Pearson Edexcel IAL Mathematics S3.3.6 Central Limit theorem for sample means

Practise using the CLT to model sample means, form approximate Normal distributions, calculate probabilities and justify large-sample assumptions.

Exam points

state the approximate Normal distribution for a sample mean using µ and σ²/n
calculate probabilities for sample means by standardising to z under the CLT
use a CLT-based critical region to decide whether a process needs action

[Maximum number: 6]

Navtej travels to work by train. A train leaves the station every 7 minutes and Navtej's arrival at the station is independent of when the train is due to leave.

(a)

Estimate the probability that the mean length of time that he has to wait for a train to leave is between 3.4 and 3.6 minutes.

[ 5 ]

By the CLT,
$\bar{T}\sim N\left(3.5,\frac{49}{552}\right)$ $P(3.4<\bar{T}<3.6)=P\left(\frac{3.4-3.5}{\sqrt{49/552}}<Z<\frac{3.6-3.5}{\sqrt{49/552}}\right)$ =P(-0.34<Z<0.34)=0.6331-(1-0.6331)=0.2662
Calculator gives $0.2628\ldots$ ; accepted range awrt 0.263 to 0.266.
M1 for using N(3.5,49/552); M1 for standardising; final answer awrt 0.263 to 0.266.

(b)

State a necessary assumption for the calculation in part (c).

[ 1 ]

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S3.3.6 - Central Limit theorem for sample means question 1

Question (a)

Question (b)