1. © 1999 Prentice-Hall, Inc. Chap. 6 - 1 Statistics for Managers Using Microsoft Excel Chapter 6 The Normal Distribution And Other Continuous Distributions

2. © 1999 Prentice-Hall, Inc. Chap. 6 - 2 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

3. © 1999 Prentice-Hall, Inc. Chap. 6 - 3 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

4. © 1999 Prentice-Hall, Inc. Chap. 6 - 4 The Normal Distribution ‘Bell Shaped’ Symmetrical Mean, Median and Mode are Equal ‘Middle Spread’ Equals 1.33  Random Variable has Infinite Range Mean Median Mode X f(X) 

5. © 1999 Prentice-Hall, Inc. Chap. 6 - 5 The Mathematical Model f(X)=frequency of random variable X  =3.14159; e = 2.71828  =population standard deviation X=value of random variable (-  < X <  )  =population mean f(X) = 1 e (-1/2) ((X-  ) 2

6. © 1999 Prentice-Hall, Inc. Chap. 6 - 6 Many Normal Distributions Varying the Parameters  and , we obtain Different Normal Distributions. There are an Infinite Number

7. © 1999 Prentice-Hall, Inc. Chap. 6 - 7 Normal Distribution: Finding Probabilities Probability is the area under the curve! c d X f(X)f(X) PcXd( ) ?  

8. © 1999 Prentice-Hall, Inc. Chap. 6 - 8 Which Table? Infinitely Many Normal Distributions Means Infinitely Many Tables to Look Up! Each distribution has its own table?

9. © 1999 Prentice-Hall, Inc. Chap. 6 - 9 ZZ Z = 0.12 Z.00.01 0.0.0000.0040.0080.0398.0438 0.2.0793.0832.0871 0.3.0179.0217.0255 The Standardized Normal Distribution.0478.02 0.1. 0478 Standardized Normal Probability Table (Portion)  = 0 and  = 1 Probabilities Shaded Area Exaggerated

10. © 1999 Prentice-Hall, Inc. Chap. 6 - 10 Z  = 0  Z = 1.12 Standardizing Example Normal Distribution Standardized Normal Distribution X  = 5  = 10 6.2 Shaded Area Exaggerated

11. © 1999 Prentice-Hall, Inc. Chap. 6 - 11 0  = 1 -.21 Z.21 Example: P(2.9 < X < 7.1) =.1664 Normal Distribution.1664.0832 Standardized Normal Distribution Shaded Area Exaggerated 5  = 10 2.97.1X

12. © 1999 Prentice-Hall, Inc. Chap. 6 - 12 Z  = 0  = 1.30 Example: P(X  8) =.3821 Normal Distribution Standardized Normal Distribution.1179.5000.3821 Shaded Area Exaggerated. X  = 5  = 10 8

13. © 1999 Prentice-Hall, Inc. Chap. 6 - 13 Z.000.2 0.0.0000.0040.0080 0.1.0398.0438.0478 0.2.0793.0832.0871.1179.1255 Z  = 0  = 1.31 Finding Z Values for Known Probabilities.1217.01 0.3 Standardized Normal Probability Table (Portion) What Is Z Given P(Z) = 0.1217? Shaded Area Exaggerated.1217

14. © 1999 Prentice-Hall, Inc. Chap. 6 - 14 Z  = 0  = 1.31 X  = 5  = 10 ? Finding X Values for Known Probabilities Normal DistributionStandardized Normal Distribution.1217 Shaded Area Exaggerated X 8.1  Z  = 5 + (0.31)(10) =

15. © 1999 Prentice-Hall, Inc. Chap. 6 - 15 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 Look for Straight Line! 30 60 90 -2012 Z X

16. © 1999 Prentice-Hall, Inc. Chap. 6 - 16 Normal Probability Plots Left-SkewedRight-Skewed RectangularU-Shaped 30 60 90 -2012 Z X 30 60 90 -2012 Z X 30 60 90 -2012 Z X 30 60 90 -2012 Z X

17. © 1999 Prentice-Hall, Inc. Chap. 6 - 17 Exponential Distributions Parrival time < X()  1 - e e = the mathematical constant 2.71828 - x  = the population mean of arrivals X = any value of the continuous random variable e.g. Drivers Arriving at a Toll Bridge Customers Arriving at an ATM Machine

18. © 1999 Prentice-Hall, Inc. Chap. 6 - 18 Describes time or distance between events  Used for queues Density function Parameters Exponential Distributions f(x) = 1 e -x   f(X) X = 0.5 = 2.0

19. © 1999 Prentice-Hall, Inc. Chap. 6 - 19 Estimation Sample Statistic Estimates Population Parameter e.g. X = 50 estimates Population Mean,  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. _

20. © 1999 Prentice-Hall, Inc. Chap. 6 - 20 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 = (Sampling Without Replacement) 1.73  10 13

21. © 1999 Prentice-Hall, Inc. Chap. 6 - 21 Population size, N = 4 Random variable, X, is Age of individuals Values of X: 18, 20, 22, 24 measured in years © 1984-1994 T/Maker Co. Developing Sampling Distributions A B C D Suppose there’s a population...

