Question

Carbon monoxide (CO) emissions for a certain kind of car vary with mean 3 g/mi and standard deviation 0.5 g/mi. A company has 81 of these cars in its fleet.

a) What’s the approximate model for the distribution of the mean CO level for the company's fleet. Explain.

b) Estimate the probability that the mean emission is between 3.0 and 3.1 g/mi.

c) There is only a 5% chance that the fleet’s mean CO level is greater than what value?

Answer 1

a) According to Central limit theorem, the sampling distribution of mean from a sample of size greater than 30 will be approximately normal, irrespective of the shape of population distribution.

P( < A) = P(Z < (A - )/)

n = 81

= 3 g/mi

= = 3 g/mi

= 0.5 g/mi

=

=

= 0.0556

b) P(mean emission is between 3.0 and 3.1 g/mi) = P(3.0 < < 3.1) = P( < 3.1) - P( < 3.0)

= P(Z < (3.1 - 3)/0.0556) - 0.5

= P(Z < 1.8) - 0.5

= 0.9641 - 0.5

= 0.4641

c) Let the fleet’s mean CO level be higher than H only 5% of the time.

P( > H) = 0.05

P( < H) = 1 - 0.05 = 0.95

P(Z < (H - 3)/0.0556) = 0.95

Take Z score corresponding to 0.95 from standard normal distribution table.

(H - 3)/0.0556 = 1.645

H = 3.09 g/mi