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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

Learn9709Paper 6 · Probability & Statistics 2Hypothesis testsBinomial and Poisson tests

Hypothesis tests · Subtopic 1 of 3

Binomial and Poisson Hypothesis Tests

Assume nothing has changed ( $H_0$ ), then ask how surprising your observation would be. If the probability of a result at least that extreme is below the significance level, the evidence says something has changed.

Cambridge 9709 Paper 6Edexcel YMA01 WST02Edexcel 9MA0 Paper 3Edexcel 9FM0 OptionAQA 7357 Paper 3≈ 40 minRevision 5 · updated 2026-10-27

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Section 1 of 14

Learning objectives

Use the language of testing: $H_0$ , $H_1$ , significance level, one- and two-tailed, critical region.
Carry out a binomial test by finding a tail probability.
Carry out a Poisson test for a change in rate.
Find a critical region and the actual significance level, and conclude in context.

Pause and think

A friend's coin lands heads $9$ times in $10$ throws. Is the coin biased? How unlikely would $9$ or more heads be with a fair coin?

Section 2 of 14

Core idea

Start hereIs the coin fair?

Toss a coin 20 times: 15 heads. Bad luck, or a biased coin? A hypothesis test asks: if the coin were fair, how surprising would 15 or more heads be? If it would be very surprising (say under 5%), we doubt fairness.

The big idea

State $H_0$ (the default) and $H_1$ (what you suspect). Assume $H_0$ ; find the probability of the observed result or more extreme. If it's below the significance level, reject $H_0$ . Conclude in context, never with certainty.

1. The procedure

Hypothesis test

Drawing diagram…

"Not rejecting" is not "proving" $H_0$ .

B(20, 0.3): the critical region for H₁: p > 0.3 at 5%

P(X\ge10)=0.048<5\%

but

P(X\ge9)=0.113>5\%

. So 10 or more successes would be significant evidence that

p>0.3

Worked exampleBinomial test, $H_1:p>0.3$

Claim $p=0.3$ ; 20 trials give 9 successes. Test at 5%.

Goal: Decision.
Strategy: Tail probability.

$H_0:p=0.3,\ H_1:p>0.3;\ X\sim B(20,0.3)$
Under $H_0$ .
$P(X\ge9)=0.113$
Observed or more extreme.
$0.113>0.05$
Compare.

AnswerDo not reject

H_0

: insufficient evidence that

p>0.3

✓ CheckCritical region

X\ge10

(

P=0.048

);

9

is outside ✓.

Worked examplePoisson test

A shop averages 4 complaints a week; this week 9. Evidence of an increase at 5%?

Goal: Decision.
Strategy: $X\sim\operatorname{Po}(4)$ .

$H_0:\lambda=4,\ H_1:\lambda>4$
One-tailed.
$P(X\ge9)=0.0214<0.05$
Tail.

AnswerReject

H_0

: evidence complaints have increased.

✓ Check9 is more than twice the mean: surprising ✓.

Common misconceptionUsing P(X = observed) Always use the tail: "this result or more extreme".

Pause and think

Two-tailed 10% test: compare each tail with?

Summary

$H_0$ , $H_1$ , level.
Tail probability vs level.
Conclude in context.

Null hypothesis: The default assumption.
Critical region: Values leading to rejecting $H_0$ .

HistoryThe lady tasting tea. In the 1920s, a colleague told the statistician Ronald Fisher that she could tell whether milk had been poured into a cup before or after the tea. Fisher designed an experiment with

8

cups and worked out how unlikely her success would be by guessing. The story, told in his 1935 book The Design of Experiments, launched modern significance testing. She reportedly got them all right.

Fun factWhy 5%? The

5\%

level is a convention, not a law of nature. Fisher suggested it in 1925 as a convenient "one in twenty" line. Particle physicists, who test millions of possibilities, demand far more: the discovery of the Higgs boson in 2012 required a "five sigma" result, about

1

3.5

million.

Section 3 of 14

Key formulae

One-tailed test (H₁: p < p₀)

\text{reject }H_0\text{ if }P(X\le x_{\text{obs}})<\alpha

One-tailed test (H₁: p > p₀)

\text{reject }H_0\text{ if }P(X\ge x_{\text{obs}})<\alpha

Two-tailed test

\text{compare the tail probability with }\tfrac{\alpha}{2}

TrickPoint H₁ at the evidence. The tail you calculate always points the way

H_1

points.

