Question

Suppose a newly released car has the average mileage of 26 mpg and a standard deviation of 7. The mileage of these cars approximately follow normal distribution. Consider a sample of size of 15 cars

a.What is the probability that the sample mean is between 22 and 32?

b.If a sample of 16 cars are considered, what is the probability that the mean mileage is

less than 27.5?

c. If a sample of 25 cars are considered, what is the probability that the average mileage

is more than 30 mph ?

Answer 1

Given,

= 26 , = 7

Using central limit theorem,

P( < x) = P( Z < x - / / sqrt(n))

a)

P( 22 < < 32) = P( < 32) - P( < 22)

= P( Z < 32 - 26 / 7 / sqrt(15) ) - P( Z < 22 - 26 / 7 / sqrt(15) )

= P( Z < 3.3197) - P( Z < -2.2131)

= 0.9995 - 0.0134

= 0.9861

b)

P( < 27.5) = P( Z < 27.5 - 26 / 7 / sqrt(16) )

= P( Z < 0.8571)

= 0.8043

c)

P( > 30) = P( Z > 30 - 26 / 7 / sqrt(25) )

= P( Z > 2.8571)

= 0.0021