Among U.S. cities with a population of more than 250,000 the mean one-way commut

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 in New York City, where the mean time is 38.9 minutes. Assume the distribution of travel times in New York City follows the normal probability distribution and the standard deviation is 7.9 minutes. a. What percent of the New York City commutes are for less than 30 minutes? (Round the intermediate values to 2 decimal places. Round your answer to 2 decimal places.)   Percent % b. What percent are between 30 and 34 minutes? (Round the intermediate values to 2 decimal places. Round your answer to 2 decimal places.)   Percent % c. What percent are between 30 and 41 minutes? (Round the intermediate values to 2 decimal places. Round your answer to 2 decimal places.)   Percent %

Explanation / Answer

Mean time = 38.9 minutes

standard deiation = 7.9 minutes

(a) Percent of Newyork CIty Commutes less than 30 minutes

Pr (x < 30; 38.9; 7.9) = ?

Z - value = (30-38.9) /7.9 = -1.1266

so Pr (x < 30; 38.9; 7.9) = (1.1266)

where is the normal cumulative standard distribution

from Z - table

Pr (x < 30; 38.9; 7.9) = 0.1300

so there are 13% of the newyork city commutes are for less than 30 minutes.

(b) Pr( 30 < x < 34 ; 38.9 ;7.9) = Pr( x <34; 38.9; 7.9) - Pr( x< 30; 38.9; 7.9)

Z- value for x = 34 is Z = (34 - 38.9)/ 7.9 = -0.62

Z - value for x = 30 is Z = (30 - 38.9)/ 7.9 = -1.1266

Pr( 30 < x < 34 ; 38.9 ;7.9) = (-0.62) - (-1.1266)

where is the normal cumulative standard distribution.

From Z - table

Pr( 30 < x < 34 ; 38.9 ;7.9) = (-0.62) - (-1.1266) = 0.2676 - 0.1300 = 0.1376

so there are 13.76% are between 30 and 34 minutes.

(c) Pr( 30 < x < 41 ; 38.9 ;7.9) = Pr( x < 41; 38.9; 7.9) - Pr( x< 30; 38.9; 7.9)

Z- value for x = 41 is Z = (41 - 38.9)/ 7.9 = 0.266

Z - value for x = 30 is Z = (30 - 38.9)/ 7.9 = -1.1266

Pr( 30 < x < 41; 38.9 ;7.9) = (0.26) - (-1.1266)

where is the normal cumulative standard distribution.

From Z - table

Pr( 30 < x < 41 ; 38.9 ;7.9) = (0.26 - (-1.1266) = 0.6026 - 0.1300 = 0.4726

so there are 47.26% are between 30 and 41 minutes.

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