H_1:p<0.3

means

P(X\le x)

;

H_1:\lambda>2.5

means

P(X\ge x)

. Calculating

P(X=x)

alone, or the wrong tail, is the most common way to lose these marks.

Section 4 of 14

Worked examples

Example 1 · A biased coin?0 / 3 steps

A coin shows

9

heads in

10

throws. Test at the

5\%

level whether it is biased towards heads.

Example 2 · A Poisson rate0 / 3 steps

A shop averaged

6

complaints a week. After new training, a randomly chosen week has

2

complaints. Test at the

5\%

level whether the rate has fallen.

Example 3 · Critical region0 / 2 steps

X\sim B(25,p)

. Find the critical region for a test of

H_0:p=0.2

against

H_1:p<0.2

at the

5\%

level.

Section 5 of 14

Explore the graph

The tail probability

Loading the interactive graph…

Try this

The curve passes through $P(X=r)$ for $X\sim B(k,p)$ under $H_0$ . The shaded region (from $c-\tfrac12$ ) suggests the upper tail $P(X\ge c)$ .
With $k=10$ , $p=0.5$ , $c=9$ , the tail is small: about $1\%$ .
Lower $c$ until the tail is clearly more than $5\%$ . That is where $H_0$ would no longer be rejected.

A test asks: is the observed value out in the tail, where results are rare if $H_0$ is true?

In the real worldSpotting a cancer cluster. If a type of cancer occurs at

2

per year in a town and

8

cases appear one year, epidemiologists ask: under

\text{Po}(2)

, how likely are

8

or more?

P(X\ge8)\approx0.001

. Such a small probability triggers an investigation, though with many towns checked, some clusters will occur by chance.

Section 6 of 14

Common mistakes

IncorrectComparing

P(X=x)

with

5\%

CorrectUse the tail

P(X\le x)

P(X\ge x)

Any single value can be unlikely.

IncorrectHypotheses about

X

\hat p

CorrectAbout the parameter:

p

\lambda

Tests are about the population.

Incorrect"

H_0

is accepted / proved"

Correct"Insufficient evidence to reject

H_0

Not rejecting is not proving.

IncorrectTwo-tailed test compared with

5\%

CorrectCompare each tail with

2.5\%

The level is split between the tails.

Comparing $P(X=3)$ with 5%

Comparing $P(X\le3)$ with 5%: tail probabilities only

Section 7 of 14

Exam tip

Exam tipConclude in context, without certainty. Finish with a sentence that uses the context and is not too definite: "There is evidence at the

5\%

level that the success rate has decreased." Avoid "proves" or "the rate has definitely changed", which lose the final mark.

Section 8 of 14

Guided practice

SA-9709-P6-HT-G014 marksL2 StandardCalculatorBinomial tests

A treatment succeeds for

30\%

of patients. In a trial of a new method on

20

patients, only

2

succeed. Test at the

5\%

significance level whether the success rate is lower with the new method. Give the probability you compare (3 s.f.) and your conclusion.

Salam Academy Original Practice · Proposed mark scheme

Section 9 of 14

Practice questions

SA-9709-P6-HT1-0014 marksL1 FoundationCalculatorBinomial tests

A coin shows

9

heads in

10

throws. Test at the

5\%

significance level whether the coin is biased towards heads. Give

P(X\ge9)

under

H_0

(3 s.f.) and your conclusion.

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P6-HT-0014 marksL2 StandardCalculatorBinomial tests

A coin is suspected of being biased towards heads. In

12

throws it shows

10

heads. Test at the

5\%

level. Give

P(X\ge10)

under

H_0

(3 s.f.) and your conclusion.

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P6-HT1-0034 marksL2 StandardCalculatorPoisson tests

A shop received an average of

6

complaints a week. After new staff training, a randomly chosen week has

2

complaints. Test at the

5\%

level whether the rate of complaints has fallen. Give

P(X\le2)

(3 s.f.) and your conclusion.

