2025 VJC P2 Q10

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

An orchard sells kiwis of two varieties, Type \(A\) and Type \(B\). The masses, in grams, of Type \(A\) and Type \(B\) kiwis follow the distributions \(N(110, 6^2)\) and \(N(85, 8^2)\) respectively. It is assumed that these two distributions are independent.

Kiwis are selected randomly and packed into bags. Type \(A\) kiwis are packed in bags of \(30\), while Type \(B\) kiwis are packed in bags of \(60\).

1. In a randomly chosen bag of Type \(A\) kiwis, find the probability that the mean mass of kiwis is between \(109\) grams and \(115\) grams.[2]

The selling prices of Type \(A\) and Type \(B\) kiwis are \(\$25\) per kilogram and \(\$15\) per kilogram respectively.

2. Find the probability that the average selling price of a kiwi from a bag of Type \(A\) kiwis is more than the average selling price of a kiwi from a bag of Type \(B\) kiwis by at most \(\$1.50\).[3]

The orchard also produces genetically modified Type \(G\) kiwis, which are packed in bags of \(50\). The masses of Type \(G\) kiwis are identically distributed with a mean of \(90\) grams and a standard deviation of \(5\) grams.

Each Assortment Box contains one bag of Type \(A\) kiwis and one bag of Type \(G\) kiwis. A box is fit for sale if the total mass of the kiwis in the box is at least \(7.85\) kilograms.

3. Ten Assortment Boxes are selected at random. Find the probability that none of the boxes are fit for sale.[3]
4. State, in context, the approximation(s) and assumption(s) needed for the distribution(s) used in part (c).[2]