Chapter 5.2
Using the Central Limit Theorem
Women’s heights are normally distributed with mean µ = 63.6 inches and standard deviation σ = 2.5 inches.
1)
a)
If one woman is randomly selected, find the
b)
If 50 women are randomly selected, find the
probability that her height is above 64
probability that they have a mean height that
inches.
is greater than 64 inches.
We want P(X > 64).
We want P( x > 64).
z = (64 – 63.6)/2.5 = 0.16
z = (64 – 63.6)/ (2.5 / √50) = 1.13
Probability = 0.4364
Probability = 0.1292
c)
If one woman is randomly selected, find the
d)
If 36 women are randomly selected, find the
probability that her height is between 63.6
probability that they have a mean height
and 64.6 inches.
between 63.6 and 64.6 inches.
We want P(63.6 < X < 64.6).
x < 64.6).
z = (63.6 – 63.6)/2.5 = 0
We want P(63.6 <
z = (64.6 – 63.6)/2.5 = 0.4
z = (63.6 – 63.6)/ (2.5 / √36) = 0
Probability = area between the two z-scores…
z = (64.6 – 63.6)/ (2.5 / √36) = 2.4
P(63.6 < X < 64.6) = 0.1554
P(63.6 <
x < 64.6) = 0.4918
e)
If one woman is randomly selected, find the
f)
If 75 women are randomly selected, find the
probability that her height is between 63.0
probability that they have a mean height
and 65.0 inches.
between 63.0 and 65.0 inches.
We want P(63 < X < 65).
x < 65).
z = (63 – 63.6)/2.5 = – 0.24
We want P(63 <
z = (65 – 63.6)/2.5 = 0.56
z = (63 – 63.6)/ (2.5 / √75) = – 2.08
Probability = area between the two z-scores…
z = (65 – 63.6)/ (2.5 / √75) = 4.85
P(63 < X < 65) = 0.3071.
Probability = 0.9811 (using 0.9999 for the area to the
left of z = 4.85)
.
2)
Replacement times for TV sets are normally distributed with a mean of 8.2 years and a standard deviation
of 1.1 years (based on data from “Getting Things Fixed,” Consumer Reports). Find the probability that 40
randomly selected TV sets will have a mean replacement time less than 8.0 years.
z = – 1.15
Probability = 0.1251
3)
The typical computer random-number generator yields numbers in a uniform distribution between 0 and 1
with mean µ = 0.500 and standard deviation σ = 0.289.
a)
If 4 random numbers are generated, we can consider them to be a random sample from the population
(which has a uniform distribution). What can we say about the distribution of
the sample size n = 4? That is, give
 x and  x ;
x , the sample mean when
can we identify the shape of the distribution of
x?
Why or why not?
 x = 0.5
 x ; = 0.1445
We cannot identify the shape of the distribution because the population is not normal and n is small.
b)
If 45 random numbers are generated, what can we say about the distribution of
when the sample size n = 45? That is, give
 x and  x ;
x , the sample mean
can we identify the shape of the distribution of
x ? Why or why not?
x
= 0.5
The distribution of
c)
 x ; = 0.04308
x is approximately normal due to the “large” sample size (more than 30).
Find the probability that the mean of 45 randomly generated numbers is below 0.565.
z = 1.51
Probability = 0.9345