Statistics for Managers Using Microsoft Excel/SPSS Chapter 6 The Normal Distribution And Other Continuous Distributions 1999 Prentice-Hall, Inc. Chap. 6-1
Chapter Topics The Normal Distribution The Standard Normal Distribution Assessing the Normality Assumption The Exponential Distribution Sampling Distribution of the Mean Sampling Distribution of the Proportion Sampling From Finite Populations 1999 Prentice-Hall, Inc. Chap. 6-2
Continuous Probability Distributions Continuous Random Variable: Values from Interval of Numbers Absence of Gaps Continuous Probability Distribution: Distribution of a Continuous Variable Most Important Continuous Probability Distribution: the Normal Distribution 1999 Prentice-Hall, Inc. Chap. 6-3
The Normal Distribution Bell Shaped Symmetrical Mean, Median and Mode are Equal Middle Spread Equals 1.33 s Random Variable has Infinite Range f() m Mean Median Mode 1999 Prentice-Hall, Inc. Chap. 6-4
The Mathematical Model f() = 1 2ps e (-1/2) ((- m)/s) 2 f() = frequency of random variable p = 3.14159; e = 2.71828 s = population standard deviation = value of random variable (- < < ) m = population mean 1999 Prentice-Hall, Inc. Chap. 6-5
Many Normal Distributions There are an Infinite Number Varying the Parameters s and m, we obtain Different Normal Distributions. 1999 Prentice-Hall, Inc. Chap. 6-6
Normal Distribution: Finding Probabilities Probability is the area under the curve! P ( c d ) =? f() c d 1999 Prentice-Hall, Inc. Chap. 6-7
Which Table? Each distribution has its own table? Infinitely Many Normal Distributions Means Infinitely Many Tables to Look Up! 1999 Prentice-Hall, Inc. Chap. 6-8
The Standardized Normal Distribution Standardized Normal Probability Table (Portion) m = 0 and s = 1 Z.00.01.02 0.0.0000.0040.0080 0.1.0398.0438.0478 Z Z.0478 0.2.0793.0832.0871 0.3.0179.0217.0255 Probabilities Z = 0.12 Shaded Area Exaggerated 1999 Prentice-Hall, Inc. Chap. 6-9
Normal Distribution s = 10 Standardizing Example m 6. 2 5 Z = = = s 10 0. 12 Standardized Normal Distribution s Z = 1 m = 5 6.2 m = 0.12 Z Shaded Area Exaggerated 1999 Prentice-Hall, Inc. Chap. 6-10
Example: P(2.9 < < 7.1) =.1664 Normal Distribution s = 10 x m z = = s 2. 9 10 5 =. 21 x m 7. 1 5 z = = =. 21 Standardized s 10 Normal Distribution s = 1.1664 Shaded Area Exaggerated.0832.0832 2.9 5 7.1 -.21 0.21 Z 1999 Prentice-Hall, Inc. Chap. 6-11
Example: P( 8) =.3821 Normal Distribution s = 10 x m 8 5 z = = = s 10.. 30 Standardized Normal Distribution.1179 s = 1.5000.3821 m = 5 8 m = 0.30 Z Shaded Area Exaggerated 1999 Prentice-Hall, Inc. Chap. 6-12
Finding Z Values for Known Probabilities What Is Z Given P(Z) = 0.1217? Standardized Normal Probability Table (Portion).1217 s = 1.01 Z.00 0.2 0.0.0000.0040.0080 0.1.0398.0438.0478 m = 0.31 Z 0.2.0793.0832.0871 Shaded Area Exaggerated 0.3.1179.1217.1255 1999 Prentice-Hall, Inc. Chap. 6-13
Finding Values for Known Probabilities Normal Distribution s = 10 Standardized Normal Distribution s = 1.1217.1217 m = 5? m = 0.31 Z = m + Zs = 5 + (0.31)(10) = 8.1 Shaded Area Exaggerated 1999 Prentice-Hall, Inc. Chap. 6-14
Assessing Normality Compare Data Characteristics to Properties of Normal Distribution Put Data into Ordered Array Find Corresponding Standard Normal Quantile Values Plot Pairs of Points Assess by Line Shape Normal Probability Plot for Normal Distribution 90 60 30-2 -1 0 1 2 Look for Straight Line! Z 1999 Prentice-Hall, Inc. Chap. 6-15
