2017 Specimen Paper P2 Q9

A type of metal bolt is manufactured with a nominal radius of \(0.8\)cm. In fact, the radii of the bolts, measured in cm, have the distribution \(\mathrm{N}\left( 0.8,{{0.01}^{2}} \right)\).

1. Find the percentage of bolts that have a radius between \(0.79\) cm and \(0.82\)cm.[1]

Metal washers are manufactured to fit on the bolts. The inside radii of the washers, measured in cm, have the distribution \(\mathrm{N}\left( 0.81,{{0.012}^{2}} \right)\).

2. Write down the distribution of the inside circumference of the washers, in cm, and find the circumference that is exceeded by \(5\%\) of the washers.[4]

A bolt and a washer are a 'good fit' if

• The inside radius of the washer is greater than the radius of the bolt and

• The inside radius of the washer is not more than \(0.04\) cm greater than the radius of the bolt.

3. A washer is chosen at random, and a bolt is chosen at random. Find the probability that the washer and bolt are a good fit.[3]

The outside radii of the washers, measured in cm, have the distribution \(\mathrm{N}\left( \mu ,{{\sigma }^{2}} \right)\). It is known that \(15\%\) of the washers have an outside radius greater than \(1.25\) cm and \(25\%\) have an outside radius of less than \(1.15\) cm.

4. Find the values of \(\mu \) and \(\sigma \).[4]