Using the Empirical Rule Remember Normal Distributions n

Using the Empirical Rule

Remember: Normal Distributions n These are special density curves. n They have the same overall shape ¨ Symmetric ¨ Single-Peaked ¨ Bell-Shaped n They are completely described by giving its mean ( ) and its standard deviation ( ). n 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 n It’s the natural measure of spread for Normal distributions. n 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) n 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). n What percent are between 85 and 135? n What percent are below 185? n What percent are below 100?

Standard Normal Distribution n It is the Normal distribution with mean 0 and standard deviation 1. n If a variable x has any Normal distribution N( , ), then the standardized variable 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 n Find P(z<1. 23) n Find P(z > 2. 01)

More examples n Find P(z< -0. 13) n Find P(z > -1. 72)

More examples n Find P(-1. 56 < z < 1. 01) n Find P(-2. 23 < z < -0. 27)

Try the following: n P(z < 1. 39) n P(z > -2. 15) n P(-0. 56 < z < 1. 81)

Find the z-score that correlates with the 20 th percentile.

For what z-score are 45% of all observations greater than z?

Homework n Correction to Assignment Sheet ¨ Page 131 (43 -52)