Normal Distributions and the Empirical Rule Learning Target: I can use percentiles and the Empirical rule to determine relative standing of data on the standard normal curve. Notes: on your own paper

Normal distributions: N (μ, σ)  Symmetric, single peaked and bell shaped.  Center of the curve are μ, M d & Mode.  Standard deviation σ controls the spread (variability) of the curve.

Normal curves are a good description of some real data:  test scores  biological measurements  also approximate chance outcomes like tossing coins

The Empirical rule ( rule) In the normal dist. with mean μ and standard deviation σ.  68% of the observations fall within of the mean.  95% of the observations fall within of the mean.  99.7% of the observations fall within of the mean. 1σ1σ 2σ2σ 3σ3σ

Normal distributions: N (μ, σ)  Inflection points: points where change of curvature takes place is located a distance σ on either side of μ.

Create a Normal Distribution Curve, N (μ, σ)

99.7% 95% 9 68%

Example: Men’s Heights The distribution of adult American men is approximately normal with mean 69 inches and standard deviation 2.5 inches. Draw the curve and mark points if inflection N(69, 2.5) Inflection Points

Example: Men’s Heights The distribution of adult American men is approximately normal with mean 69 inches and standard deviation 2.5 inches. a) What percent of men are taller than 74inches? a) What percent of men are taller than 74inches? =.025 = 2.5%

Example: Men’s Heights The distribution of adult American men is approximately normal with mean 69 inches and standard deviation 2.5 inches. b) Between what heights do the middle 95% of men fall? b) Between what heights do the middle 95% of men fall? %

Example: Men’s Heights The distribution of adult American men is approximately normal with mean 69 inches and standard deviation 2.5 inches. c) What percent of men are shorter than 66.5 inches ? c) What percent of men are shorter than 66.5 inches ? =.16 = 16%

Homework: Textbook: Page 89, problems 2.8 and 2.9 Page 89, problems 2.8 and 2.9 For each problem,  Draw the normal distribution curve  Mark the points of inflection  Label the x-values for each standard deviation  Answ er all parts of each question