AP Statistics Section 9.3B The Central Limit Theorem
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Transcript of AP Statistics Section 9.3B The Central Limit Theorem
AP Statistics Section 9.3BThe 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, , then the
sample mean, , has a Normal
distribution N(___, _____)
),N( x
n
Although many populations have roughly Normal distributions, very few are exactly Normal. So what happens to 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.
x
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 for to be close to
Normal? The farther the shape of the population is from Normal, the
more observations are required.
x
Example: The time a technician requires to perform preventative maintenance on an air-conditioning unit is an exponential distribution with
the mean time hour and the standard deviation 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?
1 1
)70
1 N(1, approx. is x of dist. theCLT, By the
700)or 10(70 units ACsuch all of Pop.
018.)25.1(
201.)1.1(
xP
xP
time theof 1.8% late
run only tech willthe
hrs/call 1.25at but time
theof 20% laterun will
tech thehrs/call 1.1At
The figure below summarizes the sampling distribution of . 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 for each sample. Collect all the and display their distribution. That’s the sampling
distribution of . Sampling distributions are the key to understanding statistical inference.
x
xsx '
x
n
case.any in samples largefor Normal approx. is dist. The
Normal. is dist. pop. theif Normal is dist. The