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Math 10 Chapter 6 Notes: The Normal Distribution Notation: X is a continuous random variable X ~ N(, ) Parameters: is the mean and is the standard deviation Graph is bell-shaped and symmetrical The mean, median, and mode are the same (in theory) Math 10 Chapter 6 Notes: The Normal Distribution Total area under the curve is equal to 1. Probability = Area P(X < x) is the cumulative distribution function or Area to the Left. A change in the standard deviation, , causes the curve to become wider or narrower A change in the mean, , causes the graph to shift Math 10 Chapter 6 Notes: The Standard Normal Distribution A normal (bell-shaped) distribution of standardized values called z-scores. Notation: Z ~ N(0, 1) A z-score is measured in terms of the standard deviation. The formula for the z-score is x z Math 10 Chapter 6 Notes: The Normal Distribution Bell-shaped curve Most values cluster about the mean Area within 4 standard deviations (+ or - 4 ) is 1 Math 10 Chapter 6 Notes: The Normal Distribution Ex. Suppose X ~ N(100, 5). Find the zscore (the standardized score) for x = 95 and for 110. x z x z = 95 – 100 = - 1 5 = 110 – 100 = 2 5 Math 10 Chapter 6 Notes: The Normal Distribution The z-score lets us compare data that are scaled differently. Ex. X~N(5, 6) and Y~N(2, 1) with x = 17 and y = 4; X = Y = weight gain 17 – 5 = 2 6 4–2 = 2 1 Math 10 Chapter 6 Notes: The Standard Normal Distribution Ex. Suppose Z ~ N(0, 1). Draw pictures and find the following. 1. P(-1.28 < Z < 1.28) 2. P(Z < 1.645) 3. P(Z > 1.645) 4. The 90th percentile, k, for Z scores. For 1, 2, 3 use the normal cdf For 4, use the inverse normal Math 10 Chapter 6 Notes: The Normal Distribution Ex: At the beginning of the term, the amount of time a student waits in line at the campus store is normally distributed with a mean of 5 minutes and a standard deviation of 2 minutes. Let X = the amount of time, in minutes, that a student waits in line at the campus store at the beginning of the term. X ~ N(5, 2) where the mean = 5 and the standard deviation = 2. Math 10 Chapter 6 Notes: The Normal Distribution Find the probability that one randomly chosen student waits more than 6 minutes in line at the campus store at the beginning of the term. P(X > 6) = 0.3085. Math 10 Chapter 6 Notes: The Normal Distribution Find the 3rd quartile. The third quartile is equal to the 75th percentile. Let k = the 75th percentile (75th %ile). P(X < k ) = 0.75. The 3rd quartile or 75th percentile is 6.35 minutes (to 2 decimal places). Seventy-five percent of the waiting times are less than 6.35 minutes.