Where: Paper 4 · Inference using normal and t-distributions
Expected: One-sample t-tests and confidence intervals for a normal mean with unknown variance; paired-sample t-tests; two-sample tests with pooled variance; large-sample normal tests.
Calculator: Permitted on all papers; exact answers are expected where the question asks.
Typical style: Typical: from summary statistics, find an unbiased variance estimate, then a 95% confidence interval for the mean using t.
Open in: 9231 · Paper 4

Section 14 of 14

Past paper questions

No past paper questions are tagged to this subtopic yet. We add papers every week.

Browse past papers

Salam Academy original content · Content status: reviewed and published. Syllabus mapping: AQA 7367, First teaching 2017 (official specification).