N2017 P2 Q10

A small component for a machine is made from two metal spheres joined by a short metal bar. The masses in grams of the spheres have the distribution \(\mathrm{N}\left( 20,{{0.5}^{2}} \right)\).

1. Find the probability that the mass of a randomly selected sphere is more than \(20.2\) grams.[1]

In order to protect them from rusting, the spheres are given a coating which increases the mass of each sphere by \(10%\).

2. Find the probability that the mass of a coated sphere is between \(21.5\) and \(22.45\) grams. State the distribution you use and its parameters.[3]
3. The masses of the metal bars are normally distributed such that \(60\%\) of them have a mass greater than \(12.2\) grams and \(25\%\) of them have a mass less than \(12\) grams. Find the mean and standard deviation of the masses of metal bars.[4]
4. The probability that the total mass of a component, consisting of two randomly chosen coated spheres and one randomly chosen bar, is more than \(k\) grams is \(0.75\). Find \(k\), stating the parameters of any distribution you use.[4]