Edexcel A-Level Mathematics A2 S3.3.2 Distribution of the Sample Mean Questions

Practise modelling the sample mean with a Normal distribution and using its reduced variance to find probabilities.

Exam points

write the distribution of x̄ as N(µ, σ²/n) before standardising a mean probability

[Maximum number: 4]

A garden centre sells bags of stones and large bags of gravel.
The weight, X kilograms, of stones in a bag can be modelled by a normal distribution with unknown mean $\mu$ and known standard deviation 0.4

The stones in each of a random sample of 36 bags from a large batch is weighed. The total weight of stones in these 36 bags is found to be 806.4 kg

(a)

Explain why the use of the Central Limit theorem is not required to answer part (a)

The manufacturer of these bags of stones claims that bags in this batch have a mean weight of 22.5 kg

[ 1 ]

(b)

Find the probability that the mean weight of gravel in the 10 large bags is less than 848 kg

[ 3 ]

$\bar{Y}\sim N\left(850,\left(\frac{5}{\sqrt{10}}\right)^2\right)$ $P(\bar{Y}<848)=P\left(Z<\frac{848-850}{\frac{5}{\sqrt{10}}}\right)=P(Z<-1.26)$ =0.1038
Calculator gives 0.10295...
ALT
N(8500,250) $P(\bar{Y}<848)=P\left(Z<\frac{8480-8500}{\sqrt{250}}\right)=P(Z<-1.26)=0.1038$

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Edexcel A-Level Mathematics A2 S3.3.2 Distribution of the Sample Mean Questions question 1

Question (a)

Question (b)