2000 PrenticeHall Inc Statistics for Business and Economics

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

Learning Objectives © 2000 Prentice-Hall, Inc. 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 -2

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

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

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

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

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

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

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

Inference Process © 2000 Prentice-Hall, Inc. Estimates & tests Sample statistic (X ) 6 - 10 Population Sample

Estimators © 2000 Prentice-Hall, Inc. 1. Random Variables Used to Estimate a Population Parameter n Sample Mean, Sample Proportion, Sample Median 2. Example: Sample Mean X Is an Estimator of Population Mean n If X = 3 then 3 Is the Estimate of 3. Theoretical Basis Is Sampling Distribution 6 - 11

© 2000 Prentice-Hall, Inc. Sampling Distributions 6 - 12

Sampling Distribution © 2000 Prentice-Hall, Inc. 1. Theoretical Probability Distribution 2. Random Variable is Sample Statistic n 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 n Sampling Distribution of Mean 6 - 13

© 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 © 1984 -1994 T/Maker Co. 6 - 14

© 2000 Prentice-Hall, Inc. Population Characteristics Summary Measures 6 - 15 Population Distribution

© 2000 Prentice-Hall, Inc. All Possible Samples of Size n = 2 16 Samples Sample with replacement 6 - 16

© 2000 Prentice-Hall, Inc. All Possible Samples of Size n = 2 16 Samples Sample with replacement 6 - 17 16 Sample Means

© 2000 Prentice-Hall, Inc. Sampling Distribution of All Sample Means 16 Sample Means 6 - 18 Sampling Distribution

© 2000 Prentice-Hall, Inc. 6 - 19 Summary Measures of All Sample Means

Comparison © 2000 Prentice-Hall, Inc. Population 6 - 20 Sampling Distribution

Standard Error of Mean © 2000 Prentice-Hall, Inc. 1. Standard Deviation of All Possible Sample Means, X n Measures Scatter in All Sample Means, X 2. Less Than Pop. Standard Deviation 6 - 21

Standard Error of Mean © 2000 Prentice-Hall, Inc. 1. Standard Deviation of All Possible Sample Means, X n Measures Scatter in All Sample Means, X 2. Less Than Pop. Standard Deviation 3. Formula (Sampling With Replacement) 6 - 22

© 2000 Prentice-Hall, Inc. Properties of Sampling Distribution of Mean 6 - 23

Properties of Sampling Distribution of Mean © 2000 Prentice-Hall, Inc. 1. Unbiasedness n Mean of Sampling Distribution Equals Population Mean 2. Efficiency n Sample Mean Comes Closer to Population Mean Than Any Other Unbiased Estimator 3. Consistency n As Sample Size Increases, Variation of Sample Mean from Population Mean Decreases 6 - 24

Unbiasedness © 2000 Prentice-Hall, Inc. Unbiased 6 - 25 Biased

Efficiency © 2000 Prentice-Hall, Inc. Sampling distribution of mean Sampling distribution of median 6 - 26

Consistency © 2000 Prentice-Hall, Inc. Larger sample size Smaller sample size 6 - 27

© 2000 Prentice-Hall, Inc. Sampling from Normal Populations 6 - 28

© 2000 Prentice-Hall, Inc. Sampling from Normal Populations Central Tendency Population Distribution Dispersion Sampling Distribution Sampling with replacement 6 - 29 n=4 X = 5 n =16 X = 2. 5

© 2000 Prentice-Hall, Inc. Standardizing Sampling Distribution of Mean Sampling Distribution 6 - 30 Standardized Normal Distribution

Thinking Challenge © 2000 Prentice-Hall, Inc. You’re an operations analyst for AT&T. Longdistance 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? 6 - 31 © 1984 -1994 T/Maker Co.

© 2000 Prentice-Hall, Inc. Sampling Distribution Solution* Sampling Distribution Standardized Normal Distribution . 3830. 1915 6 - 32

© 2000 Prentice-Hall, Inc. Sampling from Non-Normal Populations 6 - 33

© 2000 Prentice-Hall, Inc. Sampling from Non-Normal Populations Central Tendency Population Distribution Dispersion Sampling Distribution n Sampling with replacement 6 - 34 n=4 X = 5 n =30 X = 1. 8

© 2000 Prentice-Hall, Inc. Central Limit Theorem 6 - 35

Central Limit Theorem © 2000 Prentice-Hall, Inc. 6 - 36

Central Limit Theorem © 2000 Prentice-Hall, Inc. As sample size gets large enough (n 30). . . 6 - 37

Central Limit Theorem © 2000 Prentice-Hall, Inc. As sample size gets large enough (n 30). . . 6 - 38 sampling distribution becomes almost normal.

Central Limit Theorem © 2000 Prentice-Hall, Inc. As sample size gets large enough (n 30). . . 6 - 39 sampling distribution becomes almost normal.

Conclusion © 2000 Prentice-Hall, Inc. 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 6 - 40

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