Lecture3-1-1

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Examples of Normal Distributions 2
Standard Normal Distribution 3

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3 Important Areas Under Curve Empirical Rule for Normal Populations 4
Normal Probabilities 5 How to calculate these normal probabilities?

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An SAT Example Verbal SAT scores for college-bound seniors in 1999 have approximately normal distribution with μ = 505 and σ = 110. What is the proportion of scores below 615? 6
An SAT Example 7 Start by drawing a picture

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An SAT Example 8 Note: 615 is 1 SD above the mean. (615 = 505 + 110)
An SAT Example 9 68.26 % 15.87 % 15.87 % Answer = 15.87% + 68.26% = 84.13% 615 505 395

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A General Method Using z-Scores x : an individual observation from a normal distribution with mean and standard deviation z -score: describes how many SD’s x is above or below the mean 10 x z z -score
Why is z-Score Useful? If X is normally distributed with mean and standard deviation , then the random variable is normally distributed with mean 0 and standard deviation 1 ( standard normal ). So if x is an observed value from normal with mean and SD , then the z -score for x is from standard normal . 11 X Z

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xvs z-score 12
A Key Consequence The area under a normal density (mean , SD ) to the left (or right ) of x = the area under the standard normal density to the left (or right ) of z , where z is the z -score for x 13 x z

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Standard Normal Table ( p.652 ) 14
One Simple Example Find P (0 ≤ Z ≤ 1) Find the area listed in the table corresponding to

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