The mean of the commute time to work for a resident of a certain city is 28.3 minutes. Assume that the standard deviation of the commute time is 8.1 minutes

Question

What minimum percentage of commuters in the city has a commute within 2 standard deviations of the mean?

Answer 1

Problems like these are usually addressed using Chebyshev’s inequality.

The inequality states that no more than 1/k² of the distribution's values can be more than k standard deviations away from the mean. In this problem, k = 2. Therefore, we now that no more than ¼ of the distribution values can be more than 2 standard deviations away from the mean.

Put another way, we know that at least ¾ of the commuters must have a commute time within 2 standard deviations than the mean. Therefore, the minimum percentage of commuters in the city with a commute within 2 standard deviations of the mean is 75 %.