2 The fuel economy of a car varies from day to day according to weather and driving conditions. Fuel economy is measured in miles per gallon (mpg). The fuel economy of a particular petrol-fuelled type of car is known to be Normally distributed with mean 38.5 mpg and standard deviation 4.0 mpg .

1. Find the probability that on a randomly selected day the fuel economy of a car of this type will be above 45.0 mpg .
2. The manufacturer wishes to quote a fuel economy figure which will be exceeded on \(90 \%\) of days. What figure should be quoted? The daily fuel economy of a similar type of car which is diesel-fuelled is known to be Normally distributed with mean 51.2 mpg and unknown standard deviation \(\sigma \mathrm { mpg }\).
3. Given that on 75\% of days the fuel economy of this type of car is below 55.0 mpg , show that \(\sigma = 5.63\).
4. Draw a sketch to illustrate both distributions on a single diagram.
5. Find the probability that the fuel economy of either the petrol or the diesel model (or both) will be above 45.0 mpg on a randomly selected day. You may assume that the fuel economies of the two models are independent.