2025 NYJC P2 Q11

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

A machine grades apples according to their mass. Apples with mass exceeding \(120\) g are labelled as "large" and apples with mass less than \(80\) g are labelled as "small". A large batch of apples is graded and it is found that \(15\%\) are "large" and \(10\%\) are "small". It is known that the masses of the apples are normally distributed.

1. Find the mean mass of a randomly chosen apple from the batch and show that the standard deviation is \(17.3\) g, correct to \(3\) significance figures.[3]
2. Two apples are chosen in random. Find the probability that the difference in their masses is at least \(20\) g.[3]

The apples are labelled and packed into boxes. Each box contains \(20\) apples.

3. Apples with mass between \(110\) g and \(140\) g are labelled as "premium". Find the expected number of "premium" apples in a randomly chosen box.[2]
4. Find the probability that the mean mass of apples in a randomly chosen box is at most \(105\) g.[3]
5. The boxes of apples are sold by weight at a price of \(\$2.90\) per kilogram. Find the probability that the total price of \(3\) randomly chosen boxes of apples exceeds \(\$18\).[3]