The mean of the commute time to work for a resident of a certain city is 27.8 mi

Question

The mean of the commute time to work for a resident of a certain city is 27.8 minutes. Assume that the standard deviation of the commute time is 6.6 minutes to complete parts (a) through (c). What are the commute times within 2.5 standard deviations of the mean? (a) What minimum percentage of commuters in the city has a commute time within 2 standard deviations of the mean? (b) What minimum percentage of commuters in the city has a commute time within 1.5 standard deviations of the mean? What are the commute times within 1.5 standard deviations of the mean? What is the minimum percentage of commuters who have commute times between 8 minutes and 47.6 minutes?

Explanation / Answer

a)
Assume normal distribution.
Two standard deviations of the mean = 27.8-2(6.6) and 27.8+2(6.6) = (14.6, 41)
P( 14.6 < x < 41)
= 27.8
= 6.6
standardize x to z = (x - ) /
P( 14.6 < x < 41) = P[( 14.6 - 27.8) /6.6 < Z < ( 41 - 27.8) / 6.6]
P( -2 < Z < 2) = .9544
(From Normal probability table)
95.44 %
-------------------------------------
b)
Commute times : 27.8-1.5(6.6) and 27.8+1.5(6.6) = (17.9, 37.7)

= 27.8
= 6.6
standardize x to z = (x - ) /
P( 17.9 < x < 37.7) = P[( 17.9 - 27.8) / 6.6 < Z < ( 37.7 - 27.8) / 6.6]
P ( 17.9<X<37.7 )=P ( 1.5<Z<1.5 ) = 0.8664
(From Normal probability table)

86.64%
--------------------
c)
= 27.8
= 6.6
standardize x to z = (x - ) /
P( 8< x < 47.6) = P[( 8 - 27.8) / 6.6 < Z < ( 47.6 - 27.8) / 6.6]
P( -3 < Z < 3) P ( 3<Z<3 )=0.9974
(From Normal probability table)
99.74%

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