2014 MJC P2 Q10

The masses, in kilograms, for a randomly chosen pumpkin and a randomly chosen watermelon, are normally distributed with means \(4.2\) and \(8.5\), and standard deviations \(1.6\) and \(2.2\) respectively.

1. Find the range of values of \(k\) such that the average mass of five randomly chosen watermelons exceeds \(k\) kilograms occurs at most \(10\%\) of the time. [3]
2. Find the probability that the total mass of five randomly chosen pumpkins differs from three times the average mass of five randomly chosen watermelons by more than \(5\) kg. [4]
3. A fruit store has \(70\) batches, each consisting of \(n\) pumpkins. It is known that, on average, \(3\) in \(20\) pumpkins are rotten. Given that in the \(70\) batches, the probability of having an average of at least \(4\) rotten pumpkins per batch is more than \(0.7\). Determine the minimum value of \(n\) . [4]