CAIE A-Level Mathematics 6.4.4 Distribution of the Sample Mean

CAIE A-Level Mathematics 6.4.4 Distribution of the Sample Mean
Cambridge International AS & A Level Mathematics 9709 syllabus for exams in 2028, 2029 and 20302028–2030

Practise modelling a sample mean from a normal population with unchanged mean and variance σ²/n, then standardising it to calculate probabilities for sample averages.

How this is tested

  • state X̄ as normal with mean μ and variance σ²/n for a random sample of size n
  • use standard error σ/√n rather than the population deviation when standardising X̄
  • translate a condition on a sample total into its equivalent mean, or model the total directly

Question 4

[Maximum number: 6]

A population is normally distributed with mean 35 and standard deviation 8.1. A random sample of size 140 is chosen from this population and the sample mean is denoted by Xˉ\bar{X}.

Question 4(a)

(a)

Find P(Xˉ>36)\mathrm{P}(\bar{X}>36).

[ 3 ]

Question 4(b)

(b)

It is given that P(Xˉ<a)=0.986\mathrm{P}(\bar{X}<a)=0.986. Find the value of a.

[ 3 ]