2010 PJC P2 Q12

The weight, \(Y\), of a Yummy cereal bar is normally distributed with mean \(\left( 120-k \right)\)g and standard deviation 10g. The weight, F, of a Fullness cereal bar is normally distributed with mean 180 g and standard deviation 20 g.

1. Given that \(\mathrm{P}\left( Y<80 \right)=\mathrm{P}\left( Y>150 \right)\), show that the value of \(k\) is 5.[1]
2. 5 Yummy cereal bars are randomly chosen. Find the probability that exactly one bar weighs lesser than the lower quartile weight and exactly one bar weighs more than the median weight.[2]
3. Find the probability that the weight of 2 randomly chosen Yummy cereal bars differs from one and a half times the weight of a randomly chosen Fullness cereal bar by at most 5g. State the assumption needed for your working.[4]

The Fullness cereal bars are sold at $2 per 100g.

4. Find the probability that a randomly chosen Fullness cereal bar costs more than $3.50.[2]
5. A random sample of 100 Fullness cereal bars is taken. Find the probability that there are at least 55 bars but at most 65 bars with each costing more than $3.50.[3]