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The Central Limit Theorem
Today, we will learn a very powerful tool for sampling probabilities and inferential
statistics:
The Central Limit Theorem
The Central Limit Theorem
If samples of size n>29 are drawn from a population with mean, , and standard deviation, , then the
sampling distribution of the sampling means is nearly normal and also has mean and a standard deviation
Of
WTHeck?!!!
n
The Central Limit Theorem
When working with distributions of samples rather than individuatl data
points we use rather than
is called the Standard Error
n
n
The Central Limit Theorem
Example
The average fundraiser at BHS raises a mean of $550 with a standard
deviation of $35. Assume a normal distribution:
Problem we are used to: What is the probability the next fundraiser will raise
more than $600?
Sampling problem: What is the probability the next 10 fundraisers will
average more than $600
The Central Limit Theorem
The average fundraiser at BHS raises a mean of $550 with a standard
deviation of $35. Assume a normal distribution:
Problem we are used to: What is the probability the next fundraiser will raise
more than $600?
600 5501.43
35(1.43,99) .0764
x xz
snormalcdf
The Central Limit Theorem
The average fundraiser at BHS raises a mean of $550 with a standard
deviation of $35. Assume a normal distribution:
Sampling problem: What is the probability the next 30 fundraisers will
average more than $600
600 5507.82
/ 35 / 30(7.82,99) 0
x xzs n
normalcdf
The Central Limit Theorem
This makes sense: It would be much more common for a single fundraiser to vary that much from the mean, but not very likely that you get ten that average that
high.
The Central Limit Theorem
Example Two:
Mr. Gillam teachers 10,000 students. Their mean grade is 87.5 and the standard
deviation is 15.
a) What is the probability a group of 35 students has a mean less than 90?
90 87.5.9860
/ 15 / 35( 99,.9860) 0.8379
83.79%
x xzs n
normalcdf
