2022 DHS P2 Q10

In this question you should state the parameters of any distributions that you use.

A fruit stall vendor sells rock melons and watermelons. The masses (in grams) of a randomly chosen rock melon and watermelon, denoted by \(X\) and \(Y\) respectively, have independent normal distributions. The means and standard deviations of these distributions are shown in the following table.

1. Sketch the distribution of \(X\) for masses between \(558\) g to \(646\) g.[1]
2. Find the expected number of rock melons with mass more than \(600\) g from \(300\) randomly chosen rock melons.[2]
3. Comment whether the combined masses of the melons of both types is normally distributed.[1]

Rock melons and watermelons are priced at \(\$3\)/kg and \(\$2.80\)/kg respectively.

4. Find the probability that the mean selling price of \(4\) randomly chosen rock melons differs from the selling price of a randomly chosen watermelon by at most \(60\) cents.[4]
5. Ah Guan buys \(20\) melons, \(n\) of them are rock melons and the rest are watermelons. Find the greatest value of \(n\) such that the probability that the total cost of these \(20\) melons exceeding \(\$38\) is more than \(0.95\). You should show your working clearly.[4]