Question

The commute time to work in the U.S. has a bell shaped distribution with a population mean of 24.4 minutes and a population standard deviation of 6.5 minutes. What percentage of the population has a commute time (do not round answers): between 11.4 minutes and 37.4 minutes, inclusive? 95 less or equal to 11.4 minutes? 3 greater than or equal to 37.4 minutes? 3 greater than or equal to 24.4 minutes? between 11.4 minutes and 30.9 minutes, inclusive? between 17.9 minutes and 37.4 minutes, inclusive? between 37.4 minutes and 43.9 minutes, inclusive?

Answer 1

µ = 24.4, σ = 6.5

a) P(11.4 <= X <= 37.4) =
= P( (11.4-24.4)/6.5 <= (X-µ)/σ <= (37.4-24.4)/6.5 )
= P(-2 < z < 2)
= P(z < 2) - P(z < -2)
Using excel function:
= NORM.S.DIST(2, 1) - NORM.S.DIST(-2, 1)
= 0.9545 = 95.45%

b) P(X <= 11.4) =
= P( (X-µ)/σ <= (11.4-24.4)/6.5 )
= P(z <= -2)
Using excel function:
= NORM.S.DIST(-2, 1)
= 0.0228 = 2.28%

c) P(X >= 37.4) =
= P( (X-µ)/σ > (37.4-24.4)/6.5)
= P(z > 2)
= 1 - P(z < 2)
Using excel function:
= 1 - NORM.S.DIST(2, 1)
= 0.0228 = 2.28%

d) P(X > 24.4) =
= P( (X-µ)/σ > (24.4-24.4)/6.5)
= P(z > 0)
= 1 - P(z < 0)
Using excel function:
= 1 - NORM.S.DIST(0, 1)
= 0.5000 = 50%

e) P(11.4 < X < 30.9) =
= P( (11.4-24.4)/6.5 < (X-µ)/σ < (30.9-24.4)/6.5 )
= P(-2 < z < 1)
= P(z < 1) - P(z < -2)
Using excel function:
= NORM.S.DIST(1, 1) - NORM.S.DIST(-2, 1)
= 0.8186 = 81.86%

f) P(17.9 < X < 37.4) =
= P( (17.9-24.4)/6.5 < (X-µ)/σ < (37.4-24.4)/6.5 )
= P(-1 < z < 2)
= P(z < 2) - P(z < -1)
Using excel function:
= NORM.S.DIST(2, 1) - NORM.S.DIST(-1, 1)
= 0.8186 = 81.86%

g) P(37.4 < X < 43.9) =
= P( (37.4-24.4)/6.5 < (X-µ)/σ < (43.9-24.4)/6.5 )
= P(2 < z < 3)
= P(z < 3) - P(z < 2)
Using excel function:
= NORM.S.DIST(3, 1) - NORM.S.DIST(2, 1)
= 0.0214 = 2.14%