2025 RVHS P2 Q12

A factory produces metal discs and metal bars for constructing machinery parts. The mass of a randomly chosen metal disc follows a normal distribution with mean \(18\) grams and standard deviation \(0.6\) grams. The mass of a randomly chosen metal bar has an independent normal distribution with mean \(20\) grams and standard deviation \(\sigma\) grams.

1. Sketch the distribution for the masses of the metal disc between \(14\) grams and \(22\) grams.[1]
2. \(2\) metal discs are randomly chosen. Find the probability that the mass of one of the discs is less than \(19\) grams and the other metal disc has mass more than \(19\) grams.[2]
3. Find the range of values of \(k\) such that less than \(90\%\) of the metal discs have masses within \(k\) grams of the mean mass of the metal discs.[2]
4. It is known that \(7.65\%\) of the metal bars have masses more than \(21\) grams. Find the value of \(\sigma\).[2]

It is given that \(\sigma=0.5\) for the rest of the question.

5. A client places an order for a new kind of metal strips. The metal strips are modelled as similar solids to the metal bars with a \(20\%\) reduction in mass. A production worker selects \(8\) of the metal strips for inspection. Find the probability that at least \(5\) of the metal strips have masses between \(15.8\) grams and \(17.2\) grams.[3]
6. A protective case is designed for the safe transportation of the metal bar. Each metal bar is placed inside a protective case during transportation. The masses of the protective cases are modelled as \(30\%\) of the masses of the metal bars. Let \(T\) denote the total mass of \(4\) of the metal bars and their protective cases. Find the variance of \(T\).[2]