Question

Among the thirty largest U.S. cities, the mean one way commute time to work is 25.8 minutes. The longest one way travel time in New York City where the meantime is 37.5 minutes. Assume the distribution of travel times in New York City follows the normal probability distribution and the standard deviation is 6.5 minutes. What percent of the New York City commutes are for less than 26 minutes? What percent are between 26 and 32 minutes? What percent are between 26 and 40 minutes?

Answer 1

μ = 37.5, σ = 6.5; z = (x - μ)/σ

(a) x = 26

z = (26 - 37.5)/6.5 = -1.7692

P(x < 26) = P(z < -1.7692) = 0.0384 (3.84%)

(b) z1 = -1.7692, z2 = (32 - 37.5)/6.5 = -0.8462

P(26 < x < 32) = P(-1.7692 < z < -0.8462) = 0.1603 (16.03%)

(c) z1 = -1.7692, z2 = (40 - 37.5)/6.5 = 0.3846

P(26 < x < 40) = P(-1.7692 < z < 0.3846) = 0.6113 (61.13%)

Answer 2

To solve this problem, we need to standardize the values using the z-score formula and then use a standard normal distribution table or a calculator to find the probabilities.

a) To find the probability that the commute time is less than 26 minutes:

z = (26 - 37.5) / 6.5 = -1.77

Using a standard normal distribution table or calculator, we find that the probability of a z-score being less than -1.77 is approximately 0.0384.

So, the percent of New York City commutes that are less than 26 minutes is approximately 3.84%.

b) To find the probability that the commute time is between 26 and 32 minutes:

z1 = (26 - 37.5) / 6.5 = -1.77 z2 = (32 - 37.5) / 6.5 = -0.85

Using a standard normal distribution table or calculator, we find that the probability of a z-score being between -1.77 and -0.85 is approximately 0.2289.

So, the percent of New York City commutes that are between 26 and 32 minutes is approximately 22.89%.

c) To find the probability that the commute time is between 26 and 40 minutes:

z1 = (26 - 37.5) / 6.5 = -1.77 z2 = (40 - 37.5) / 6.5 = 0.38

Using a standard normal distribution table or calculator, we find that the probability of a z-score being between -1.77 and 0.38 is approximately 0.6465.

So, the percent of New York City commutes that are between 26 and 40 minutes is approximately 64.65%.