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MATH 264 QUIZ #3
December 29, 2004
QUESTION
The fuel consumption in miles per gallon of all cars of a particular model has mean 25 and standard
deviation 2. The population distribution can be assumed to be normal.
(a) If a car is randomly selected from this population, what is the probability that its fuel consumption
will be less than 24.8 miles per gallon?
(b) If a sample of 4 cars is randomly selected from this population, what is the probability that the
sample mean will be less than 24.8 miles per gallon?
(c) If a sample of 100 cars is randomly selected from this population, what is the probability that the
sample mean will be less than 24.8 miles per gallon?
ANSWER
The population mean (µ) is equal to 25, and the population standard deviation (σ) is equal to 2.
(a) Let X be the fuel consumption of a randomly selected car, in miles per gallon. We are asked to
X −µ
find P (X < 24.8). Since X is normally distributed
is the standard normal distribution, and
σ
we have
X −µ
24.8 − 25
X −µ
P (X < 24.8) = P
<
=P
< −0.1 = 0.5 − 0.0398 = 0.4602.
σ
2
σ
(b) Let x̄ be the mean (average) fuel consumption of randomly selected sample of size n = 4. We are
x̄ − µ
asked to find P (x̄ < 24.8). By Central Limit Theorem, we know that σ is the standard normal
√
n
distribution, Thus
P (x̄ < 24.8) = P
x̄ − µ
√σ
n
<
24.8 − 25
!
√2
4
=P
x̄ − µ
√σ
n
!
< −0.2
= 0.5 − 0.0793 = 0.4207.
(c) Let x̄ be the mean (average) fuel consumption of randomly selected sample of size n = 100. We are
x̄ − µ
asked to find P (x̄ < 24.8). By Central Limit Theorem, we know that σ is the standard normal
√
n
distribution, Thus
P (x̄ < 24.8) = P
x̄ − µ
√σ
n
<
24.8 − 25
√2
100
!
=P
x̄ − µ
√σ
n
!
< −1
= 0.5 − 0.3413 = 0.1587.