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