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Chapter Six
Normal Curves and
Sampling Probability
Distributions
SECTION 5
The Central Limit
Theorem
The Central Limit Theorem is
applied in the following ways:
A given population has fixed parameters µ and  .
For each population, there are very many samples
of size n, which can be taken. Each of these samples
has a sample mean x. The x statistic varies from
sample to sample. The Central Limit Theorem tells
us what to expect about the sample means.
If x is normally distributed for any size sample, or if the sample
size is 30 or more, the sample means will be normally
distributed.
When working with any normal distribution, we have to
know the mean of the distribution and the standard
deviation. The Central Limit Theorem relates the mean
and the standard deviation of the original x to the mean
and standard deviation of x.
When working with normal distribution probabilities, we
must convert all values to z-scores.
The Central Limit Theorem is used to investigate the probability
of a sample mean being in a given interval.
Since we will be working with normal
distributions, the x values in the given
interval must be converted to z-scores.
There are three formulas, all which are
equivalent, which may be used to
convert from an x value to a z-score.
The simplest formula is
z
x
x
The deviation of x value from the mean is divided by the
standard deviation of the distribution  x . To use this
version, first we calculate  x by dividing  by n. Then
substitute.
You may prefer to directly substitute the fraction

for
n
 x in the denominator and use either of the two formulas
that follow. You could use ...
z
x

n
or invert and multiply to get the formula
you are to use in this class:
x  

z

n
To find the probability that a sample
mean is within a given interval:
First, check to see if the distribution of x (from which the
sample mean was taken) is either a normal distribution, or
if not, the sample size is 30 or more. If either of these are
true, the sample distribution is normal.
Find the mean and the standard deviation for the sample
distribution of x using the formulas for the Central Limit
Theorem.
Convert each endpoint of the given interval to the standard
normal z-score.
Rewrite the problem with the z-score interval.
Sketch a standard normal distribution curve and shade the
area you wish to calculate.
Use the table for standard normal probability distribution
to calculate the area, and thus, the probability.
Case 1
When a variable x has a mean of  x ,
and a standard deviation of  x , and
is normally distributed. For a random
sample of any size n, the following
statements about the sampling
distribution of x are true.
1.
The distribution of x is normal.
2. The mean of the sample means is equal to the
mean of the population. That is:  x  
3. The standard deviation of the distributions
of the sample means is called the standard
error of the mean and is smaller than the
1
distribution of x by a factor of
.
n
That is:  x 

n
.
CASE 2
When a variable x comes from any
type of distribution, no matter how
unusual, as long as the random sample
has at least 30 members, then the
following statements about the
distribution of the sample size are true.
1.
The distribution of x is normal.
2. The mean of the sample means is equal to the
mean of the population. That is:  x  
3. The standard deviation of the distributions
of the sample means is called the standard
error of the mean and is smaller than the
1
distribution of x by a factor of
.
n
That is:  x 

n
.
Sampling Distribution
A probability distribution
for the sample statistic
we are using.
Example of a Sampling
Distribution
Select samples with two elements
each (in sequence with
replacement) from the set
{1, 2, 3, 4, 5, 6}.
Constructing a Sampling Distribution
of the Mean for Samples of Size n = 2
List all samples and compute the mean of each sample.
sample: mean:
sample: mean
{1,1}
1.0
{1,6}
3.5
{1,2}
1.5
{2,1}
1.5
{1,3}
2.0
{2,2}
2.0
{1,4}
2.5
…
...
{1,5}
3.0
How many different samples are there? 36
Sampling Distribution of the Mean
x
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
p
1/36
2/36
3/36
4/36
5/36
6/36
5/36
4/36
3/36
2/36
1/36
Sampling Distribution
Histogram
Let x be a random variable with a
normal distribution with mean 
and standard deviation  . Let x be
the sample mean corresponding to
random samples of size n taken
from the distribution.
The following are true:
1. The x distribution is a normal distribution.
2. The mean of the x distribution is  (the
same mean as the original distribution).
3. The standard deviation of the x distribution
is

(the standard deviation of the original
n
distribution, divided by the square root of the
sample size).
We can use this theorem to draw conclusions
about means of samples taken from normal
distributions.
If the original distribution is normal, then the
sampling distribution will be normal.
x
The Mean of the Sampling
Distribution
The mean of the sampling distribution is equal
to the mean of the original distribution.
x  
x
The Standard Deviation of the
Sampling Distribution
The standard deviation of the sampling
distribution is equal to the standard deviation of
the original distribution divided by the square
root of the sample size.
x 

n
The Central
Limit Theorem
As the sample size continue to
increase closer and closer
to the population size the
following statements are true.
1. The sample will have
a normal distribution.
2. The sample mean will
approach the population

mean. lim x  
nN

3. The standard deviation will take on the
intermediate value of the population
standard deviation divided by the
square root of the sample size. However,
because of the increasing sample size
this value will approach zero.



