normal

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