Normal Probability Plots Left-Skewed Right-Skewed 90 90 60 60 30-2 -1 0 1 2 Z 30-2 -1 0 1 2 Z Rectangular U-Shaped 90 90 60 60 30-2 -1 0 1 2 Z 30-2 -1 0 1 2 Z 1999 Prentice-Hall, Inc. Chap. 6-16
Exponential Distributions P ( arrival time < ) = 1 - e -l x e = the mathematical constant 2.71828 l = the population mean of arrivals = any value of the continuous random variable e.g. Drivers Arriving at a Toll Bridge Customers Arriving at an ATM Machine 1999 Prentice-Hall, Inc. Chap. 6-17
Exponential Distributions Describes time or distance between events Used for queues f() l = 0.5 l = 2.0 Density function f(x) = 1 l Parameters e -x/l m = l, s = l 1999 Prentice-Hall, Inc. Chap. 6-18
Estimation Sample Statistic Estimates Population Parameter _ e.g. = 50 estimates Population Mean, m Problems: Many samples provide many estimates of the Population Parameter. Determining adequate sample size: large sample give better estimates. Large samples more costly. How good is the estimate? Approach to Solution: Theoretical Basis is Sampling Distribution. 1999 Prentice-Hall, Inc. Chap. 6-19
Sampling Distributions Theoretical Probability Distribution Random Variable is Sample Statistic: Sample Mean, Sample Proportion Results from taking All Possible Samples of the Same Size Comparing Size of Population and Size of Sampling Distribution Population Size = 100 Size of Samples = 10 Sampling Distribution Size = 1.73 10 13 (Sampling Without Replacement) 1999 Prentice-Hall, Inc. Chap. 6-20
Developing Sampling Distributions Suppose there s a population... B C Population size, N = 4 Random variable,, is Age of individuals Values of : 18, 20, 22, 24 measured in years A D 1984-1994 T/Maker Co. 1999 Prentice-Hall, Inc. Chap. 6-21
Population Characteristics Summary Measure Population Distribution m = N i = 1 N i P().3 s = = 18 + 20 + 22 + 4 N i = 1 m ) i N 2 24 = 21 = 2. 236.2.1 0 A B C D (18) (20) (22) (24) Uniform Distribution 1999 Prentice-Hall, Inc. Chap. 6-22
All Possible Samples of Size n = 2 1 st 2 nd Observation Obs 18 20 22 24 18 18,18 18,20 18,22 18,24 20 20,18 20,20 20,22 20,24 22 22,18 22,20 22,22 22,24 24 24,18 24,20 24,22 24,24 16 Samples Samples Taken with Replacement 16 Sample Means 1st 2nd Observation Obs 18 20 22 24 18 18 19 20 21 20 19 20 21 22 22 20 21 22 23 24 21 22 23 24 1999 Prentice-Hall, Inc. Chap. 6-23
Sampling Distribution of All Sample Means 16 Sample Means 1st 2nd Observation Obs 18 20 22 24 18 18 19 20 21.3 P() Sample Means Distribution 20 19 20 21 22.2 22 20 21 22 23 24 21 22 23 24.1 0 18 19 20 21 22 23 24 _ # in sample = 2, # in Sampling Distribution = 16 1999 Prentice-Hall, Inc. Chap. 6-24
Summary Measures for the Sampling Distribution m x N 18+ 19+ 19+ + 24 = = i i = N 16 1 = 21 s x N i 1 = = = m ) i N 2 x 2 18 21) + 19 21) + + 24 21) 16 2 2 = 1. 58 1999 Prentice-Hall, Inc. Chap. 6-25
Comparing the Population with its Sampling Distribution P().3.2 Population N = 4 m = 21, s = 2.236 Sample Means Distribution n = 2 P().3.2 m = 21 s x = 1. 58 x.1 0 A B C D.1 0 18 19 20 21 22 23 24 _ (18) (20) (22) (24) 1999 Prentice-Hall, Inc. Chap. 6-26
