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Normal Distribution Def'n: NORMAL DISTRIBUTION is one type of distribution of data, which describes how data clusters around the mean. The graph of a Normal Distribution is bell-shaped and symmetric, with the peak at the mean. Oct 283:17 PM The EMPIRICAL RULE (also known as the 68 - 95 - 99.7 rule) applies to all Normal Distributions. It dictates the following: - 68% of all observations fall within one standard deviation of the mean - 95% of all observations fall within two standard deviations of the mean - 99.7% of all observations fall within three standard deviations of the mean Oct 283:19 PM 1 The Normal Distribution Curve: Oct 283:20 PM Oct 283:35 PM 2 Ex #1 A normal distribution has a mean of 27 and a standard deviation of 5. Find the probability that a randomly selected x-value from the distribution is in the interval of 17 and 37. Oct 283:21 PM Ex #2 A normal distribution has a mean of x and a standard deviation of σ. Find the probability that a randomly selected x-value from the distribution is in the interval x < x - 2 σ. Oct 283:26 PM 3 Ex #3 The distribution of heights of young women aged 18 to 24 is approximately normal with a mean ( x ) of 64.5 inches and a standard deviation ( σ ) of 2.5 inches. a) Draw a Normal Distribution curve to represent this data, clearly showing the application of the empirical rule. Oct 283:26 PM b) What percent of woman are taller than 69.5 inches? c) Between what heights do the middle 95% of women fall? d) What percent of woman are shorter than 62 inches? e) A height of 67 inches corresponds to what percentile of adult female American heights? Oct 2810:13 PM 4 Ex #4 Scores for a professional exam are normally distributed with a mean ( x ) of 650 and a st dev (σ) of 100. a) Draw a Normal Distribution curve for the problem. Oct 283:35 PM b) What is the probability that a randomly selected test score is between 450 and 850? c) Out of 1000 randomly selected test scores, how many would you expect to be between 450 and 850? d) Out of 2300 randomly selected test scores, how many would you expect to be between 650 and 950? Oct 2810:19 PM 5 Def'n: An OUTLIER is any value that is more than 2 standard deviations away from the mean Oct 283:52 PM Ex #5 Jesse is the manager of a guitar shop. He recorded the number of guitars sold each week for a period of 10 weeks. His data is shown in the following table: week # sold 1 12 2 15 3 20 4 8 5 15 6 18 7 17 8 21 9 10 10 24 a) What is the mean of Jesse's data? b) What is the standard deviation of Jesse's data? c) Which, if any, values in Jesse's data would be considered an outlier? Oct 283:37 PM 6 Jan 191:33 PM HOMEWORK Oct 288:58 PM 7