2025 RI P2 Q7

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

A snack company produces two types of potato chips, Rays and Luffles. The mass, in grams, of a regular packet of Rays follows the distribution \(\mathrm{N}(100, \sigma^2)\) and the mass, in grams, of a regular packet of Luffles follows the distribution \(\mathrm{N}(120, 16)\). The masses of all regular packets of potato chips are independent of one another.

1. If the probability of the mass of a randomly chosen packet of Rays not exceeding \((100-\sigma^2)\) grams is less than \(0.2\), find the possible range of values of \(\sigma\).[2]

It is given that \(\sigma=3\) for the rest of this question.

2. Find the probability that the total mass of \(2\) randomly chosen regular packets of Rays and \(3\) randomly chosen packets of Luffles is greater than \(0.55\) kg.[3]
3. The snack company decides to launch a new product called Mega Jumbo Pack, which consists of \((24-n)\) regular packets of Rays and \(n\) regular packets of Luffles. It is given that the probability of the mass of a Mega Jumbo Pack exceeding \(20\) times the mass of a regular packet of Luffles by more than \(500\) g is at least \(0.1\). Find the minimum value of \(n\).[4]