A farm in the west of Singapore grows turnips for sale to the local market. $5$ turnips are random

Question

A farm in the west of Singapore grows turnips for sale to the local market. $5$ turnips are randomly chosen. Find the probability that exactly one turnip weighs less than the lower quartile weight and exactly two turnips weigh more than the median weight.[1] The mass of a randomly chosen turnip has mean $40$ g and standard deviation of $3$ g. If the probability that the mean mass of a large sample of $n$ turnips is greater than $39.6$ g exceeds $0.95$, find the least value of $n$. [3] A random variable $X$ has the distribution $X\sim N\left( 40,{{3}^{2}} \right)$. The random variable $Y$ is related to $X$ by the formula $Y=aX-\frac{1}{b}$, where $a$ and $b$ are constants and $a>0$. Given that $\mathrm{P}(Y<85)=\mathrm{P}(Y>155)=0.075$, find the values of $\mathrm{E}(Y)$ and $\mathrm{Var}(Y)$, and hence find the values of $a$ and $b$.[5]

Accepted Answer

(a)(i) $\frac{15}{128}$ (a)(ii) $n=153$ (b) $E(Y)=120,\ \operatorname{Var}(Y)=591\text{ (3 s.f.)},\ a=8.10,\ b=0.00490\text{ (3 s.f.)}$

2015 SRJC P2 Q11

2015 SRJC P2 Q11

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