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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 ex
( )( ) /
1
2
2 2 2
for all xNote: e is the mathematical constant, 2.718282
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Standard Normal Distribution
f t e t( ) / 1
2
2 2
for all x.
The normal distribution with =0 and =1 is called the standard normal
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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.
zx
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Standard Normal Table
Use the table in the text to verify the following.
P(z < -2) = F(2) = 0.0228F(2) = 0.9773F(1.42) = 0.9222F(-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 = 2and find
P(z < 2) = .9773So 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 = 3and find
P(z > 3) = 1- P(x < 3) = 1- .9987= 0.0013So 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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