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Normal Distributions

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. for all x Note: e is the mathematical constant, 2.718282.

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Normal Distributions

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  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 normal

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

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

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

  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? normal

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

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

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

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