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STATISTICS FOR MANAGERS University of Management and Technology 1925 North Lynn Street Arlington, VA 22209 Voice: (703) 516-0035 Fax: (703) 516-0985 Website: www.umtweb.edu © Prentice Hall 2003 Basic Business Statistics Visit UMT online at www.umtweb.edu 1 of 26 Chapter 7, STAT125 CHAPTER 7 Sampling Distributions © Prentice Hall 2003 Basic Business Statistics Visit UMT online at www.umtweb.edu 2 of 26 Chapter 7, STAT125 Chapter Topics Sampling Distribution of the Mean The Central Limit Theorem Sampling Distribution of the Proportion Sampling from Finite Population © Prentice Hall 2003 Basic Business Statistics Visit UMT online at www.umtweb.edu 3 of 26 Chapter 7, STAT125 Why Study Sampling Distributions Sample Statistics are Used to Estimate Population Parameters E.g., X 50 estimates the population mean Problem: Different Samples Provide Different Estimates Large sample gives better estimate; large sample costs more How good is the estimate? Approach to Solution: Theoretical Basis is Sampling Distribution © Prentice Hall 2003 Basic Business Statistics Visit UMT online at www.umtweb.edu 4 of 26 Chapter 7, STAT125 Sampling Distribution Theoretical Probability Distribution of a Sample Statistic Sample Statistic is a Random Variable Sample mean, sample proportion Results from Taking All Possible Samples of the Same Size © Prentice Hall 2003 Basic Business Statistics Visit UMT online at www.umtweb.edu 5 of 26 Chapter 7, STAT125 Developing Sampling Distributions Suppose There is a Population … Population Size N=4 Random Variable, X, is Age of Individuals Values of X: 18, 20, 22, 24 Measured in Years B C D A © Prentice Hall 2003 Basic Business Statistics Visit UMT online at www.umtweb.edu 6 of 26 Chapter 7, STAT125 Developing Sampling Distributions (continued) Summary Measures for the Population Distribution N X i 1 P(X) i .3 N 18 20 22 24 21 4 N X i 1 © Prentice Hall 2003 Basic Business Statistics i N .2 .1 0 2 2.236 A B C D (18) (20) (22) (24) X Uniform Distribution Visit UMT online at www.umtweb.edu 7 of 26 Chapter 7, STAT125 Developing Sampling Distributions (continued) All Possible Samples of Size n=2 1st Obs 2nd Observation 18 20 22 24 18 18,18 18,20 18,22 18,24 16 Sample Means 20 20,18 20,20 20,22 20,24 1st 2nd Observation Obs 18 20 22 24 22 22,18 22,20 22,22 22,24 18 18 19 20 21 24 24,18 24,20 24,22 24,24 20 19 20 21 22 22 20 21 22 23 24 21 22 23 24 16 Samples Taken with Replacement © Prentice Hall 2003 Basic Business Statistics Visit UMT online at www.umtweb.edu 8 of 26 Chapter 7, STAT125 Developing Sampling Distributions (continued) Sampling Distribution of All Sample Means Sample Means Distribution 16 Sample Means 1st 2nd Observation Obs 18 20 22 24 18 18 19 20 21 .3 20 19 20 21 22 .2 22 20 21 22 23 .1 24 21 22 23 24 0 © Prentice Hall 2003 Basic Business Statistics P X _ 18 19 20 21 22 23 Visit UMT online at www.umtweb.edu 24 X 9 of 26 Chapter 7, STAT125 Developing Sampling Distributions (continued) Summary Measures of Sampling Distribution N X X i 1 N i 18 19 19 16 N X X i 1 i X © Prentice Hall 2003 Basic Business Statistics 21 2 N 18 21 19 21 2 24 2 24 21 16 Visit UMT online at www.umtweb.edu 2 1.58 10 of 26 Chapter 7, STAT125 Comparing the Population with Its Sampling Distribution Population N=4 21 P X 2.236 Sample Means Distribution n=2 X 21 .3 .3 .2 .2 .1 .1 0 A B C D (18) (20) (22) (24) © Prentice Hall 2003 Basic Business Statistics X 0 P X X 1.58 _ 18 19 20 21 22 23 Visit UMT online at www.umtweb.edu 24 X 11 of 26 Chapter 7, STAT125 Properties of Summary Measures X I.e., X is unbiased Standard Error (Standard Deviation) of the Sampling Distribution X is Less Than the Standard Error of Other Unbiased Estimators For Sampling with Replacement or without Replacement from Large or Infinite Populations: X n As n increases, © Prentice Hall 2003 Basic Business Statistics X decreases Visit UMT online at www.umtweb.edu 12 of 26 Chapter 7, STAT125 Unbiasedness ( X ) f X Unbiased © Prentice Hall 2003 Basic Business Statistics Biased X Visit