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The Central Limit Theorem
Rio Hondo College Teaching Demonstration
Greg Miller 3/18/2011
Recall: Random Variables
Let X be a random variable.
Just to remind you what a random variable is, just think of it as the output of a procedure.
Some procedure happens and the output (a number) is called X.
Example 1
Recall: Random Variables
Let X be a random variable.
Just to remind you what a random variable is, just think of it as the output of a procedure.
Some procedure happens and the output (a number) is called X.
Example 1
Recall: Random Variables
Each random variable comes with its own probability distribution.
A probability distribution tells you any probability you want to know pertaining to your random
variable.
Recall: Random Variables
Each random variable comes with its own probability distribution.
A probability distribution tells you any probability you want to know pertaining to your random
variable.
Example 2
Procedure: Roll a single die once
X= the number that is face up on the die.
Recall: Random Variables
Each random variable comes with its own probability distribution.
A probability distribution tells you any probability you want to know pertaining to your random
variable.
Example 2
Procedure: Roll a single die once
X= the number that is face up on the die.
From this you can find any probability you care about, such as
P(X=3) = 1/6
or
P(X>1) = 5/6
The Random Variable
X
Let X be a random variable and define a NEW random variable X by running the X random
variable’s procedure a few times and take the average of those results (the number of times X ’s
procedure is run will depend on the problem)
The Random Variable
X
Let X be a random variable and define a NEW random variable X by running the X random
variable’s procedure a few times and take the average of those results (the number of times X ’s
procedure is run will depend on the problem)
Example 3
Run X’s procedure 3 times and X is the average of the 3 results.
QUESTION: What is the distribution of the new random variable X ?
QUESTION: What is the distribution of the new random variable X ?
Example 4 Define the random variables X , X 2 , X 3 as follows:
QUESTION: What is the distribution of the new random variable X ?
Example 4 Define the random variables X , X 2 , X 3 as follows:
X = number face up on a die if rolled once
QUESTION: What is the distribution of the new random variable X ?
Example 4 Define the random variables X , X 2 , X 3 as follows:
X = number face up on a die if rolled once
X 2 = mean of the face up numbers if the die is rolled 2 times
QUESTION: What is the distribution of the new random variable X ?
Example 4 Define the random variables X , X 2 , X 3 as follows:
X = number face up on a die if rolled once
X 2 = mean of the face up numbers if the die is rolled 2 times
X 3 = mean of the face up numbers if the die is rolled 3 times
QUESTION: What is the distribution of the new random variable X ?
Example 4 Define the random variables X , X 2 , X 3 as follows:
X = number face up on a die if rolled once
X 2 = mean of the face up numbers if the die is rolled 2 times
X 3 = mean of the face up numbers if the die is rolled 3 times
QUESTION: What is the distribution of the new random variable X ?
Example 4 Define the random variables X , X 2 , X 3 as follows:
X = number face up on a die if rolled once
X 2 = mean of the face up numbers if the die is rolled 2 times
X 3 = mean of the face up numbers if the die is rolled 3 times
Answer:
Central Limit Theorem
Let X be a random variable with any distribution whatsoever with mean
standard deviation
mean
X
 X  X
 X and
. If n is very large, then the distribution of X is NORMAL with
and standard deviation
X  X
n
.
Answer:
Central Limit Theorem
Let X be a random variable with any distribution whatsoever with mean
standard deviation
mean
X
 X  X
 X and
. If n is very large, then the distribution of X is NORMAL with
and standard deviation
X  X
n
.
Answer:
Central Limit Theorem
Let X be a random variable with any distribution whatsoever with mean
standard deviation
mean
X
 X  X
 X and
. If n is very large, then the distribution of X is NORMAL with
and standard deviation
X  X
.
n
mean =
X
mean =
standard deviation =
X
standard deviation =
 X  X
X  X
n
Central Limit Theorem
Rule of thumb: If n is larger than 30, then that is large enough for us to assume that the
distribution of X is normal
Exercises
Exercise 1: The Coca-Cola company claims that the mean amount of soda in their cans is 12
ounces with a standard deviation of 0.5 ounces.
a) If 100 cans of Coke are randomly selected, what is the probability that the mean amount of
soda in them is less than 11.88 ounces?
Exercises
Exercise 1: The Coca-Cola company claims that the mean amount of soda in their cans is 12
ounces with a standard deviation of 0.5 ounces.
a) If 100 cans of Coke are randomly selected, what is the probability that the mean amount of
soda in them is less than 11.88 ounces?
Answer: P = 0.0082
b) If 100 cans of Coke are randomly selected and the mean amount of soda in this sample is 11.84
ounces, what can you conclude?
Exercises
Exercise 1: The Coca-Cola company claims that the mean amount of soda in their cans is 12
ounces with a standard deviation of 0.5 ounces.
a) If 100 cans of Coke are randomly selected, what is the probability that the mean amount of
soda in them is less than 11.88 ounces?
Answer: P = 0.0082
b) If 100 cans of Coke are randomly selected and the mean amount of soda in this sample is 11.84
ounces, what can you conclude?
Answer: Their claim is most likely incorrect. Either their machines are not working properly
or they are trying to cheat their customers.
Thank you for having me!