In this question you should state clearly all the distributions that you use, together with the valu

Question

In this question you should state clearly all the distributions that you use, together with the values of the appropriate parameters. A beauty company sells their best-selling perfume in two sizes. The volume of perfume, in millilitres, in a small bottle and large bottle is denoted by $S$ and $L$ respectively. $S$ and $L$ can be modelled using normal distributions with the means and standard deviations as follows. SizeMean (millilitres)Standard deviation (millilitres) Small$\mu$$\sigma$ Large$75$$3.5$ Deduce an approximate value for $\mathrm{P}\left( \mu \le S\le \mu +\sigma \right)$.[1] $10\%$ of the small bottles contain less than $~56$ millilitres of perfume and $30\%$ of the small bottles contain more than $62$ millilitres of perfume. Find the values of $\mu $ and $\sigma $.[3] Use $\mu =50$ and $\sigma =2$ for the rest of the question. Find the probability that the total volume of perfume in $2$ randomly chosen small bottles and $3$ randomly chosen large bottles is at least $330$ millilitres.[3] Find the probability that the total volume of perfume of $2$ randomly chosen large bottles is within $\pm 1$ millilitres of three times the volume of perfume in a randomly chosen small bottle.[4] Find the probability that the average volume of perfume of $10$ randomly chosen large bottles is less than $74$ millilitres.[2]

Accepted Answer

(i) ${0.341}$ (ii) ${\sigma \approx 3.32}$ ($3$ s.f.) (iii) ${\mathrm{P}(T \geq 330) \approx 0.227}$ ($3$ s.f.) (iv) ${\mathrm{P}(-1 \leq X \leq 1) \approx 0.102}$ ($3$ s.f.) (v) ${\mathrm{P}(Y < 740) \approx 0.183}$ ($3$ s.f.)

2023 MI P2 Q12

2023 MI P2 Q12

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Solution:

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Size	Mean (millilitres)	Standard deviation (millilitres)
Small	\(\mu\)	\(\sigma\)
Large	\(75\)	\(3.5\)