6 - 1 © 2000 Prentice-Hall, Inc. Statistics for Business and Economics Sampling Distributions Chapter 6

6 - 2 © 2000 Prentice-Hall, Inc. Learning Objectives 1.Describe the Properties of Estimators 2.Explain Sampling Distribution 3.Describe the Relationship between Populations & Sampling Distributions 4.State the Central Limit Theorem 5.Solve Probability Problems Involving Sampling Distributions

6 - 3 © 2000 Prentice-Hall, Inc. Inferential Statistics

6 - 4 © 2000 Prentice-Hall, Inc. Statistical Methods

6 - 5 © 2000 Prentice-Hall, Inc. Inferential Statistics 1.Involves: Estimation Estimation Hypothesis Testing Hypothesis Testing 2.Purpose Make Decisions about Population Characteristics Make Decisions about Population Characteristics Population?

6 - 6 © 2000 Prentice-Hall, Inc. Inference Process

6 - 7 © 2000 Prentice-Hall, Inc. Inference Process Population

6 - 8 © 2000 Prentice-Hall, Inc. Inference Process Population Sample

6 - 9 © 2000 Prentice-Hall, Inc. Inference Process Population Sample Sample statistic (X)

© 2000 Prentice-Hall, Inc. Inference Process Population Sample Sample statistic (X) Estimates & tests

© 2000 Prentice-Hall, Inc. 1.Random Variables Used to Estimate a Population Parameter Sample Mean, Sample Proportion, Sample Median Sample Mean, Sample Proportion, Sample Median 2.Example: Sample Mean  X Is an Estimator of Population Mean  If  X = 3 then 3 Is the Estimate of  If  X = 3 then 3 Is the Estimate of  3.Theoretical Basis Is Sampling Distribution Estimators

© 2000 Prentice-Hall, Inc. Sampling Distributions

© 2000 Prentice-Hall, Inc. 1.Theoretical Probability Distribution 2.Random Variable is Sample Statistic Sample Mean, Sample Proportion etc. Sample Mean, Sample Proportion etc. 3.Results from Drawing All Possible Samples of a Fixed Size 4.List of All Possible [  X, P(  X) ] Pairs Sampling Distribution of Mean Sampling Distribution of Mean Sampling Distribution

© 2000 Prentice-Hall, Inc. Developing Sampling Distributions Suppose There’s a Population... Population Size, N = 4 Random Variable, x, Is # Errors in Work Values of x: 1, 2, 3, 4 Uniform Distribution © T/Maker Co.

© 2000 Prentice-Hall, Inc. Population Characteristics Population Distribution Summary Measures

© 2000 Prentice-Hall, Inc. All Possible Samples of Size n = 2 16 Samples Sample with replacement

© 2000 Prentice-Hall, Inc. All Possible Samples of Size n = 2 16 Samples 16 Sample Means Sample with replacement

© 2000 Prentice-Hall, Inc. Sampling Distribution of All Sample Means 16 Sample Means Sampling Distribution

© 2000 Prentice-Hall, Inc. Summary Measures of All Sample Means

© 2000 Prentice-Hall, Inc. Comparison Population Sampling Distribution

© 2000 Prentice-Hall, Inc. Standard Error of Mean 1.Standard Deviation of All Possible Sample Means,  X Measures Scatter in All Sample Means,  X Measures Scatter in All Sample Means,  X 2.Less Than Pop. Standard Deviation

© 2000 Prentice-Hall, Inc. Standard Error of Mean 1.Standard Deviation of All Possible Sample Means,  X Measures Scatter in All Sample Means,  X Measures Scatter in All Sample Means,  X 2.Less Than Pop. Standard Deviation 3.Formula (Sampling With Replacement)

© 2000 Prentice-Hall, Inc. Properties of Sampling Distribution of Mean

© 2000 Prentice-Hall, Inc. Properties of Sampling Distribution of Mean 1.Unbiasedness Mean of Sampling Distribution Equals Population Mean Mean of Sampling Distribution Equals Population Mean 2.Efficiency Sample Mean Comes Closer to Population Mean Than Any Other Unbiased Estimator Sample Mean Comes Closer to Population Mean Than Any Other Unbiased Estimator 3.Consistency As Sample Size Increases, Variation of Sample Mean from Population Mean Decreases As Sample Size Increases, Variation of Sample Mean from Population Mean Decreases

© 2000 Prentice-Hall, Inc. Unbiasedness  UnbiasedBiased

© 2000 Prentice-Hall, Inc. Efficiency  Sampling distribution of median Sampling distribution of mean

© 2000 Prentice-Hall, Inc. Consistency Smaller sample size Larger sample size 

© 2000 Prentice-Hall, Inc. Sampling from Normal Populations

© 2000 Prentice-Hall, Inc. Sampling from Normal Populations Central Tendency Dispersion Sampling with replacement Central Tendency Dispersion Sampling with replacement Population Distribution Sampling Distribution n =16   X = 2.5 n = 4   X = 5

© 2000 Prentice-Hall, Inc. Standardizing Sampling Distribution of Mean Sampling Distribution Standardized Normal Distribution

© 2000 Prentice-Hall, Inc. Thinking Challenge You’re an operations analyst for AT&T. Long- distance telephone calls are normally distribution with  = 8 min. &  = 2 min. If you select random samples of 25 calls, what percentage of the sample means would be between 7.8 & 8.2 minutes? © T/Maker Co.

© 2000 Prentice-Hall, Inc. Sampling Distribution Solution* Sampling Distribution Standardized Normal Distribution

© 2000 Prentice-Hall, Inc. Sampling from Non-Normal Populations

© 2000 Prentice-Hall, Inc. Sampling from Non-Normal Populations Central Tendency Dispersion Sampling with replacement Sampling with replacement Central Tendency Dispersion Sampling with replacement Sampling with replacement Population Distribution Sampling Distribution n =30   X = 1.8 n = 4   X = 5

© 2000 Prentice-Hall, Inc. Central Limit Theorem

© 2000 Prentice-Hall, Inc. Central Limit Theorem

© 2000 Prentice-Hall, Inc. Central Limit Theorem As sample size gets large enough (n  30)...

© 2000 Prentice-Hall, Inc. Central Limit Theorem As sample size gets large enough (n  30)... sampling distribution becomes almost normal.

© 2000 Prentice-Hall, Inc. Central Limit Theorem As sample size gets large enough (n  30)... sampling distribution becomes almost normal.

© 2000 Prentice-Hall, Inc. Conclusion 1.Described the Properties of Estimators 2.Explained Sampling Distribution 3.Described the Relationship between Populations & Sampling Distributions 4.Stated the Central Limit Theorem 5.Solved Probability Problems Involving Sampling Distributions

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