Question

II. The ages of cars owned by all people living in a city have a bell-shaped distribution with a mean of 7.3 years and a standard deviation of 2.2 years.

1. Find the (approximate) percentage of cars in this city that are 7 to 13.9 years old

2. Find the interval that contains the ages of (approximate) 95% of the cars owned by all people in this city.

Answer 1

Part 1)

X ~ N ( µ = 7.3 , σ = 2.2 )
P ( 7 < X < 13.9 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 7 - 7.3 ) / 2.2
Z = -0.1364
Z = ( 13.9 - 7.3 ) / 2.2
Z = 3
P ( -0.14 < Z < 3 )
P ( 7 < X < 13.9 ) = P ( Z < 3 ) - P ( Z < -0.14 )
P ( 7 < X < 13.9 ) = 0.9987 - 0.4458
P ( 7 < X < 13.9 ) = 0.5529

Part 2)

X ~ N ( µ = 7.3 , σ = 2.2 )
P ( a < X < b ) = 0.95
Dividing the area 0.95 in two parts we get 0.95/2 = 0.475
since 0.5 area in normal curve is above and below the mean
Area below the mean is a = 0.5 - 0.475
Area above the mean is b = 0.5 + 0.475
Looking for the probability 0.025 in standard normal table to calculate Z score = -1.96
Looking for the probability 0.975 in standard normal table to calculate Z score = 1.96
Z = ( X - µ ) / σ
-1.96 = ( X - 7.3 ) / 2.2
a = 2.988
1.96 = ( X - 7.3 ) / 2.2
b = 11.612
P ( 2.988 < X < 11.612 ) = 0.95