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Business Statistics, 5th ed. by Ken Black Chapter 6 Discrete Distributions Continuous Distributions PowerPoint presentations prepared by Lloyd Jaisingh, Morehead State University Learning Objectives • Understand concepts of the uniform distribution. • Appreciate the importance of the normal distribution. • Recognize normal distribution problems, and know how to solve them. • Decide when to use the normal distribution to approximate binomial distribution problems, and know how to work them. • Decide when to use the exponential distribution to solve problems in business, and know how to work them. Uniform Distribution 1 b a f ( x) 0 for a xb for all other values 1 ba f (x) Area = 1 a x b Uniform Distribution of Lot Weights 1 47 41 f ( x) 0 for for 41 x 47 all other values 1 1 47 41 6 f (x) Area = 1 41 47 x Uniform Distribution Probability P( x1 X x 2 ) x1 ba x 2 45 42 1 P(42 X 45) 47 41 2 45 42 1 47 41 2 f (x) Area = 0.5 41 42 45 47 x Uniform Distribution Mean and Standard Deviation Mean = Mean a + 2 b Standard Deviation ba 12 41 + 47 88 = 44 2 2 Standard Deviation 47 41 6 1. 732 12 3. 464 Uniform Distribution of Assembly of Plastic Modules 1 1 39 27 12 f ( x) 0 for 27 x 39 for all other values 1 1 39 27 12 f ( x) Area = 1 27 39 x Uniform Distribution Mean and Standard Deviation Mean = Mean a + 2 b 39 + 27 33 = 2 Standard Deviation Standard Deviation ba 12 39 27 12 3 .464 3.464 12 Uniform Distribution of Assembly of Plastic Modules P(30 X 35) 35 30 5 39 27 12 0.4167 35 30 0 . 4167 39 27 f (x) 42 27 30 35 39 x 45 Uniform Distribution of Assembly of Plastic Modules P ( X 30 ) 30 27 39 27 3 0 . 2500 12 35 30 0 . 4167 39 27 f (x) 27 30 39 x Normal Distribution • Probably the most widely known and used of all distributions is the normal distribution. • It fits many human characteristics, such as height, weight, length, speed, IQ scores, scholastic achievements, and years of life expectancy, among others. • Many things in nature such as trees, animals, insects, and others have many characteristics that are normally distributed. Normal Distribution • Many variables in business and industry are also normally distributed. For example variables such as the annual cost of household insurance, the cost per square foot of renting warehouse space, and managers’ satisfaction with support from ownership on a five-point scale, amount of fill in soda cans, etc. • Because of the many applications, the normal distribution is an extremely important distribution. Normal Distribution • Discovery of the normal curve of errors is generally credited to mathematician and astronomer Karl Gauss (1777 – 1855), who recognized that the errors of repeated measurement of objects are often normally distributed. • Thus the normal distribution is sometimes referred to as the Gaussian distribution or the normal curve of errors. • In addition, some credit were also given to Pierre-Simon de Laplace (1749 – 1827) and Abraham de Moivre (1667 – 1754) for the discovery of the normal distribution. Properties of the Normal Distribution • The normal distribution exhibits the following characteristics: • It is a continuous distribution. • It is symmetric about the mean. • It is asymptotic to the horizontal axis. • It is unimodal. • It is a family of curves. • Area under the curve is 1. • It is bell-shaped. Graphic Representation of the Normal Distribution Probability Density of the Normal Distribution This image cannot currently be display ed. Family of Normal Curves Standardized Normal Distribution • Since there is an infinite number of combinations for and , then we can generate an infinite family of curves. • Because of this, it would be impractical to deal with all of these normal distributions. • Fortunately, a mechanism was developed by which all normal distributions can be converted into a single distribution called the z distribution. • This process yields the standardized normal distribution (or curve). Standardized Normal Distribution • The conversion formula for any x value of a given normal distribution is given below. It is called the z-score. x z • A z-score gives the number of standard deviations that a value x, is above or below the mean. Standardized Normal Distribution • If x is normally distributed with a mean of and a standard deviation of , then the z-score will also be normally distributed with a mean of 0 and