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Announcements Finite Probability Wednesday, November 16th I MyMathLab 9 is Monday Nov 28 I Problem Set 9 is due Monday Nov 28 Today: Sec. 7.6: The Normal Distribution Find area below a standard normal curve using a table Transform a normally distributed random variable to the standard normal distribution. Next Class: Sec. 7.6: The Normal Distribution II Cherveny Nov 16 Math 1004: Probability Normal Curve A normal curve (a.k.a. “bell curve”) approximates many situations: I The heights of adult men in a population. I The number of heads obtained in 1000 flips of a coin. I The weight of cereal put in a box by a factory machine. A normal curve is symmetric about x = µ and σ controls its width. Cherveny Nov 16 Math 1004: Probability Looking Ahead: Normal Approximations Goal: Sec. 7.7 will use normal curves to approximate probabilities. It turns out the total area under any normal curve is 1. If a random variable is “normally distributed”, the probability of an outcome within a range is the area under the curve on that range. Cherveny Nov 16 Math 1004: Probability Standard Normal Distribution Definition The standard normal curve is normal with µ = 0 and σ = 1. We want to: I Find areas of regions under the standard normal curve. I Find areas for other normal curves by translating the problem to the standard normal curve. Cherveny Nov 16 Math 1004: Probability Finding Areas on the Standard Normal Curve The area under the standard normal curve to the left of a given z-value is given in Appendix A. Example (a) What is the area to the left of z = .7? (b) What is the area to the right of z = −1.5? (c) What is the area between z = .5 and z = 2.2? Answers: (a) .7580 (b) 1 - .0668 = .9332 (c) .9861 - .6915 = 0.2946 Cherveny Nov 16 Math 1004: Probability “68-95-99.7 Rule” The “68-95-99.7” rule says that on any normal curve roughly.. I 68% of the area is within 1 standard deviation of the mean. I 95% of the area is within 2 standard deviations of the mean. I 99.7% of the area is within 3 standard deviations of the mean. Cherveny Nov 16 Math 1004: Probability Actual GRE Question Example The standard deviation on a test was 12 points, and the mean was 70. If the scores fell along a normal distribution and student X scored 95 points, then student X scored higher than approximately what percent of students? (a) 2% (b) 13% (c) 48% (d) 96% (e) 98% Answer: E Cherveny Nov 16 Math 1004: Probability Finding Areas on Normal Curves If X follows a normal curve with mean µ and standard deviation σ, then Z= X −µ σ follows a standard normal curve. Example (a) What is the area to the left of X = 4 on a normal curve with mean 5 and standard deviation 2? (b) The amount of contaminant in a water sample is normally distributed with mean .5 grams and standard deviation .15 grams. The sample is a health violation if more than .7 grams are measured. What is the probability a sample is in violation of health code? Answer: (a) .3085 (b) ≈ .0885 Cherveny Nov 16 Math 1004: Probability Practice 1. What is the area of each shaded region on the standard normal curve? 2. What is the area of each shaded region on these normal curves? 3. If test scores are normal with mean 34 and standard deviation 4, what percent of the class made above a 30? Cherveny Nov 16 Math 1004: Probability Practice Answers 1. (a) .8944 (b) .4013 (c) .2417 2. (a) .9772 (b) .6247 3. 84% (Can you do this with only the 68-95-99.7 Rule?) Cherveny Nov 16 Math 1004: Probability