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
:)
