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IE 440 PROCESS IMPROVEMENT THROUGH PLANNED EXPERIMENTATION The Normal Distribution and Other Continuous Distributions Dr. Xueping Li University of Tennessee © 2003 Prentice-Hall, Inc. Chap 6-1 Chapter Topics The Normal Distribution The Standardized Normal Distribution Evaluating the Normality Assumption The Uniform Distribution The Exponential Distribution © 2003 Prentice-Hall, Inc. Chap 6-2 Continuous Probability Distributions Continuous Random Variable Continuous Probability Distribution Values from interval of numbers Absence of gaps Distribution of continuous random variable Most Important Continuous Probability Distribution The normal distribution © 2003 Prentice-Hall, Inc. Chap 6-3 The Normal Distribution “Bell Shaped” Symmetrical Mean, Median and Mode are Equal Interquartile Range Equals 1.33 s Random Variable Has Infinite Range © 2003 Prentice-Hall, Inc. f(X) X Mean Median Mode Chap 6-4 The Mathematical Model 2 1 (1/ 2) X / s f X e 2s f X : density of random variable X 3.14159; e 2.71828 : population mean s : population standard deviation X : value of random variable X © 2003 Prentice-Hall, Inc. Chap 6-5 Many Normal Distributions There are an Infinite Number of Normal Distributions Varying the Parameters s and , We Obtain Different Normal Distributions © 2003 Prentice-Hall, Inc. Chap 6-6 The Standardized Normal Distribution When X is normally distributed with a mean and a standard deviation s, Z X follows a s standardized (normalized) normal distribution with a mean 0 and a standard deviation 1. f(Z) s f(X) sZ 1 © 2003 Prentice-Hall, Inc. Z 0 X Z Chap 6-7 Finding Probabilities Probability is the area under the curve! P c X d ? f(X) c © 2003 Prentice-Hall, Inc. d X Chap 6-8 Which Table to Use? Infinitely Many Normal Distributions Means Infinitely Many Tables to Look Up! © 2003 Prentice-Hall, Inc. Chap 6-9 Solution: The Cumulative Standardized Normal Distribution Cumulative Standardized Normal Distribution Table (Portion) Z .00 .01 Z 0 sZ 1 .02 .5478 0.0 .5000 .5040 .5080 0.1 .5398 .5438 .5478 0.2 .5793 .5832 .5871 Probabilities 0.3 .6179 .6217 .6255 © 2003 Prentice-Hall, Inc. 0 Z = 0.12 Only One Table is Needed Chap 6-10 Standardizing Example Z X s Standardized Normal Distribution Normal Distribution s 10 sZ 1 6.2 © 2003 Prentice-Hall, Inc. 6.2 5 0.12 10 5 X 0.12 Z 0 Z Chap 6-11 Example: P 2.9 X 7.1 .1664 Z X s 2.9 5 .21 10 Z X s 7.1 5 .21 10 Standardized Normal Distribution Normal Distribution s 10 .0832 sZ 1 .0832 2.9 7.1 5 © 2003 Prentice-Hall, Inc. X 0.21 0.21 Z 0 Z Chap 6-12 Example: P 2.9 X 7.1 .1664(continued) Cumulative Standardized Normal Distribution Table (Portion) Z .00 .01 Z 0 .02 sZ 1 .5832 0.0 .5000 .5040 .5080 0.1 .5398 .5438 .5478 0.2 .5793 .5832 .5871 0.3 .6179 .6217 .6255 © 2003 Prentice-Hall, Inc. 0 Z = 0.21 Chap 6-13 Example: 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 -0.2 .4207 .4168 .4129 -0.1 .4602 .4562 .4522 0.0 .5000 .4960 .4920 © 2003 Prentice-Hall, Inc. 0 Z = -0.21 Chap 6-14 Normal Distribution in PHStat PHStat | Probability & Prob. Distributions | Normal … Example in Excel Spreadsheet © 2003 Prentice-Hall, Inc. Chap 6-15 Example: P X 8 .3821 Z X s 85 .30 10 Standardized Normal Distribution Normal Distribution s 10 sZ 1 .3821 5 © 2003 Prentice-Hall, Inc. 8 X 0.30 Z 0 Z Chap 6-16 Example: P X 8 .3821 Cumulative Standardized Normal Distribution Table (Portion) Z .00 .01 Z 0 .02 (continued) sZ 1 .6179 0.0 .5000 .5040 .5080 0.1 .5398 .5438 .5478 0.2 .5793 .5832 .5871 0.3 .6179 .6217 .6255 © 2003 Prentice-Hall, Inc. 0 Z = 0.30 Chap 6-17 Finding Z Values for Known Probabilities What is Z Given Probability = 0.6217 ? Z 0 sZ 1 Cumulative Standardized Normal Distribution Table (Portion) Z .00 .01 0.2 0.0 .5000 .5040 .5080 .6217 0.1 .5398 .5438 .5478 0.2 .5793 .5832 .5871 0 Z .31 © 2003 Prentice-Hall, Inc. 0.3 .6179 .6217 .6255 Chap 6-18 Recovering X Values for Known Probabilities Standardized Normal Distribution Normal Distribution s 10 sZ 1 .6179 .3821 5 ? X Z 0 0.30 Z X Zs 5 .3010 8 © 2003 Prentice-Hall, Inc. Chap 6-19 More Examples of Normal Distribution Using PHStat A set of final exam grades was found to be normally distributed with a mean of 73 and a standard deviation of 8. What is the probability of getting a grade no higher than 91 on this exam? X N 73,8 2 Mean Standard Deviation P X 91 ? 