Download Lesson on the Central Limit Theorem

Unit 6
Section 5.4
Section 5.4
5.4: The Central Limit Theorem

Sampling Distribution – the probability
distribution of a sample statistic that is formed
when samples of size n are repeatedly taken
from a population.


If the sample statistic is the sample mean, then it
is a Sampling Distribution of Sample Means
Sampling Error – the difference between the
sample measure and the corresponding
population measure due to the fact that the
sample is not a perfect representation of the
population.
Section 5.4
Properties of the Distribution of Sample
Means
 The
mean of the sample means will be
the same as the population mean.
 The
standard deviation of the sample
means will be equal to the population
standard deviation divided by the square
root of the sample size.
s
sx =
n
Section 5.4
 Example
1:
Suppose a professor gave an 8 point
quiz to a small class of 4 students. The
results of the quiz were 2, 4, 6, and 8.
a)
Determine the mean of the
population.
b)
Determine the standard deviation of
the population.
Section 5.4
c)
Now, if all samples of size 2 are taken with
replacement, construct a a table to represent
all the possibilities and their means.
d)
Use your table to create a frequency
distribution for the sample means.
e)
Determine the mean for the samples
f)
Determine the standard deviation for the
samples.
g)
Take the standard deviation for the
population and divide it by the square root of
the sample size. What do you notice?
Section 5.4
The standard deviation of the sample
means is also known as the standard error of
the mean.
The formula for the standard error of the
mean is:
sX =
s
n
Section 5.4
The Central Limit Theorem

If samples of size n (where n is greater than or
equal to 30) are drawn from any population
(with mean μ and standard deviation σ), then
the sampling distribution of sample means
approximates a normal distribution.

If the population is normally distributed, then
the sampling distribution of sample means is
normally distributed for ANY sample size n.
This distribution will have a mean μ and a
standard deviation
s
n
Section 5.4

Example 2:
Cell phone bills for residents of a city
have a mean of $47 and a standard
deviation of $9, as shown in the figure.
Random samples of 100 cell phone
bills are drawn from this population,
and the mean of each sample is
determined.
Find the mean and standard
deviation of the sample means. Then
sketch a graph of the sampling
distribution.
Section 5.4
When finding the probabilities for a sample
mean within a sampling distribution, you will
need to find z-scores.
When the population is sufficiently large,
you can locate a z-score using the following
formula:
z =
X -m
s/ n
Section 5.4
Things to remember when using the
Central Limit Theorem:

When the original variable is normally
distributed, the distribution of the sample
means will be normally distributed for any
sample size n. (Normal Method)

When the distribution of the original value
might not be normal, a sample size of 30 or
more is needed to use a normal distribution to
approximate the distribution of sample
means.

The larger the sample, the better the
approximation
Section 5.4
 Example
3:
A.C. Neilsen reported that children
between the ages of 2 and 5 watch an
average of 25 hours of television per week.
Assume the variable is normally distributed
and the standard deviation is 3 hours. If 30
children between the ages of 2 and 5 are
randomly selected, find the probability that
the mean of the number of hours they
watch television will be greater than 26.3
hours.
Section 5.4
 Step
1: Determine the standard deviation
of the sample means.
 Step
2: Draw a picture to represent the
situation.
 Step
3: Find the z-score
 Step
4: Determine the probability
Section 5.4
 Example
4:
The average age of a vehicle registered in
the United States is 8 years, or 96 months.
Assume the standard deviation is 16
months. If a random sample of 36 vehicles is
selected, find the probability that the mean
of their age is between 90 and 100 months.
Section 5.4
Homework:
 Pg
269-271
 #’s
1 – 29 ODD