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?

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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