CAIE S2 2005 November — Question 7

Exam BoardCAIE
ModuleS2 (Statistics 2)
Year2005
SessionNovember
TopicLinear combinations of normal random variables
TypeMultiple stage process probability

7 A journey in a certain car consists of two stages with a stop for filling up with fuel after the first stage. The length of time, \(T\) minutes, taken for each stage has a normal distribution with mean 74 and standard deviation 7.3. The length of time, \(F\) minutes, it takes to fill up with fuel has a normal distribution with mean 5 and standard deviation 1.7. The length of time it takes to pay for the fuel is exactly 4 minutes. The variables \(T\) and \(F\) are independent and the times for the two stages are independent of each other.
  1. Find the probability that the total time for the journey is less than 154 minutes.
  2. A second car has a fuel tank with exactly twice the capacity of the first car. Find the mean and variance of this car's fuel fill-up time.
  3. This second car's time for each stage of the journey follows a normal distribution with mean 69 minutes and standard deviation 5.2 minutes. The length of time it takes to pay for the fuel for this car is also exactly 4 minutes. Find the probability that the total time for the journey taken by the first car is more than the total time taken by the second car.