Police measure the speed of cars passing a particular point on a motorway. The random variable \(X\) is the speed of a car.
\(X\) is modelled by a normal distribution with mean 55 mph (miles per hour).
Draw a sketch to illustrate the distribution of \(X\). Label the mean on your sketch.
The speed limit on the motorway is 70 mph . Car drivers can choose to travel faster than the speed limit but risk being caught by the police.
The distribution of \(X\) has a standard deviation of 20 mph .
Find the percentage of cars that are travelling faster than the speed limit.
The fastest \(1 \%\) of car drivers will be banned from driving.
Show that the lowest speed, correct to 3 significant figures, for a car driver to be banned is 102 mph . Show your working clearly.
Car drivers will just be given a caution if they are travelling at a speed \(m\) such that
$$\mathrm { P } ( 70 < X < m ) = 0.1315$$
Find the value of \(m\). Show your working clearly.