2021 NJC P2 Q9 [Modified]

A shop sells two models of ovens produced by Factory \(A\) and Factory \(B\). The lifespans of ovens produced by Factory \(A\) have the normal distribution with mean \(13\) years and standard deviation \(6\) months, while the lifespans of ovens produced by Factory \(B\) have the normal distribution with mean \(15\) years and standard deviation \(k\) months. The lifespan of any oven is independent of one another.

1. Given that \(90\%\) of the ovens produced by Factory \(B\) exceeds a lifespan of \(14\) years. Show that \(k=9.3636\), correct to \(5\) significant figures.[3]
2. Find the probability that the lifespan of a randomly chosen oven produced by Factory \(B\) exceeds the lifespan of a randomly chosen oven produced by Factory \(A\) by less than \(3\) years[2]
3. There is a probability of at least \(0.4\) that the lifespan of a randomly chosen oven produced by Factory \(A\) is within \(n\) years of the mean lifespan. Find the least value of \(n\), correct to \(3\) decimal places.[3]
4. Every oven produced by Factory \(A\) is wrapped in a box. A carton contains \(20\) of such boxes. If there are at least \(3\) ovens in a carton with lifespans of less than \(12\) years, the carton will be rejected. Find the probability that a carton is rejected.[3]