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ELEMENTARY Section 5-4 STATISTICS Normal Distributions: Finding Values EIGHTH Chapter 5. Section 5-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman EDITION MARIO F. TRIOLA 1 Cautions to keep in mind 1. Don’t confuse z scores and areas. Z scores are distances away from the mean along the horizontal scale, but areas are regions under the normal curve. 2. Use invNorm() to find a z-score for a given percentage to the left(percentile). 3. A z score must be negative whenever it is located to the left of the centerline of 0. Chapter 5. Section 5-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 2 Finding z Scores when Given Probabilities 95% z = invNorm(.95) 0.95 0 1.64 (z score will be positive) FIGURE 5-11 Finding the 95th Percentile Chapter 5. Section 5-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 3 Finding z Scores when Given Probabilities 80% 20% Bottom 20% 0.80 0.20 -.84 0 z = invNorm(.20) (z score will be negative) Finding the top 80% Chapter 5. Section 5-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 4 Procedure for Finding Values (x) 1. Sketch a normal distribution curve, enter the given probability or percentage in the appropriate region of the graph, and identify the value(s) being sought. 2. x Use invNorm() to find the z score corresponding to the region bounded by x. Cautions: • You must input the percentile (area below) into invNorm() • If the area(%) is above you must enter its complement into invNorm() • If the area(%) is the middle you must enter area(%) in the tail into invNorm(). You will have two opposite z-scores to define the interval. 3. Enter the values for µ, , and the z score found in step 2, then solve for x. x = µ + (z • ) (z-score formula solved for x) 4. Refer to the sketch of the curve to verify that the solution makes sense in the context of the graph and the context of the problem. Chapter 5. Section 5-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 5 Example What weight denotes the 10th percentile of women’s weight? Assume women’s weights are normally distributed with a mean of 143 pounds and standard deviation of 29 pounds. Chapter 5. Section 5-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 6 Finding P10 for Weights of Women 10% .10 x=? 143 Weight FIGURE 5-17 Chapter 5. Section 5-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 7 Finding P10 for Weights of Women z = invNorm(.10) = -1.28 0.10 x=? 143 Weight z -1.28 0 Chapter 5. Section 5-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 8 Finding P10 for Weights of Women x = + (z• ) x = 143 + (-1.28 • 29) = 105.88 0.10 x=? 143 -1.28 0 Chapter 5. Section 5-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman Weight 9 Finding P10 for Weights of Women The weight of 106 lb (rounded) separates the lowest 10% from the highest 90%. 0.10 x = 106 FIGURE 5-17 -1.28 143 Weight 0 Chapter 5. Section 5-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 10 Forgot to make z score negative??? x = 143 + (1.28 • 29) = 180 0.10 x = 180 1.28 143 Weight 0 Chapter 5. Section 5-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 11 Forgot to make z score negative??? UNREASONABLE ANSWER! 0.10 x = 180 1.28 143 Weight 0 Chapter 5. Section 5-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 12 z x x normalcdf(zleft, zright) z x = +z % invNormal(% to Left) Chapter 5. Section 5-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 13