Question

Among the thirty largest U.S. cities, the mean one-way commute time to work is 25.8 minutes. https://deepblue.lib.umich.edu/bitstream/handle/2027.42/112057/103196.pdf?sequence=1&isAllowed=y. The longest one-way travel time is in New York City, where the mean time is 38.2 minutes. Assume the distribution of travel times in New York City follows the normal probability distribution and the standard deviation is 7.0 minutes.

1. What percent of the New York City commutes are for less than 28 minutes? (Round your intermediate calculations and final answer to 2 decimal places.)

2. What percent are between 28 and 33 minutes? (Round your intermediate calculations and final answer to 2 decimal places.)

3. What percent are between 28 and 46 minutes? (Round your intermediate calculations and final answer to 2 decimal places.)

Answer 1

Here' the answer to the question. Please let me know in case you've doubts.

We have been given parameters of normal distribution :

Mean = 25.8 min

Stdev = 7 min

We will normalize the distribution using the formula: Z = (X-Mean)/Stdev

To convert the Z value into cumulative probability use the Excel formula = NORMSDIST(Z), where Z = Z-score.

a. P(X<28) = P(Z< (28-25.8)/7 ) = P(Z< .31) = 0.6217

= 0.62 ( 2 digit round off)

b. P(28<X<33) = P(Z< ( 33-25.8)/7) - P(Z< (28-25.8)/7) = P(Z<1.03) - P(Z<.31) = .85 - 0.62 = .23

= 0.23 ( 2 digit roundoff)

c. P(28<X<46) = P(Z<(46-25.8)/7) - P(Z< (28-25.8)/7)) = P(Z<2.89) - P(Z<.31) = .9981- .6233 = .3747

= 0.37 ( 2 digit roundoff)