The mean travel time to work is 25.2 minutes (U.S.Census 2010).Further,suppose that commute time follows a log-normal distribution with a standard deviation of 10 minutes. What is the probability a randomly selected U.S.worker has a commute time of less than 20 minutes?
A) 30.15%
B) 34.09%
C) 65.91%
D) 69.85%

Question

Quizplus · Accepted Answer

In order to solve this problem, we need to first standardize the distribution using the formula: Z = (ln(x) - ln(mu)) / sigma, where x is the commute time we are interested in (in this case, 20 minutes), mu is the mean (25.2 minutes), and sigma is the standard deviation (10 minutes).

Plugging in the values, we get:

Z = (ln(20) - ln(25.2)) / 10 = -0.354

Next, we need to find the area to the left of this standardized value on the standard normal distribution. Using a calculator or a Z-table, we find this area to be 0.3632.

Finally, we convert back to the original log-normal distribution using the formula:

P(X < x) = P(ln(X) < ln(x)) = P(Z < (ln(x) - ln(mu)) / sigma) = 0.3632

Solving for x, we get:

ln(x) = -0.354 * 10 + ln(25.2) = 3.006

x = e^3.006 = 20.25

Therefore, the probability that a randomly selected U.S. worker has a commute time of less than 20 minutes is approximately 0.3409, or 34.09%. The best choice is B.

The Mean Travel Time to Work Is 25