UMT online at www.umtweb.edu X 13 of 26 Chapter 7, STAT125 Less Variability Standard Error (Standard Deviation) of the Sampling Distribution is Less Than the Standard Error of Other X Unbiased Estimators f X Sampling Distribution of Median © Prentice Hall 2003 Basic Business Statistics Sampling Distribution of Mean Visit UMT online at www.umtweb.edu X 14 of 26 Chapter 7, STAT125 Effect of Large Sample For sampling with replacement: As n increases, X decreases f X Larger sample size Smaller sample size © Prentice Hall 2003 Basic Business Statistics Visit UMT online at www.umtweb.edu X 15 of 26 Chapter 7, STAT125 When the Population is Normal Population Distribution 10 Central Tendency X 50 Variation X © Prentice Hall 2003 Basic Business Statistics n Sampling Distributions n4 X 5 n 16 X 2.5 Visit UMT online at www.umtweb.edu X 50 X 16 of 26 Chapter 7, STAT125 When the Population is Not Normal Population Distribution Central Tendency 10 X 50 Variation X © Prentice Hall 2003 Basic Business Statistics n Sampling Distributions n4 X 5 Visit UMT online at www.umtweb.edu n 30 X 1.8 X 50 X 17 of 26 Chapter 7, STAT125 Central Limit Theorem Sampling Distribution Becomes Almost Normal Regardless of Shape of Population As Sample Size Gets Large Enough © Prentice Hall 2003 Basic Business Statistics Visit UMT online at www.umtweb.edu X 18 of 26 Chapter 7, STAT125 How Large is Large Enough? For Most Distributions, n>30 For Fairly Symmetric Distributions, n>15 For Normal Distribution, the Sampling Distribution of the Mean is Always Normally Distributed Regardless of the Sample Size This is a property of sampling from a normal population distribution and is NOT a result of the central limit theorem © Prentice Hall 2003 Basic Business Statistics Visit UMT online at www.umtweb.edu 19 of 26 Chapter 7, STAT125 Example: 8 =2 n 25 P 7.8 X 8.2 ? 7.8 8 X X 8.2 8 P 7.8 X 8.2 P X 2 / 25 2 / 25 P .5 Z .5 .3830 Sampling Distribution 2 X .4 25 Standardized Normal Distribution Z 1 .1915 0.5 0.5 8.2 X X 8 Visit UMT online at www.umtweb.edu Z 0 © Prentice Hall 2003 7.8 Basic Business Statistics Z 20 of 26 Chapter 7, STAT125 Population Proportions p Categorical Variable E.g., Gender, Voted for Bush, College Degree Proportion of Population Having a Characteristic Sample Proportion Provides an Estimate p X number of successes pS n sample size If Two Outcomes, X Has a Binomial Distribution Possess or do not possess characteristic © Prentice Hall 2003 Basic Business Statistics Visit UMT online at www.umtweb.edu 21 of 26 Chapter 7, STAT125 Sampling Distribution of Sample Proportion Approximated by Normal Distribution np 5 n 1 p 5 Mean: p p S Standard error: p S p 1 p n Sampling Distribution f(ps) .3 .2 .1 0 0 .2 .4 .6 8 1 ps p = population proportion © Prentice Hall 2003 Basic Business Statistics Visit UMT online at www.umtweb.edu 22 of 26 Chapter 7, STAT125 Standardizing Sampling Distribution of Proportion Z pS pS p S p 1 p n Standardized Normal Distribution Sampling Distribution p pS p Z 1 S p © Prentice Hall 2003 Basic Business Statistics S pS Visit UMT online at www.umtweb.edu Z 0 Z 23 of 26 Chapter 7, STAT125 n 200 Example: p .4 P pS .43 ? p .43 .4 S pS P pS .43 P pS .4 1 .4 200 Standardized Normal Distribution Sampling Distribution p P Z .87 .8078 Z 1 S p .43 S © Prentice Hall 2003 Basic Business Statistics pS Visit UMT online at www.umtweb.edu 0 .87 Z 24 of 26 Chapter 7, STAT125 Sampling from Finite Sample Modify Standard Error if Sample Size (n) is Large Relative to Population Size (N ) n .05N or n / N .05 Use Finite Population Correction Factor (fpc) Standard Error with FPC X P S © Prentice Hall 2003 Basic Business Statistics n N n N 1 p 1 p N n n N 1 Visit UMT online at www.umtweb.edu 25 of 26 Chapter 7, STAT125 Chapter Summary Discussed Sampling Distribution of the Sample Mean Described the Central Limit Theorem Discussed Sampling Distribution of the Sample Proportion Described Sampling from Finite Populations © Prentice Hall 2003 Basic Business Statistics Visit UMT online at www.umtweb.edu 26 of 26 Chapter 7, STAT125