CAIE A-Level Mathematics 6.4.4 Distribution of the Sample Mean

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.

Exam points

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

[Maximum number: 3]

Wendy's journey to work consists of three parts: walking to the train station, riding on the train and then walking to the office. The times, in minutes, for the three parts of her journey are independent and have the distributions $\mathrm{N}\left(15.0,1.1^{2}\right), \mathrm{N}\left(32.0,3.5^{2}\right)$ and $\mathrm{N}\left(8.6,1.2^{2}\right)$ respectively.

Find the probability that the mean of Wendy's journey times over 15 randomly chosen days will be less than 54.5 minutes.

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6.4.4—Normal distribution question 1