 x  lim
 0
 lim
nN
nN

n
4. The probability of an interval that
contains the population mean
will approach 1.
lim P x    x  1
nN
1
2
5. The probability of an interval
that does NOT contain the
population mean will approach 0.
lim P x    0
nN
1
or
lim P x     0
nN
***Note ***
As the sample size increases to the population size
Graph 1  Graph 2  Graph 3
the graphs get closer and closer to the population mean.
The Central Limit Theorem states that
if the probability interval contains the
population mean and the sample size
continues to increase closer and closer
to the population size then the probability
will continue to increase and get closer
and closer to 1.
or
The Central Limit Theorem states that
if the probability interval does not
contain the population mean and the
sample size continues increase closer
and closer to the population size then
the probability will continue to decrease
and get closer and closer to 0.
Sample
Questions
1.
Suppose that it is known that the time
spent by customers in the local coffee
shop is normally distributed with a
mean of 24 minutes and a standard
deviation of 6 minutes.
Find the probability that an individual
a.
customer will spend more than 26
minutes in the coffee shop.
P x  26 
26  24 

P z 


6

P z 

0.33
2

6
P z  0.33
0.5000  0.1293
0.3707
Find the probability that a random sample
b.
of 9 customers will have a mean stay of
more than 26 minutes in the coffee shop.
P x  26 

26  24  9 

P z 

6


2 3

P z 

6 
1.00
6

P z  

6
P z  1.00 
0.5000  0.3413
0.1587
Find the probability that a random sample
c.
of 64 customers will have a mean stay of
more than 26 minutes in the coffee shop.
P x  26 

26  24  64 

P z 

6


2 8 

P z 

6 
2.67
16 

P z  

6
P z  2.67 
0.5000  0.4962
0.0038
Find the probability that a random sample
d.
of 100 customers will have a mean stay of
more than 26 minutes in the coffee shop.
P x  26 

26  24  100 

P z 

6


2 10 

P z 

6 
20 

P z  

6
3.33
P z  3.33
0.5000  0.4996
0.0004
Find the probability that a random sample
e.
of 144 customers will have a mean stay of
more than 26 minutes in the coffee shop.
P x  26 

26  24  144 

P z 

6


2 12 

P z 

6 
4.00
24 

P z  

6
P z  4.00 
0.5000  0.4999
0.0001
e. The Central Limit Theorem states that
if the probability interval does not
contain the population mean and the
sample size continues increase closer
and closer to the population size then
the probability will continue to decrease
and get closer and closer to 0.
2.
Suppose that it is known that the time
spent by customers in the local coffee
shop is normally distributed with a
mean of 24 minutes and a standard
deviation of 6 minutes.
Find the probability that an individual
a.
customer will spend between 22 to 25
minutes in the coffee shop.
P 22  x  25 
25  24 
 22  24
P
z

 6
6 
1
 2
P
z 
 6
6
-0.33 0.17
P 0.33  z  0.17 
0.1293  0.0675
0.1968
Find the probability that a random sample
b. of 9 customers will have a mean stay in
the coffee shop between 22 to 25 minutes.

P 22  x  25 

 22  24 9
25  24  9 

P
z

6
6


13
 2 3

P
z
 6
6 
-1.00
0.50
 6
P
z
 6
3

6
P 1.00  z  0.50 
0.3413  0.1915
0.5328
Find the probability that a random sample
c. of 64 customers will have a mean stay in
the coffee shop between 22 to 25 minutes.

P 22  x  25 

 22  24 64
25  24  64 

P
z

6
6


 2 8 
18 
P
z

6
6 
-2.67
1.33
 16
P
z
 6
8

6
P 2.67  z  1.33
0.4962  0.4082
0.9044
Find the probability that a random sample
d. of 100 customers will have a mean stay in
the coffee shop between 22 to 25 minutes.

P 22  x  25 

 22  24 100
25  24  100 

P
z

6
6


110 
 2 10 

P
z

6
6 
-3.33
1.67
10 
 20
P
z 
 6
6
P 3.33  z  1.67 
0.4996  0.4525
0.9521
Find the probability that a random sample
e. of 144 customers will have a mean stay in
the coffee shop between 22 to 25 minutes.

P 22  x  25 

 22  24 144
25  24  144 

P
z

6
6


 2 12 
112 
P
z

6
6 
-4.00
2.00
12 
 24
P
z 
 6
6
P 4.00  z  2.00 
0.4999  0.4772
0.9771
e. The Central Limit Theorem states that
if the probability interval contains the
population mean and the sample size
continues to increase closer and closer
to the population size then the probability
will continue to increase and get closer
and closer to 1.
THE END