Question

Among U.S. cities with a population of more than 250,000, the mean one-way commute time to work is 24.3 minutes. The longest one-way travel time is New York City, where the mean time 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.

1. What percent of the New York City commutes are for less than 26 minutes?

2. What percent are between 26 and 32 minutes?

3. What percent are between 26 and 40 minutes?

Answer 1

Solution :

Given that ,

mean = = 37.5

standard deviation = = 6.5

a) P(x < 26) = P[(x - ) / < (26 - 37.5) / 6.5]

= P(z < -1.77)

Using z table,

= 0.0384

The percentage is = 3.84%

b) P(26 < x < 32) = P[(26 - 37.5)/6.5 ) < (x - ) /  < (32 - 37.5) / 6.5) ]

= P(-1.77 < z < -0.85)

= P(z < -0.85) - P(z < -1.77)

Using z table,

= 0.1977 - 0.0384

= 0.1593

The percentage is = 15.93%

c) P(26 < x < 40) = P[(26 - 37.5)/6.5 ) < (x - ) /  < (40 - 37.5) / 6.5) ]

= P(-1.77 < z < 0.38)

= P(z < 0.38) - P(z < -1.77)

Using z table,

= 0.648 - 0.0384

= 0.6096

The percentage is = 60.96%