The time taken to cycle into College on any day is an independent random variable, which is normally distributed with mean 20 minutes and standard deviation 4 minutes. The time taken to cycle home is an independent random variable, which is normally distributed with mean 15 minutes and standard deviation 3 minutes. The total time cycling is the sum of the time cycling in and cycling out of College.
(i) What is the probability that the total time taken cycling on Tuesday is less than three quarters of the time it took on Monday?
(ii) What is the probability that the time taken cycling on Wednesday is at least four minutes more than that on Monday?
(iii) Over a period of a week, what is the probability that the seven day average of cycling time is more than 40 minutes?
(iv) The student claims that the average cycling time each day is 30 minutes. How many days will it take before the student can be 95% confident that this claim is too low?
(v) Without doing the calculations, explain what would happen, in the context of your answer to (i) above, if the time cycling into College was not independent of the time cycling out of College.