Download pp Section 9.3B

AP Statistics Section 9.3B
The Central Limit Theorem
In Section 9.3A, we saw that if we
draw an SRS of size n from a
population with a Normal
distribution, N(  ,  ), then the
sample mean, , has a Normal
x

 _____)
distribution N(___,
n
Note : For  x  
n
, N  10n
Although many populations have roughly
Normal distributions, very few are exactly
Normal. So what happens to x when the
population distribution is not Normal? In
Activity 9B, the distribution of the ages of
pennies should have been right- skewed, but as
the sample size increased from 1 to 5 to 10 and
then to 25, the distribution should have gotten
closer and closer to a Normal distribution.
This is true no matter what shape
the population distribution has, as
long as the population has a finite
standard deviation . This famous
fact of probability is called the
central limit theorem.
Central Limit Theorem
Draw an SRS of size n from any
population whatsoever with
mean  and standard deviation  .
When n is large, the sampling
distribution of the sample mean is
close to the Normal
distribution N (  ,  ) .
n
There are 3 situations to consider
when discussing the shape of the
sampling distribution of x .
1. If the population has a Normal
distribution, then the shape of the
sampling distribution is Normal,
regardless of the sample size.
2. If the population has any shape
and the sample size is small, then
the shape of the sampling
distribution is similar to the
shape of the parent population.
3. If the population has any shape
and the sample size is large, then
the shape of the sampling
distribution is approximately
Normal.
**How large a sample size is
needed x for to be close to
Normal? The farther the shape of
the population is from Normal, the
more observations are required.
Example: The time a technician requires to perform preventative
maintenance on an air-conditioning unit is an exponential distribution with
the mean time   1 hour and the standard deviation   1 hour. Your
company has a contract to maintain 70 of these units in an apartment
building. You must schedule technicians’ times for a visit. Is it safe to budget
an average of 1.1 hours for each unit? Or should you budget an average of
1.25 hours?
By the CLT, the dist. of x is approx. N(1, 1
70
)
Pop. of all such AC units  10(70 )or 700
P( x  1.1)  .201
P( x  1.25)  .018
At 1.1 hrs/call the tech
will run late 20% of the
time but at 1.25 hrs/call
the tech will only run
late 1.8% of the time
The figure below summarizes the sampling distribution of x . It reminds us of
the big idea of a sampling distribution. Keep taking random samples of size n
from a population with mean  . Find the sample mean x for each sample.
Collect all the x' s and display their distribution. That’s the sampling
distribution of x . Sampling distributions are the key to understanding
statistical inference.
N  10n

n

The dist. is Normal if the pop. dist. is Normal.
The dist. is approx. Normal for large samples in any case.