Further Mathematics · Pearson Edexcel 9FM0 · Option FS1 · Topic 6

Chi-squared tests

χ² (chi-squared) tests: checking whether data fits a model (goodness of fit) and whether two factors are linked (contingency tables).

2 sub-topics
1 in Year 12, 1 in Year 13
21 practice questions
2 animated lessons

Practice questions (21) Lessons Worksheets All Edexcel Further Mathematics topics

What the specification says

Every sub-topic Pearson Edexcel can examine in this topic, with its reference number from the official specification. The Y12 and Y13 tags show when each sub-topic is usually taught.

6.1Chi-squared tests (discrete uniform, binomial, Poisson)Y12
6.1Chi-squared tests for geometric distributionsY13

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Easy, medium and hard questions from this topic, each with a worked solutions and the mark for every step. All 21 questions in this topic, free, no account needed.

Chi-squared tests

Not needed for Year 12Easy

Degrees of freedom for a geometric model · Year 13 · 1 mark

A geometric distribution is fitted to data, with $p$ estimated from the data. After combining there are $n$ classes. How many degrees of freedom does the chi-squared test have?
(1)

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One restriction for the total and one for the estimated parameter. $B$ : $n - 2$

Total1

MMethod mark: for a correct method, even with a slip.
AAccuracy mark: for a correct answer from a correct method.
BIndependent mark: for a correct fact or result on its own, with no method needed.

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Chi-squared tests

Medium

Degrees of freedom in a goodness of fit test · Year 12 · 1 mark

Data are tested against a discrete uniform distribution. After any classes have been combined there are 6 classes. How many degrees of freedom are there?
(1)

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Degrees of freedom = number of classes − number of restrictions: one for the total only. $C$ : 5

Total1

MMethod mark: for a correct method, even with a slip.
AAccuracy mark: for a correct answer from a correct method.
BIndependent mark: for a correct fact or result on its own, with no method needed.

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Chi-squared tests

Hard

Chi-squared goodness of fit: Poisson · Year 12 · 8 marks

The numbers of emails received in 100 randomly chosen five-minute intervals are: 0 emails: 46; 1: 34; 2: 15; 3: 4; 4 or more: 1. Treat '4 or more' as 4.

(a)
Show that the mean number of emails per interval is $0.8$ to 2 decimal places.
(1)
(b)
Test, at the 5% level, whether a Poisson distribution is a suitable model.
(7)

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Part (a)1 mark

$\frac{\sum xf}{100} = 0.800$

Part (b)7 marks

H₁: it is not. $H_0: \text{a Poisson distribution is a suitable model}$
$N \times \mathrm{P}(X = x)$ with $\lambda = 0.8$ ; the last class is $\mathrm{P}(X \ge 4)$ . $E$ ${}= 44.93, 35.95, 14.38, 3.83, 0.91$
Classes with $E < 5$ are combined: 0, 1, 2+.
$X^2$ ${}= \sum\frac{(O - E)^2}{E}$ ${}= 0.171$
Subtract 1 for the total and 1 for estimating $\lambda$ . $\nu$ ${}= 3 - 2$ ${}= 1, \ \text{cv}$ ${}= 3.841$
Do not reject H₀: a Poisson distribution is a suitable model.

Total8

MMethod mark: for a correct method, even with a slip.
AAccuracy mark: for a correct answer from a correct method.
BIndependent mark: for a correct fact or result on its own, with no method needed.

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