Further Mathematics · Pearson Edexcel 9FM0 · Option FS1 · Topic 5

The central limit theorem

The Central Limit Theorem: averages of large samples are approximately normally distributed, whatever the original distribution.

1 sub-topic
Taught in Year 13
18 practice questions
1 animated lesson

Practice questions (18) 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.

5.1The central limit theoremY13

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Easy and medium questions from this topic, each with a worked solutions and the mark for every step. All 18 questions in this topic, free, no account needed.

The central limit theorem

Not needed for Year 12Easy

The central limit theorem · Year 13 · 1 mark

What does the Central Limit Theorem say?
(1)

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$\bar{X}$ ${}\approx N\!\left(\mu, \dfrac{\sigma^2}{n}\right)$ for large $n$ , whatever the shape of the population. $B$ : For a large sample from any population, the sample mean is approximately Normally distributed

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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The central limit theorem

Not needed for Year 12Medium

The central limit theorem · Year 13 · 5 marks

A random sample of 80 observations is taken from a continuous uniform distribution on [2, 8].

(a)
Use the Central Limit Theorem to find the approximate distribution of the sample mean.
(3)
(b)
Hence find the probability that the sample mean is greater than 5.3.
(2)

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Part (a)3 marks

$\mu = 5, \ \sigma^2 = 3$
For large n, X̄ ~ N(μ, σ²/n) approximately. $\bar{X}$ ${}\approx \mathrm{N}\left(5, \frac{3}{80}\right)$

Part (b)2 marks

Standardise. $P(\bar{X} > 5.3)$
$= 0.0607$

Total5

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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