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

© 1999 Prentice-Hall, Inc. Chap 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 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 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) 

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

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

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

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

© 1999 Prentice-Hall, Inc. Chap ZZ Z = 0.12 Z The Standardized Normal Distribution Standardized Normal Probability Table (Portion)  = 0 and  = 1 Probabilities Shaded Area Exaggerated

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

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

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

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

© 1999 Prentice-Hall, Inc. Chap 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) =

© 1999 Prentice-Hall, Inc. Chap 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! Z X

© 1999 Prentice-Hall, Inc. Chap Normal Probability Plots Left-SkewedRight-Skewed RectangularU-Shaped Z X Z X Z X Z X

© 1999 Prentice-Hall, Inc. Chap Exponential Distributions Parrival time < X()  1 - e e = the mathematical constant 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

© 1999 Prentice-Hall, Inc. Chap 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

© 1999 Prentice-Hall, Inc. Chap 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. _

© 1999 Prentice-Hall, Inc. Chap 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

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

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

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

© 1999 Prentice-Hall, Inc. Chap P(X) X Sample Means Distribution 16 Sample Means Sampling Distribution of All Sample Means # in sample = 2, # in Sampling Distribution = 16 _

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

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

© 1999 Prentice-Hall, Inc. Chap 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 _ _

© 1999 Prentice-Hall, Inc. Chap 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

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

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

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

© 1999 Prentice-Hall, Inc. Chap 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 = _ _

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

© 1999 Prentice-Hall, Inc. Chap 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

© 1999 Prentice-Hall, Inc. Chap Example: Sampling Distribution Sampling Distribution Standardized Normal Distribution  = 0 Z  =

© 1999 Prentice-Hall, Inc. Chap 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

© 1999 Prentice-Hall, Inc. Chap 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 ) psps

© 1999 Prentice-Hall, Inc. Chap 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

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

© 1999 Prentice-Hall, Inc. Chap 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

© 1999 Prentice-Hall, Inc. Chap 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