9.3 The Central Limit Theorem for Sample Means
Consider a normal distribution
with mean µ x and standard
A sampling distribution from
this x-distribution will be
deviation σ x
normal regardless of the size of n
x
x
σ
Note: µ x = µ x and σ x = x
n
Consider a right skewed distribution
with mean µ x and standard
deviation σ x
A sampling distribution from
this x-distribution will be
normal if and only if n is large
x
x
σx
n
Can be computed
regardless of the shape
Note: µ x = µ x and σ x =
Consider a left skewed distribution
with mean µ x and standard
deviation σ x
A sampling distribution from
this x-distribution will be
normal if and only if n is large
x
σ
Note: µ x = µ x and σ x = x
n
Can be computed
regardless of the shape
x
9.3 The Central Limit Theorem for Sample Means
The time that a technician requires to perform preventative maintenance
on an air conditioning unit is modeled by the exponential distribution.
µ x = 1 hour
σ x = 1 hour
1
a.) What is the probability that five randomly selected units takes
the technician an average time of more than 50 minutes to
perform preventative maintenance on ?
b.) What is the probability that a randomly selected sample of 70
of these units take an average time of more than 50 minutes
to perform preventative maintenance on ?
9.3 The Central Limit Theorem for Sample Means
The heights of young women vary normally with parameters µ x =64.5 and
σ x = 2.5.
a.) What is the probability that a randomly selected young woman is
shorter than 66.5 inches ?
b.) What is the probability that the mean height of an SRS of 4 women is
less than 66.5 inches ?
As reported by “Runner’s World” magazine, times of finishers in the NYC
10k run are distributed normally with a mean of 61 minutes and a standard
deviation of 9 minutes. 50 runners are randomly selected. What is the
probability that the average time of these 50 runners is less than 58
minutes ?