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

normal Normal Probability Density for all x Note: e is the mathematical constant, 

normal Standard Normal Distribution for all x. The normal distribution with  =0 and  =1 is called the standard normal

normal 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.

normal Standard Normal Table Use the table in the text to verify the following. P(z < -2) = F(2) = F(2) = F(1.42) = F(-0.95) =

normal 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 oz. What proportion of the jar contain: a) less than 6.06 oz? b) more than 6.09 oz? c) less than 6 oz?

normal 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 = ( )/.03 = 2 and find P(z < 2) =.9773 So 97.73% of the jar have less than 6.06 oz.

normal 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 = ( )/.03 = 3 and find P(z > 3) = 1- P(x < 3) = = So 0.13% of the jar havemore than 6.09oz.

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