Question

Ex 7. Michael and Greg share an apartment 10 miles from campus. Michael thinks that the fastest way to get to campus is to drive the shortest route, which involves taking several side streets. Greg thinks the fastest way is to take the route with the highest speed limits, which involves taking the highway most of the way but is two miles longer than Michael’s route. You recruit 50 college friends who are willing to take either route and time themselves. After compiling all the results, you found that the travel time for Michael’s route follows a Normal distribution with a mean equal to 30 minutes and a standard deviation equal to 5 minutes. Greg’s route follows a Normal distribution with a mean equal to 26 minutes and a standard deviation of 9.5 minutes. 1)Which route is faster and why? 2)Which route is more reliable and why? 3) Suppose that you leaving home headed for a University exam. Obviously, you don’t want to be late. You are leaving home at 5:15 and the exam is at 6:00 PM. Which route would you take to avoid being late and why? Show your calculations.

Answer 1

Ans-1) Greg's route is faster because the mean travel time is equal to 26 minutes which is less than the mean travel time of 30 minutes on Michael's route.

Ans-2) Michael's route is more reliable than Greg's route. It is because standard deviation of 5 minutes for travel time at Michael's route is less than the standard deviation of 9.5 minutes for travel time at Greg's route. A lower standard deviation means confidence in the data, meaning the values are less spread out, whereas a higher value of standard deviation means the values are more spread out, meaning that the outliers play a role in calculating the mean value of the data set.

Ans-3) Distance to be travelled, d = 10 miles

Time available, t = 45 minutes

Mean time for Greg's route, g = 26 minutes

Mean time for Michael's route, m = 30 minutes

For an approximately normal data set, the values within one standard deviation of the mean account for about 68% of the set; while within two standard deviations account for about 95%; and within three standard deviations account for about 99.7%.

Thus, we will consider third standard deviation to calculate the maximum time that can be taken on both the routes.

Maximum time on Greg's route, G = 26 minutes + 3 * 9.5 minutes = 26 + 28.5 = 54.5 minutes

Maximum time on Michael's route, M = 30 minutes + 3 * 5 minutes = 45 minutes

Since, M < G and M = t, also, G > t,

we conclude that, Michael's route is to followed to avoid being late for the University Exam.