Transcript the central limit theorem

Section 6-5
The Central Limit Theorem
THE CENTRAL LIMIT THEOREM
Given:
1. The random variable x has a distribution
(which may or may not be normal) with
mean µ and standard deviation σ.
2. Samples all of the same size n are randomly
selected from the population of x values.
THE CENTRAL LIMIT THEOREM
Conclusions:
1. The distribution of sample means x will, as
the sample size increases, approach a
normal distribution.
2. The mean of the sample means will be the
population mean µ.
3. The standard deviation of the sample
means will approach  / n .
COMMENTS ON THE CENTRAL
LIMIT THEOREM
The Central Limit Theorem involves two
distributions.
1. The population distribution. (This is what
we studied in Sections 6-1 through 6-3.)
2. The distribution of sample means. (This is
what we studied in the last section,
Section 6-4.)
PRACTICAL RULES
COMMONLY USED
1. For samples of size n larger than 30, the
distribution of the sample means can be
approximated reasonably well by a normal
distribution. The approximation gets better
as the sample size n becomes larger.
2. If the original population is itself normally
distributed, then the sample means will be
normally distributed for any sample size n
(not just the values of n larger than 30).
NOTATION FOR THE SAMPLING
DISTRIBUTION OF x
If all possible random samples of size n are selected from
a population with mean μ and standard deviation σ, the
mean of the sample means is denoted by  x , so
x 
Also, the standard deviation of the sample means is
denoted by  x , so

x
n

x
is often called the standard error of the mean.
A NORMAL DISTRIBUTION
As we proceed from
n = 1 to n = 50, we
see that the
distribution of
sample means is
approaching the
shape of a normal
distribution.
A UNIFORM DISTRIBUTION
As we proceed from
n = 1 to n = 50, we
see that the
distribution of
sample means is
approaching the
shape of a normal
distribution.
A U-SHAPED DISTRIBUTION
As we proceed from
n = 1 to n = 50, we
see that the
distribution of
sample means is
approaching the
shape of a normal
distribution.
As the sample size increases, the
sampling distribution of sample means
approaches a normal distribution.
CAUTIONS ABOUT THE
CENTRAL LIMIT THEOREM
•
When working with an individual value from a
normally distributed population, use the methods of
Section 6-3. Use
z
•
x

When working with a mean for some sample (or
group) be sure to use the value of  / n for the
standard deviation of sample means. Use
x
z
/ n
RARE EVENT RULE
If, under a given assumption, the
probability of a particular observed event
is exceptionally small, we conclude that
the assumption is probably not correct.