22. © 1999 Prentice-Hall, Inc. Chap. 6 - 22 Population Characteristics Summary MeasurePopulation Distribution.3.2.1 0 A B C D (18) (20) (22) (24) Uniform Distribution P(X) X

23. © 1999 Prentice-Hall, Inc. Chap. 6 - 23 16 Samples Samples Taken with Replacement 16 Sample Means All Possible Samples of Size n = 2

24. © 1999 Prentice-Hall, Inc. Chap. 6 - 24 18 19 20 21 22 23 24 0.1.2.3 P(X) X Sample Means Distribution 16 Sample Means Sampling Distribution of All Sample Means # in sample = 2, # in Sampling Distribution = 16 _

25. © 1999 Prentice-Hall, Inc. Chap. 6 - 25 Summary Measures for the Sampling Distribution

26. © 1999 Prentice-Hall, Inc. Chap. 6 - 26 18 19 20 21 22 23 24 0.1.2.3 P(X) X Sample Means Distribution n = 2 Comparing the Population with its Sampling Distribution A B C D (18) (20) (22) (24) 0.1.2.3 Population N = 4  = 21,  = 2.236 P(X) X _

27. © 1999 Prentice-Hall, Inc. Chap. 6 - 27 Population Mean Equal to Sampling Mean The Standard Error (standard deviation) of the Sampling distribution is Less than Population Standard Deviation Formula (sampling with replacement): Properties of Summary Measures As n increase, decrease.  x = x _ _

28. © 1999 Prentice-Hall, Inc. Chap. 6 - 28 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 Properties of the Mean

29. © 1999 Prentice-Hall, Inc. Chap. 6 - 29  Unbiasedness BiasedUnbiased P(X) X

30. © 1999 Prentice-Hall, Inc. Chap. 6 - 30  Efficiency Sampling Distribution of Median Sampling Distribution of Mean X P(X)

31. © 1999 Prentice-Hall, Inc. Chap. 6 - 31  Larger sample size Smaller sample size Consistency X P(X) A B

32. © 1999 Prentice-Hall, Inc. Chap. 6 - 32 n =16   X = 2.5 n = 4   X = 5 When the Population is Normal Central Tendency Variation Sampling with Replacement Population Distribution Sampling Distributions  x =  x = _ _

33. © 1999 Prentice-Hall, Inc. Chap. 6 - 33 Central Limit Theorem As Sample Size Gets Large Enough Sampling Distribution Becomes Almost Normal regardless of shape of population

34. © 1999 Prentice-Hall, Inc. Chap. 6 - 34 n =30   X = 1.8 n = 4   X = 5 When The Population is Not Normal Central Tendency Variation Sampling with Replacement Population Distribution Sampling Distributions  = 50  = 10 X

35. © 1999 Prentice-Hall, Inc. Chap. 6 - 35 Example: Sampling Distribution Sampling Distribution Standardized Normal Distribution.1915 7.8 8 8.2  = 0 Z  = 1.3830.1915

36. © 1999 Prentice-Hall, Inc. Chap. 6 - 36 Categorical variable (e.g., gender) % population having a characteristic If two outcomes, binomial distribution  Possess or don’t possess characteristic Sample proportion (p s ) Population Proportions

37. © 1999 Prentice-Hall, Inc. Chap. 6 - 37 Approximated by normal distribution  n·p  5  n·(1 - p)  5 Mean Standard error Sampling Distribution of Proportion p = population proportion Sampling Distribution P(p s ).3.2.1 0 0. 2.4.6 8 1 psps

38. © 1999 Prentice-Hall, Inc. Chap. 6 - 38 Standardizing Sampling Distribution of Proportion Sampling Distribution Standardized Normal Distribution Z p p s  s -  p  p = p- psps Z  = 0 pp  p  = 1

39. © 1999 Prentice-Hall, Inc. Chap. 6 - 39 Example: Sampling Distribution of Proportion Sampling Distribution Normal Distribution Standardized Z  p s - - p =.43 -.40 =.87  p =.0346 psps  = 1  = 0.87 Z..3078  p =.40.43

40. © 1999 Prentice-Hall, Inc. Chap. 6 - 40 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: Sampling from Finite Populations

41. © 1999 Prentice-Hall, Inc. Chap. 6 - 41 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