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BA 201 Lecture 9 The Normal Distribution © 2001 Prentice-Hall, Inc. Chap 6-1 Topics The Normal Distribution The Standardized Normal Distribution © 2001 Prentice-Hall, Inc. Chap 6-2 Continuous Probability Distributions Continuous Random Variable Values from interval of numbers Absence of gaps Continuous Probability Distribution p.198 Distribution of continuous random variable Most Important Continuous Probability Distribution The normal distribution © 2001 Prentice-Hall, Inc. Chap 6-3 p.199 The Normal Distribution “Bell Shaped” Symmetrical Mean, Median and Mode are Equal Interquartile Range Equals 1.33 s Random Variable has Infinite Range © 2001 Prentice-Hall, Inc. f(X) X Mean Median Mode Chap 6-4 p.200 The Mathematical Model (Called the Probability Density Function) f X 1 2s 2 X e 1 2s X2 X X 2 f X : density of random variable X 3.14159; e 2.71828 X : population mean s X : population standard deviation X : value of random variable X © 2001 Prentice-Hall, Inc. Chap 6-5 p.201 Many Normal Distributions There are an Infinite Number of Normal Distributions Varying the Parameters sX and X, We Obtain Different Normal Distributions © 2001 Prentice-Hall, Inc. Chap 6-6 p.201 The Standardized Normal Distribution Given X N X ,s 2 X , Z X X has a sX standardized (normalized) normal distribution where Z f(Z) N 0,1 sX f(X) sZ 1 X © 2001 Prentice-Hall, Inc. Z 0 X Z Chap 6-7 pp.203-212 Finding Probabilities Probability is the area under the curve! P c X d ? f(X) a © 2001 Prentice-Hall, Inc. b c d X Chap 6-8 Example: pp.203-212 P 2.9 X 7.1 .1664 Z X X sX 2.9 5 .21 10 Z X X sX 7.1 5 .21 10 Standardized Normal Distribution Normal Distribution s X 10 .0832 sZ 1 .0832 2.9 X 5 © 2001 Prentice-Hall, Inc. 7.1 X 0.21 Shaded Area Exaggerated Z 0 0.21 Z Chap 6-9 Example: pp.203-212 P 2.9 X 7.1 .1664(continued) Cumulative Standardized Normal Distribution Table (Portion) Z .00 .01 Z 0 sZ 1 .02 .5832 0.0 .5000 .5040 .5080 Shaded Area Exaggerated 0.1 .5398 .5438 .5478 0.2 .5793 .5832 .5871 0.3 .6179 .6217 .6255 © 2001 Prentice-Hall, Inc. 0 Z = 0.21 Chap 6-10 Example: pp.203-212 P 2.9 X 7.1 .1664(continued) Cumulative Standardized Normal Distribution Table (Portion) Z .00 .01 .02 Z 0 sZ 1 .4168 -0.3 .3821 .3783 .3745 Shaded Area Exaggerated -0.2 .4207 .4168 .4129 -0.1 .4602 .4562 .4522 0.0 .5000 .4960 .4920 © 2001 Prentice-Hall, Inc. 0 Z = -0.21 Chap 6-11 p.247 Normal Distribution in PHStat PHStat | Probability & Prob. Distributions | Normal … Example in Excel Spreadsheet © 2001 Prentice-Hall, Inc. Chap 6-12 Summary Discussed the Normal Distribution Described the Standard Normal Distribution © 2001 Prentice-Hall, Inc. Chap 6-13