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5.1 Introduction to Normal Distributions AP Statistics Recall: Continuous Random Variable: ~ number of possible values ~represented by an on a number line ~its probability distribution is called a probability distribution NORMAL DISTRIBUTION: ~ probability distribution for random variable, ~Used to model many in nature, business, and industry ~Graph is called PROPERTIES OF NORMAL DISTRIBUTION: 1.) Mean, Median, Mode are equal 2.) Normal curve is bell-shaped and symmetrical about the mean 3.) Total area under the normal curve is 1 4.) Normal curve approaches, but never touches, the x-axis as it extends farther and farther away from the mean 5.) Between 𝜇 − 𝜎 and 𝜇 + 𝜎 (center of curve) graph curves downward Graph curves upward left of 𝜇 − 𝜎 and right of 𝜇 + 𝜎 Points at which the curving changes from downward to upward are called inflection points. ~Graph is a probability density function: 𝑦 = 𝜎 x 2 2 1 𝑒 −(𝑥−𝜇) ⁄2𝜎 2𝜋 √ ~Graphing Calculator! ~Two parameters, 𝜇 and 𝜎 completely determine shape of normal curve ~any mean and any positive standard deviation ~mean gives the location of the line of symmetry ~standard deviation describes the spread of the data Example: Which normal curve has the greatest mean? Which normal curve has the greatest standard deviation? Example: Use the normal curve to estimate the following: Mean Standard Deviation Location of line of symmetry Location of points of inflection Normal Curves and Probability Recall: Example: Adult IQ scores are normally distributed with 𝜇 = 100 and 𝜎 = 15. Estimate the probability that a randomly chosen adult has an IQ between 70 and 115.