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Normal Distributions

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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 ... – PowerPoint PPT presentation

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Title: Normal Distributions


1
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

2
Normal Probability Density
  • for all x
  • Note e is the mathematical constant,
  • 2.718282 ...

3
Standard Normal Distribution
  • for all x.
  • The normal distribution with m 0 and s 1 is
    called the standard normal

4
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.

5
Standard Normal Table
  • Use the table in the text to verify the
    following.
  • P(z lt -2) F(2) 0.0228
  • F(2) 0.9773
  • F(1.42) 0.9222
  • F(-0.95) 0.1711

6
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?

7
Normal Example - part a)
  • Assume m 6 and s .03.
  • The problem requires us to find
  • P(X lt 6.06)
  • Convert x 6.06 to a z-score
  • z (6.06 - 6)/.03 2
  • and find
  • P(z lt 2) .9773
  • So 97.73 of the jar have less than 6.06 oz.

8
Normal Example - part b)
  • Again m 6 and s .03.
  • The problem requires us to find
  • P(X gt 6.09)
  • Convert x 6.09 to a z-score
  • z (6.09 - 6)/.03 3
  • and find
  • P(z gt 3) 1- P(x lt 3) 1- .9987 0.0013
  • So 0.13 of the jar havemore than 6.09oz.

9
Preview
  • Probabiltiy Plots
  • Normal Approximation of the Binomial
  • Random Sampling
  • The Central Limit Theorem
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