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AP Statistics Empirical Rule What does a population that is normally distributed look like? = 80 and = 10 X 50 60 70 80 90 100 110 Normal Distribution Features • Symmetric, single-peaked, and bell-shaped. • Reported: N , • Changing µ changes the location of the curve on the horizontal axis with no change in spread. • Changing changes the spread of the curve. • is located at the inflection point of the curve. Empirical Rule 68% 95% 99.7% 68-95-99.7% RULE Empirical Rule—restated 68% of the observations fall within 1 standard deviation of the mean in either direction. 95% of the observations fall within 2 standard deviations of the mean in either direction. 99.7% of the observations fall within 3 standard deviations of the mean in either direction. Remember values in a data set must appear to be a normal bell-shaped histogram, dotplot, or stemplot to use the Empirical Rule! Empirical Rule 34% 34% 68% 47.5% 47.5% 95% 49.85% 49.85% 99.7% Average American adult male height is 69 inches (5’ 9”) tall with a standard deviation of 2.5 inches. Empirical Rule-- Let H~N(69, 2.5) What is the likelihood that a randomly selected adult male would have a height less than 69 inches? Answer: P(h < 69) = .50 P represents Probability h represents one adult male height Using the Empirical Rule Let H~N(69, 2.5) What is the likelihood that a randomly selected adult male will have a height between 64 and 74 inches? P(64 < h < 74) = .95 Using Empirical Rule-- Let H~N(69, 2.5) What is the likelihood that a randomly selected adult male would have a height of less than 66.5 inches? P(h < 66.5) = 1 – (.50 + .34) = .16 OR .50 - .34 = .16 Using Empirical Rule--Let H~N(69, 2.5) What is the likelihood that a randomly selected adult male would have a height of greater than 74 inches? P(h > 74) = 1 – P(h 74) = 1 – (.50 + .47.5) = 1 - .975 = .025 Using Empirical Rule--Let H~N(69, 2.5) What is the probability that a randomly selected adult male would have a height between 64 and 76.5 inches? P(64 < h < 76.5) = .475 + .4985 = .9735 Empirical Rule More about Normal Curves • Percentiles divide the area of the density curve into fractions out of 100. • Median: 50th percentile. • Q1: 25th percentile. • Q3: 75th percentile. Why are Normal Curves Important? • They provide good descriptions of some distributions of real data (SAT, ACT, characteristics of biological populations). • They are a good approximation of the results of chance outcomes (sum the two die rolls). • Many statistical procedures are based on normal distributions. However! • Many distributions are not normal (income). Don’t assume every situation is approximated by a normal distribution. • HW: Textbook 2.7, 2.8, 2.13, 2.15