73 8 Probability for X <= X Value 91 Z Value 2.25 P(X<=91) 0.9877756 © 2003 Prentice-Hall, Inc. s 8 X 73 91 0 2.25 Z Chap 6-20 More Examples of Normal Distribution Using PHStat (continued) What percentage of students scored between 65 and 89? X N 73,82 P 65 X 89 ? Probability for a Range From X Value 65 To X Value 89 Z Value for 65 -1 Z Value for 89 2 P(X<=65) 0.1587 P(X<=89) 0.9772 P(65<=X<=89) 0.8186 X 65 73 89 -1 0 © 2003 Prentice-Hall, Inc. Z 2 Chap 6-21 More Examples of Normal Distribution Using PHStat (continued) Only 5% of the students taking the test scored higher than what grade? X N 73,8 2 P ? X .05 Find X and Z Given Cum. Pctage. Cumulative Percentage 95.00% Z Value 1.644853 X Value 86.15882 X 73 ? =86.16 0 © 2003 Prentice-Hall, Inc. Z 1.645 Chap 6-22 More Examples of Normal Distribution Using PHStat (continued) The middle 50% of the students scored between what two scores? X N 73,82 P a X b .50 Find X and Z Given Cum. Pctage. Cumulative Percentage 25.00% Z Value -0.67449 X Value 67.60408 Find X and Z Given Cum. Pctage. Cumulative Percentage 75.00% Z Value 0.67449 X Value 78.39592 © 2003 Prentice-Hall, Inc. .25 .25 X 67.6 73 78.4 -0.67 0 Z 0.67 Chap 6-23 Assessing Normality Not All Continuous Random Variables are Normally Distributed It is Important to Evaluate How Well the Data Set Seems to Be Adequately Approximated by a Normal Distribution © 2003 Prentice-Hall, Inc. Chap 6-24 Assessing Normality Construct Charts (continued) For small- or moderate-sized data sets, do the stem-and-leaf display and box-and-whisker plot look symmetric? For large data sets, does the histogram or polygon appear bell-shaped? Compute Descriptive Summary Measures Do the mean, median and mode have similar values? Is the interquartile range approximately 1.33 s? Is the range approximately 6 s? © 2003 Prentice-Hall, Inc. Chap 6-25 Assessing Normality Observe the Distribution of the Data Set (continued) Do approximately between mean Do approximately between mean Do approximately between mean 2/3 of the observations lie 1 standard deviation? 4/5 of the observations lie 1.28 standard deviations? 19/20 of the observations lie 2 standard deviations? Evaluate Normal Probability Plot Do the points lie on or close to a straight line with positive slope? © 2003 Prentice-Hall, Inc. Chap 6-26 Assessing Normality (continued) Normal Probability Plot Arrange Data into Ordered Array Find Corresponding Standardized Normal Quantile Values Plot the Pairs of Points with Observed Data Values on the Vertical Axis and the Standardized Normal Quantile Values on the Horizontal Axis Evaluate the Plot for Evidence of Linearity © 2003 Prentice-Hall, Inc. Chap 6-27 Assessing Normality (continued) Normal Probability Plot for Normal Distribution 90 X 60 Z 30 -2 -1 0 1 2 © 2003 Prentice-Hall, Inc. Look for Straight Line! Chap 6-28 Normal Probability Plot Left-Skewed Right-Skewed 90 90 X 60 X 60 Z 30 -2 -1 0 1 2 -2 -1 0 1 2 Rectangular U-Shaped 90 90 X 60 X 60 Z 30 -2 -1 0 1 2 © 2003 Prentice-Hall, Inc. Z 30 Z 30 -2 -1 0 1 2 Chap 6-29 Obtaining Normal Probability Plot in PHStat PHStat | Probability & Prob. Distributions | Normal Probability Plot Enter the range of the cells that contain the data in the Variable Cell Range window © 2003 Prentice-Hall, Inc. Chap 6-30 The Uniform Distribution Properties: The probability of occurrence of a value is equally likely to occur anywhere in the range between the smallest value a and the largest value b Also called the rectangular distribution © 2003 Prentice-Hall, Inc. a b 2 2 b a 2 s 12 Chap 6-31 The Uniform Distribution (continued) The Probability Density Function 1 f X if a X b b a Application: Selection of random numbers E.g., A wooden wheel is spun on a horizontal surface and allowed to come to rest. What is the probability that a mark on the wheel will point to somewhere between the North and the East? 90 P 0 X 90 0.25 360 © 2003 Prentice-Hall, Inc. Chap 6-32 Exponential Distributions P arrival time X 1 e X X : any value of continuous random variable : the population average number of arrivals per unit of time 1/: average time between arrivals e 2.71828 E.g., Drivers arriving at a toll bridge; customers arriving at an ATM machine © 2003 Prentice-Hall, Inc. Chap 6-33 Exponential Distributions (continued) Describes Time or Distance between Events f(X) Density Function Used for queues f x Parameters © 2003 Prentice-Hall, Inc. 1 e x = 0.5 = 2.0 X s Chap 6-34 Example E.g., Customers arrive at the checkout line of a supermarket at the rate of 30 per hour. What is the probability that the arrival time between consecutive customers will be greater than 5 minutes? 30 X 5 / 60 hours P arrival time >X 1 P arrival time X 1 1 e 30 5/ 60 .0821 © 2003 Prentice-Hall, Inc. Chap 6-35 Exponential Distribution in PHStat PHStat | Probability & Prob. Distributions | Exponential Example in Excel Spreadsheet © 2003 Prentice-Hall, Inc. Chap 6-36 Chapter Summary Discussed the Normal Distribution Described the Standard Normal Distribution Evaluated the Normality Assumption Defined the Uniform Distribution Described the Exponential Distribution © 2003 Prentice-Hall, Inc. Chap 6-37