a standard deviation of 1. • Since we can convert to the standard normal distribution, tables have been generated for this standard normal distribution which will enable us to determine probabilities for normal variables. • The tables in the text are set up to give the probabilities between z = 0 and some other z value, z0 say, which is depicted on the next slide. Standardized Normal Distribution Z Table Second Decimal Place in Z Z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.00 0.10 0.20 0.30 0.0000 0.0398 0.0793 0.1179 0.0040 0.0438 0.0832 0.1217 0.0080 0.0478 0.0871 0.1255 0.0120 0.0517 0.0910 0.1293 0.0160 0.0557 0.0948 0.1331 0.0199 0.0596 0.0987 0.1368 0.0239 0.0636 0.1026 0.1406 0.0279 0.0675 0.1064 0.1443 0.0319 0.0714 0.1103 0.1480 0.0359 0.0753 0.1141 0.1517 0.90 1.00 1.10 1.20 0.3159 0.3413 0.3643 0.3849 0.3186 0.3438 0.3665 0.3869 0.3212 0.3461 0.3686 0.3888 0.3238 0.3485 0.3708 0.3907 0.3264 0.3508 0.3729 0.3925 0.3289 0.3531 0.3749 0.3944 0.3315 0.3554 0.3770 0.3962 0.3340 0.3577 0.3790 0.3980 0.3365 0.3599 0.3810 0.3997 0.3389 0.3621 0.3830 0.4015 2.00 0.4772 0.4778 0.4783 0.4788 0.4793 0.4798 0.4803 0.4808 0.4812 0.4817 3.00 3.40 3.50 0.4987 0.4997 0.4998 0.4987 0.4997 0.4998 0.4987 0.4997 0.4998 0.4988 0.4997 0.4998 0.4988 0.4997 0.4998 0.4989 0.4997 0.4998 0.4989 0.4997 0.4998 0.4989 0.4997 0.4998 0.4990 0.4997 0.4998 0.4990 0.4998 0.4998 Applying the Z Formula X is normallydistributed with = 485, and = 105 P(485 X 600) P(0 Z 1.10) .3643 For X = 485, X- 485 485 Z= 0 105 Z 0.00 0.01 0.02 0.00 0.10 0.0000 0.0040 0.0080 0.0398 0.0438 0.0478 For X = 600, 1.00 0.3413 0.3438 0.3461 X- 1.10 0.3643 0.3665 0.3686 1.20 0.3849 0.3869 0.3888 600 485 Z= 1.10 105 Applying the Z Formula X is normallydistributed with = 494, and = 100 P( X 550) P(Z 0.56) .7123 For X = 550 X - 550 494 Z= 0.56 100 0.5 + 0.2123 = 0.7123 Applying the Z Formula X is normallydistributed with = 494, and = 100 P( X 700) P(Z 2.06) .0197 For X = 700 X - 700 494 Z= 2.06 100 0.5 – 0.4803 = 0.0197 Applying the Z Formula X is normallydistributed with = 494, and = 100 P(300 X 600) P(1.94 Z 1.06) .8292 For X = 300 X - 300 494 Z= 1.94 100 For X = 600 X - 600 494 Z= 1.06 100 0.4738+ 0.3554 = 0.8292 Demonstration Problem 6.9 • These types of problems can be solved quite easily with the appropriate technology. The output shows the MINITAB solution. Normal Approximation of the Binomial Distribution • The normal distribution can be used to approximate binomial probabilities. • Procedure – Convert binomial parameters to normal parameters. – Does the interval 3 lie between 0 and n? If so, continue; otherwise, do not use the normal approximation. – Correct for continuity. – Solve the normal distribution problem. Normal Approximation of Binomial: Parameter Conversion • Conversion equations n p n pq • Conversion example: Given that X has a binomial distribution , find P( X 25| n 60 and p . 30 ). n p (60 )(. 30 ) 18 n p q (60 )(. 30 )(. 70 ) 3. 55 Normal Approximation of Binomial: Interval Check . ) 18 1065 . 3 18 3(355 3 7.35 . 3 2865 0 10 20 30 40 50 60 n 70 Graph of the Binomial Problem: n = 60, p = 0.3 0.12 0.10 P(x) 0.08 0.06 0.04 0.02 0.00 10 15 20 x 25 30 Normal Approximation of Binomial: Correcting for Continuity Values Being Determined Correction X X X X X X +.50 -.50 -.50 +.05 -.50 and +.50 +.50 and -.50 The binomial probability, P( X 25| n 60 and p . 30) is approximated by the normal probabilit P(X 24.5| 18 and 3. 55). Normal Approximation of Binomial: Computations X P(X) 25 26 27 28 29 30 31 32 33 Total 0.0167 0.0096 0.0052 0.0026 0.0012 0.0005 0.0002 0.0001 0.0000 0.0361 The normal approximation, . ) P(X 24.5| 18 and 355 24.5 18 P Z . 355 . ) P( Z 183 . .5 P 0 Z 183 .5.4664 .0336 Exponential Distribution • • • • • • • Continuous Family of distributions Skewed to the right X varies from 0 to infinity Apex is always at X = 0 Steadily decreases as X gets larger Probability function X f ( X) e for X 0, 0 Different Exponential Distributions Exponential Distribution: Probability Computation 1.2 1.0 0.8 X 0 P X X 0 e (12 . )(2) P X 2| 12 . e .0907 0.6 0.4 0.2 0.0 0 1 2 3 4 5 Copyright 2008 John Wiley & Sons, Inc. All rights reserved. Reproduction or translation of this work beyond that permitted in section 117 of the 1976 United States Copyright Act without express permission of the copyright owner is unlawful. 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