1 of 26Visit UMT online at Prentice Hall 2003 Chapter 7, STAT125Basic Business Statistics STATISTICS FOR MANAGERS University of Management and Technology 1925 North Lynn Street Arlington, VA Voice: (703) Fax: (703) Website:

2 of 26Visit UMT online at Prentice Hall 2003 Chapter 7, STAT125Basic Business Statistics CHAPTER 7 Sampling Distributions

3 of 26Visit UMT online at Prentice Hall 2003 Chapter 7, STAT125Basic Business Statistics Chapter Topics Sampling Distribution of the Mean The Central Limit Theorem Sampling Distribution of the Proportion Sampling from Finite Population

4 of 26Visit UMT online at Prentice Hall 2003 Chapter 7, STAT125Basic Business Statistics Why Study Sampling Distributions Sample Statistics are Used to Estimate Population Parameters E.g., 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

5 of 26Visit UMT online at Prentice Hall 2003 Chapter 7, STAT125Basic Business Statistics 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

6 of 26Visit UMT online at Prentice Hall 2003 Chapter 7, STAT125Basic Business Statistics 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 A B C D

7 of 26Visit UMT online at Prentice Hall 2003 Chapter 7, STAT125Basic Business Statistics A B C D (18) (20) (22) (24) Uniform Distribution P(X) X Developing Sampling Distributions (continued) Summary Measures for the Population Distribution

8 of 26Visit UMT online at Prentice Hall 2003 Chapter 7, STAT125Basic Business Statistics 16 Samples Taken with Replacement 16 Sample Means (continued) All Possible Samples of Size n=2 Developing Sampling Distributions

9 of 26Visit UMT online at Prentice Hall 2003 Chapter 7, STAT125Basic Business Statistics Developing Sampling Distributions X Sample Means Distribution 16 Sample Means _ (continued) Sampling Distribution of All Sample Means

10 of 26Visit UMT online at Prentice Hall 2003 Chapter 7, STAT125Basic Business Statistics (continued) Summary Measures of Sampling Distribution Developing Sampling Distributions

11 of 26Visit UMT online at Prentice Hall 2003 Chapter 7, STAT125Basic Business Statistics Comparing the Population with Its Sampling Distribution X Sample Means Distribution n = 2 A B C D (18) (20) (22) (24) Population N = 4 X _

12 of 26Visit UMT online at Prentice Hall 2003 Chapter 7, STAT125Basic Business Statistics Properties of Summary Measures I.e., is unbiased Standard Error (Standard Deviation) of the Sampling Distribution is Less Than the Standard Error of Other Unbiased Estimators For Sampling with Replacement or without Replacement from Large or Infinite Populations: As n increases, decreases

13 of 26Visit UMT online at Prentice Hall 2003 Chapter 7, STAT125Basic Business Statistics Unbiasedness ( ) BiasedUnbiased

14 of 26Visit UMT online at Prentice Hall 2003 Chapter 7, STAT125Basic Business Statistics Less Variability Standard Error (Standard Deviation) of the Sampling Distribution is Less Than the Standard Error of Other Unbiased Estimators Sampling Distribution of Median Sampling Distribution of Mean

15 of 26Visit UMT online at Prentice Hall 2003 Chapter 7, STAT125Basic Business Statistics Effect of Large Sample Larger sample size Smaller sample size

16 of 26Visit UMT online at Prentice Hall 2003 Chapter 7, STAT125Basic Business Statistics When the Population is Normal Central Tendency Variation Population Distribution Sampling Distributions

17 of 26Visit UMT online at Prentice Hall 2003 Chapter 7, STAT125Basic Business Statistics When the Population is Not Normal Central Tendency Variation Population Distribution Sampling Distributions

18 of 26Visit UMT online at Prentice Hall 2003 Chapter 7, STAT125Basic Business Statistics Central Limit Theorem As Sample Size Gets Large Enough Sampling Distribution Becomes Almost Normal Regardless of Shape of Population

19 of 26Visit UMT online at Prentice Hall 2003 Chapter 7, STAT125Basic Business Statistics 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

20 of 26Visit UMT online at Prentice Hall 2003 Chapter 7, STAT125Basic Business Statistics Example: Sampling Distribution Standardized Normal Distribution

21 of 26Visit UMT online at Prentice Hall 2003 Chapter 7, STAT125Basic Business Statistics Population Proportions Categorical Variable E.g., Gender, Voted for Bush, College Degree Proportion of Population Having a Characteristic Sample Proportion Provides an Estimate If Two Outcomes, X Has a Binomial Distribution Possess or do not possess characteristic

22 of 26Visit UMT online at Prentice Hall 2003 Chapter 7, STAT125Basic Business Statistics Sampling Distribution of Sample Proportion Approximated by Normal Distribution Mean: Standard error: p = population proportion Sampling Distribution f(p s ) psps

23 of 26Visit UMT online at Prentice Hall 2003 Chapter 7, STAT125Basic Business Statistics Standardizing Sampling Distribution of Proportion Sampling Distribution Standardized Normal Distribution

24 of 26Visit UMT online at Prentice Hall 2003 Chapter 7, STAT125Basic Business Statistics Example: Sampling Distribution Standardized Normal Distribution

25 of 26Visit UMT online at Prentice Hall 2003 Chapter 7, STAT125Basic Business Statistics Sampling from Finite Sample Modify Standard Error if Sample Size (n) is Large Relative to Population Size (N ) Use Finite Population Correction Factor (fpc) Standard Error with FPC

26 of 26Visit UMT online at Prentice Hall 2003 Chapter 7, STAT125Basic Business Statistics 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