Download AP Statistics Section 9.3B The Central Limit Theorem

AP Statistics Section 7.3B
The Central Limit Theorem
In Section 7.3A, we saw that if we
draw an SRS of size n from a
population with a Normal
distribution, N(  ,  ), then the
sample mean, x , has a Normal

 _____)
distribution N(___,
n
Note : For  x  
n
, N  10n
Although many populations have roughly
Normal distributions, there are certainly
some population distributions that are not
Normal. So what happens to the shape of
the distribution of x when the population
distribution is not Normal?
The website lock5stat.com/statkey/ is
awesome for visualizing distributions.
Notice the bar for sampling distributions,
in particular the “mean”.
If you click on the mean, the screen at the right appears. In
the upper left hand corner
there is a drop down menu with preloaded
sets of data. Let’s use the salaries of baseball
players in the year 2012.
One you have clicked on the baseball salaries the distribution
of the population will appear at the upper right along with the
mean of the distribution and the standard deviation.
a) Describe the shape of
the population distribution.
Severely right skewed
b) Give the mean and standard
deviation of the population
distribution.
  3.439
  4.698
Now let’s look at the sampling distribution of x .
Notice Statkey allows you to choose samples of
different size – let’s start with samples of size 10
and generate 1 sample.
What does this one dot on
the distribution represent?
The mean of a random
sample of 10 baseball
salaries
Now let’s generate 1000 samples of size 10.
a) Describe the shape of the sampling
distribution.
Slightly right skewed
b) Give the mean and standard deviation of
the sampling distribution.
  3.512
  1.517
Now let’s generate 1000 samples of size 30.
a) Describe the shape of the sampling
distribution.
Bell  shaped
Approx.Normal
b) Give the mean and standard deviation of
the population distribution.
  3.399
  0.822
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?
Because 70  30, 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.