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Announcements Tuesday, May 2 is our last class Final exam review Homework 6 2e) will be graded as extra credit The last homework (homework 7) is due on Friday, May 5 Moodle submission Normal Distribution Both the binomial distribution and the Poisson distribution describe discrete numerical values. The normal distribution provides probabilities for continuous numerical variables. Height of adult males (..., 70, 70.1, 70.12, ...) Body temperature (..., 98.6, 98.66, 98.661, ...) ... Bell-shape curve Symmetrical Many things closely follow a Normal Distribution: Height of adult males Body temperature Blood pressure Marks on a test ... We say the data is "normally distributed“ if .. symmetry about the center 50% of values less than the mean and 50% greater than the mean Measures There are two ways to characterize any data distribution. Central tendency: average value Dispersion: how spread out numbers are how far values in the data set are from the average standard deviation The Normal Distribution Assume that the height of U.S. males is normally distributed with an average of 70 inches and with a standard deviation of 3 inches. Generate the normal distribution probabilities for the height of adult males in one-inch units, including at least 3 standard deviations on each side of the mean. Let’s start the spreadsheet! The NORMDIST() Function = NORMDIST(k, μ, σ, cumulative?) where k is the value of the point of interest (65, 71, ...) μ is the average of the distribution (70 inches) σ is the standard deviation of the distribution (3 inches) and 0 or 1 to indicate whether it is actual probability or the cumulative probability Let’s try to answer them: What is the proportion of men who are less than 65 inches (including 65) tall? How about less than 69 (including 69) inches tall? How about greater than 71 inches tall? How about greater than 68 inches tall? What is the proportion of men between 67 and 70 inches tall? Attention: Cumulative probabilities must be used here! Facts about Normal Distribution Approximately 68% of the values lie between +- 1 standard deviation from the average. Approximately 95% of the values lie between +- 2 standard deviation from the average. Approximately 99% of the values lie between +- 3 standard deviation from the average. 68% of values are within 1 standard deviation of the mean \\ 95% are within 2 standard deviations 99.7% are within 3 standard deviations Standard Normal Distribution Average μ=0 Standard deviation σ =1