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 is in New York City, where the mean time is 39.7 minutes. Assume the distribution of travel times in New York City follows the normal probability distribution and the standard deviation is 7.5 minutes.
a. What percent of the New York City commutes are for less than 30 minutes?
b. What percent are between 30 and 35 minutes?
c. What percent are between 30 and 50 minutes?

Answer 1

Answer:

A. 0.9015

B. 0.1658

C. 0.0132

Step-by-step explanation:

Given

Mean, μ of commuting time in New York is 39.7 minutes

Standard Deviation, σ = 7.5 minutes

Let x represent the commute time

For Normal Distribution, z = (x - μ) /σ

A. x is less than 30 minutes

P(x<30) = P(x - μ < 30 - μ)

P(x<30) = P((x - μ)/ σ < (30 - μ)/ σ)

P(x<30) = P(z < (30 - μ)/ σ)

P(x<30) = P(z < (30 - 39.7) / 7.5)

P(x<30) = P(z < (-9.7/7.5)

P(x<30) = P(z < -1.29)

P(x<30) = P(z > 1.29)

P(x<30) = 0.9015 -------- From z table

B. x is between 30 and 35

P(30>x<35) = P(30 -μ > x - μ < 35 - μ)

P(30>x<35) = P((30 -μ)/σ < (x - μ)/σ < (35 - μ) / σ)

P(30>x<35) = P((30 -μ)/σ < z < (35 - μ) / σ)

P(30>x<35) = P((30 -39.7)/7.5 < z < (35 - 39.7) / 7.5)

P(30>x<35) = P(-9.7/7.5 < z < -4.7/7.5)

P(30>x<35) = P(-1.29 < z < -0.63)

There are two points on the same side here; we calculate both probabilities and subtract to give

P(30>x<35) = P(-1.29 < z < 0) - P(0 < z < -0.63)

P(30>x<35) = P(0 < z < 1.29) - P(0 < z < 0.63)

P(30 > x < 35) = 0.9015 - 0.7357

P(30 > x < 35) = 0.1658

C. x is between 30 and 50

P(30>x<50) = P(30 -μ > x - μ < 50 - μ)

P(30>x<50) = P((30 -μ)/σ < (x - μ)/σ < (50 - μ) / σ)

P(30>x<50) = P((30 -μ)/σ < z < (50 - μ) / σ)

P(30>x<50) = P((30 -39.7)/7.5 < z < (50 - 39.7) / 7.5)

P(30>x<50) = P(-9.7/7.5 < z < 10.3/7.5)

P(30>x<50) = P(-1.29 < z < 1.37)

There are two points on different sides here; calculate both probabilities and add to give

P(30>x<50) = P(-1.29 < z < 0) + P(0 < z < 1.37)

P(30>x<50) = -P(0 < z < 1.29) + P(0 < z < 1.37)

P(30 > x < 50) = -0.9015 + 0.9147

P(30 > x < 50) = 0.0132