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P6-HT-0024 marksL2 StandardCalculatorPoisson tests

Accidents at a junction occur at an average of

4

per day. After changes, there is

1

accident on a randomly chosen day. Test at the

5\%

level whether the rate has fallen. Give

P(X\le1)

(3 s.f.) and your conclusion.

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P6-HT1-0023 marksL2 StandardCalculatorBinomial tests

X\sim B(25,p)

. A test of

H_0:p=0.2

against

H_1:p<0.2

is carried out at the

5\%

level. The critical region is

X\le c

. Find

c

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P6-HT-0084 marksL4 ChallengingCalculatorBinomial tests

X\sim B(20,p)

. A test of

H_0:p=0.4

against

H_1:p>0.4

is carried out at the

5\%

level. Find the critical region

X\ge c

, giving the value of

c

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P6-HT1-0052 marksL2 StandardCalculatorPoisson tests

A test of

H_0:\lambda=5

against

H_1:\lambda>5

rejects

H_0

X\ge10

, where

X\sim\text{Po}(\lambda)

. Find the actual significance level of the test, correct to 3 significant figures.

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

SA-9709-P6-HT1-0044 marksL3 ExamCalculatorBinomial tests

It is claimed that a coin is fair. In

20

throws it shows

15

heads. Test at the

5\%

level whether the coin is biased (in either direction). Give

P(X\ge15)

(3 s.f.) and your conclusion.

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P6-HT-0094 marksL3 ExamCalculatorPoisson tests

Emails arrive at

2.5

per hour on average. On a day after a website change,

17

arrive in

4

hours. Test at the

5\%

level whether the rate has increased. Give

P(X\ge17)

(3 s.f.) and your conclusion.

Salam Academy Original Practice · Proposed mark scheme

Section 11 of 14

Summary

$H_0$ : the usual parameter value; $H_1$ : what you suspect (one- or two-tailed).
Find the tail probability in the direction of $H_1$ , assuming $H_0$ .
Below the significance level: reject $H_0$ . Two-tailed: compare with half of it.
Critical region: the values that lead to rejection; conclude in context.

Section 12 of 14

Quick quiz

Q1 · SA-9709-P6-HT-Q021 markL1 FoundationCalculatorBinomial tests

The null hypothesis

H_0

usually states that

Salam Academy Original Practice · Proposed mark scheme

Q2 · SA-9709-P6-HT1-0061 markL1 FoundationCalculatorBinomial tests

A manufacturer claims

10\%

of its products are faulty. A consumer group believes the proportion has changed. The alternative hypothesis is

Salam Academy Original Practice · Proposed mark scheme

Section 13 of 14

Same topic, different board

Specification: 9709 AS & A Level Mathematics
Where: Paper 6, Probability & Statistics 2 · 6.5 Hypothesis tests
Expected: Understand the nature of a hypothesis test, the difference between one-tailed and two-tailed tests, and the terms null hypothesis, alternative hypothesis, significance level, rejection region (or critical region), acceptance region and test statistic; formulate hypotheses and carry out a hypothesis test in the context of a single observation from a population that has a binomial or Poisson distribution, using direct evaluation of probabilities or a normal approximation.
Calculator: Permitted. Give non-exact answers to 3 significant figures; probabilities may be fractions or decimals.
Typical style: A claim about a proportion or a rate: state hypotheses, find the tail probability, compare with the significance level and conclude in context.
Open in: 9709 · Paper 6

Section 14 of 14

Past paper questions

4 questions (28 marks) from real exams test this subtopic. Try each one under timed conditions, then mark it with the official mark scheme.

9709/62 · Oct/Nov 2025Question 47 marks
Binomial test for a proportion, and why a Poisson approximation does not suit.
Question paper, page 5 Mark scheme
9709/62 · May/June 2025Question 810 marks
Binomial test on the chance of rolling a six, then Type I and Type II error probabilities.
Question paper, page 12 Mark scheme
9709/62 · Oct/Nov 2024Question 36 marks
Binomial hypothesis test at 10%, and whether the model suits the context.
Question paper, page 4 Mark scheme
9709/62 · May/June 2023Question 65 marks
Type I error from a binomial test and a Type II error.
Question paper, page 11 Mark scheme

All 9709 past paper questions by topic

Salam Academy original content · Content status: reviewed and published. Syllabus mapping: Cambridge International 9709, 2026–2027 (official specification).