Properties of Summary Population Mean Equal to Sampling Mean Measures m = m x The Standard Error (standard deviation) of the Sampling distribution is Less than Population Standard Deviation Formula (sampling with replacement): s _ x = s n As n increase, s _ x decrease. 1999 Prentice-Hall, Inc. Chap. 6-27
Properties of the Mean Unbiasedness Mean of sampling distribution equals population mean Efficiency Sample mean comes closer to population mean than any other unbiased estimator Consistency As sample size increases, variation of sample mean from population mean decreases 1999 Prentice-Hall, Inc. Chap. 6-28
Unbiasedness P() Unbiased Biased m 1999 Prentice-Hall, Inc. Chap. 6-29
Efficiency P() Sampling Distribution of Median Sampling Distribution of Mean m 1999 Prentice-Hall, Inc. Chap. 6-30
Consistency P() Smaller sample size A B Larger sample size m 1999 Prentice-Hall, Inc. Chap. 6-31
When the Population is Normal Central Tendency s m _ x = m Variation _ s x = n Sampling with Replacement n = 4 s` = 5 Population Distribution m = 50 s = 10 Sampling Distributions m - = 50 n =16 s` = 2.5 1999 Prentice-Hall, Inc. Chap. 6-32
Central Limit Theorem As Sample Size Gets Large Enough Sampling Distribution Becomes Almost Normal regardless of shape of population 1999 Prentice-Hall, Inc. Chap. 6-33
When The Population is Not Normal Central Tendency m = x s = x m Variation s n Sampling with Replacement Population Distribution m = 50 Sampling Distributions n = 4 s` = 5 s = 10 n =30 s` = 1.8 1999 Prentice-Hall, Inc. Chap. 6-34 m =50
Example: Sampling Distribution Sampling Distribution s =.4 Z m = = s / n 7. 8 8 2 / 25 m 8. 2 8 Z = = = s / n 2 / 25 =. 50. 50 Standardized Normal Distribution s = 1.3830.1915.1915 7.8 8 8.2 m = 0 Z 1999 Prentice-Hall, Inc. Chap. 6-35
Population Proportions Categorical variable (e.g., gender) % population having a characteristic If two outcomes, binomial distribution Possess or don t possess characteristic Sample proportion (p s ) P s = = n number of successes sample size 1999 Prentice-Hall, Inc. Chap. 6-36
Sampling Distribution of Proportion Approximated by normal distribution Mean n p 5 n (1 - p) 5 m P = Standard error s P = p p 1 p) n P(p s ).3.2.1 0 Sampling Distribution 0. 2.4.6 8 1 p = population proportion p s 1999 Prentice-Hall, Inc. Chap. 6-37
Standardizing Sampling Distribution of Proportion Sampling Distribution Z @ p s - m p s = p - s p p( 1 n p p ) Standardized Normal Distribution s p s = 1 m p p s m = 0 Z 1999 Prentice-Hall, Inc. Chap. 6-38
Example: Sampling Distribution of Proportion np 5 n( 1 p ) 5 Sampling Distribution s p =.0346 Z @ p s - p p( 1 p) n - =.43 -.40. 40 ( 1. 40) 200 =.87 Standardized Normal Distribution s = 1..3078 m p =.40.43 p s m = 0.87 1999 Prentice-Hall, Inc. Chap. 6-39 Z
Sampling from Finite Populations Modify Standard Error if Sample Size (n) is Large Relative to Population Size (N) n >.05 N (or n/n >.05) Use Finite Population Correction Factor (fpc) Standard errors if n/n >.05: s x = s n N N n 1 s P = p 1 p) N n) n N 1) 1999 Prentice-Hall, Inc. Chap. 6-40
Chapter Summary Discussed The Normal Distribution Described The Standard Normal Distribution Assessed the Normality Assumption Defined The Exponential Distribution Discussed Sampling Distribution of the Mean Described Sampling Distribution of the Proportion Defined Sampling From Finite Populations 1999 Prentice-Hall, Inc. Chap. 6-41