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Using the Empirical Rule Remember: Normal Distributions These are special density curves. They have the same overall shape Symmetric Single-Peaked Bell-Shaped They are completely described by giving its mean () and its standard deviation (). We abbreviate it N(,) Normal Curves…. •Changing the mean without changing the standard deviation simply moves the curve horizontally. •The Standard deviation controls the spread of a Normal Curve. Standard Deviation It’s the natural measure of spread for Normal distributions. It can be located by eye on a Normal curve. It’s the point at which the curve changes from concave down to concave up. Empirical Rule (68-95-99.7 Rule) In the Normal distribution with mean () and standard deviation (): 1 of ≈ 68% of the observations Within 2 of ≈ 95% of the observations Within 3 of ≈ 99.7% of the observations Within Scores on the Wechsler adult Intelligence Scale (a standard IQ test) for the 20 to 34 age group are approximately Normally distributed. N(110, 25). What percent are between 85 and 135? What percent are below 185? What percent are below 100? Standard Normal Distribution It is the Normal distribution with mean 0 and standard deviation 1. If a variable x has any Normal distribution N(, ), then the standardized variable z x has the standard Normal distribution. A standard Normal table give the area under the curve to the left of z. Find the area to the left of z = 0.21 Using the chart Find P(z<1.23) Find P(z > 2.01) More examples Find P(z< -0.13) Find P(z > -1.72) More examples Find P(-1.56 < z < 1.01) Find P(-2.23 < z < -0.27) Try the following: P(z < 1.39) P(z > -2.15) P(-0.56 < z < 1.81) Find the z-score that correlates with the 20th percentile. For what z-score are 45% of all observations greater than z? Homework Correction to Assignment Sheet Page 131 (43-52)