Transcript Central Limit Theorem

Normal Distributions
Family of distributions, all with the same
general shape.
 Symmetric about the mean
 The y-coordinate (height) specified in
terms of the mean and the standard
deviation of the distribution

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Normal Probability Density
f ( x) 
1
( x   )2 / 2  2
e
2
for all x
Note: e is the mathematical constant,
2.718282 ...
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Standard Normal Distribution
f (t ) 
1
2
t /2
e
2 
for all x.
The normal distribution with  =0
and  =1 is called the standard
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Transformations
Normal distributions can be transformed
to the standard normal.
We use what is called the z-score which is
a value that gives the number of
standard deviations that X is from the
mean.
z 
x  

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Standard Normal Table
Use the table in the text to verify the
following.
P(z < -2) = F(2) = 0.0228
F(2) = 0.9773
F(1.42) = 0.9222
F(-0.95) = 0.1711
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Example of the Normal
The amount of instant coffee that is put
into a 6 oz jar has a normal distribution
with a standardard deviation of 0.03.
oz. What proportion of the jar contain:
a) less than 6.06 oz?
b) more than 6.09 oz?
c) less than 6 oz?
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Normal Example - part a)
Assume  = 6 and  = .03.
The problem requires us to find
P(X < 6.06)
Convert x = 6.06 to a z-score
z = (6.06 - 6)/.03 = 2
and find
P(z < 2) = .9773
So 97.73% of the jar have less than 6.06
oz.
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Normal Example - part b)
Again  = 6 and  = .03.
The problem requires us to find
P(X > 6.09)
Convert x = 6.09 to a z-score
z = (6.09 - 6)/.03 = 3
and find
P(z > 3) = 1- P(x < 3) = 1- .9987=
0.0013
So 0.13% of the jar havemore than 6.09oz.
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Preview
Probabiltiy Plots
Normal Approximation of the Binomial
Random Sampling
The Central Limit Theorem
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