Download Measures of Central Tendancy

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 
n
WTHeck?!!!
The Central Limit Theorem
When working with distributions of
samples rather than individuatl data
points we use  rather than 
n

n
is called the Standard Error
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?
x  x 600  550
z

 1.43
s
35
normalcdf (1.43,99)  .0764
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
x  x 600  550
z

 7.82
s/ n
35 / 30
normalcdf (7.82,99)  0
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?
x  x 90  87.5
z

 .9860
s / n 15 / 35
normalcdf (99,.9860)  0.8379
 83.79%