The service manager of a garage wishes to determine the cost (labour plus parts) of performing a standard car service on a typical family car. The manager knows that the more mileage a car has done after its last service the greater the cost of a service will be. He assumes that the mileage that the typical family car does when it is serviced is normally distributed with a mean of 10 000 miles and a standard deviation of 2000 miles.

If the car when serviced has done:

? 6000 miles or less, the cost of the service to the garage will be £75.
? Between 6000 and 8000 miles, the cost of the service to the garage will be £125.
? Between 8000 and 11 000 miles, the cost of the service to the garage will be £190.
? Over 11 000 miles, the cost of the service to the garage will be £250.

a. Determine the following probabilities:

i) a car has done 6000 miles or less when serviced;
ii) a car has done between 6000 and 8000 miles when serviced;
iii) a car has done between 8000 and 11 000 miles when serviced;
iv) a car has done 11 000 or more miles when serviced.

b. Determine the expected cost of a service for a typical family car.

c. If the Service Manager wants to make a 25 per cent profit on servicing a typical family car, how much should he charge a customer for a standard service?