Transcript 9.3.2 - GEOCITIES.ws

The Central Limit Theorem
Section 9.3.2
Starter
• Assume I have 1000 pennies in a jar
• Let X = the age of a penny in years
– If the date is 2007, X = 0
– If the date is 2006, X = 1, etc
• Draw a smooth curve that is your best
guess of the shape of the distribution of X
• Write a verbal description of your guess of
the distribution
Today’s Objectives
• Perform an activity that demonstrates the effect
of the Central Limit Theorem
• Write a statement of The Central Limit Theorem
California Standard 9.0
Students know the central limit theorem and can use it to
obtain approximations for probabilities in problems of
finite sample spaces in which the probabilities are
distributed binomially.
How Old Are Pennies in Circulation?
• That’s a hard question to answer. Let’s try an
easier one:
• What’s the average age of 1000 randomly
collected pennies?
– We could agree that the answer to this question is a
reasonable approximation to the answer to the main
question.
• We could now actually record the ages of all
1000 pennies and calculate the mean, or we
could approach the problem by sampling.
• I will give you 25 pennies in a cup that are a
sample of the 1000 pennies. We will take
samples of increasing size and see what
happens.
Sample the population
n=2
• Shake up your pennies in the cup and
select a sample of 2 pennies
• Calculate the mean age of the 2 pennies
– Record the sample mean you found
• Replace the pennies, shake again, and
choose another sample of 2 pennies
– Record the new sample mean
• Plot your sample means on the whiteboard
– You should be plotting two points
• Draw a sketch and write a description of
the sampling distribution
Sample the population
n = 10
• Shake up your pennies in the cup and
select a sample of 10 pennies
• Calculate the mean age of the 10 pennies
– Record the sample mean you found
• Replace the pennies, shake again, and
choose another sample of 10 pennies
– Record the new sample mean
• Plot your sample means on the whiteboard
– You should be plotting two points
• Draw a sketch and write a description of
the sampling distribution
Sample the population
n = 25
• Combine your pennies with one or more
partners and draw a new sample of 25.
• Record the ages of all 25 of your pennies
and calculate the sample mean.
• Plot your sample mean on the whiteboard.
• Draw a sketch and write a description of
the sampling distribution.
– NOTE: KEEP YOUR SAMPLE MEAN IN
YOUR NOTES. WE WILL USE IT AGAIN IN
CHAPTER 10.
The Central Limit Theorem
If an SRS of size n is drawn from a
population of any shape with mean μ and
standard deviation σ, and if n is large, then
the sampling distribution of the sample
means will be approximately normal with
mean μ and standard deviation σ/ √n
– How large is large? For most situations, n=30
will be large enough.
– The more “non-normal” the population, the
larger n should be to get good results
Today’s Objectives
• Perform an activity that demonstrates the effect
of the Central Limit Theorem
• Write a statement of The Central Limit Theorem
California Standard 9.0
Students know the central limit theorem and can use it to
obtain approximations for probabilities in problems of
finite sample spaces in which the probabilities are
distributed binomially.
Homework
• Read pages 487 - 491
• Do problems 30 - 34