1. Behavioral Statistics
Sampling Distributions Chapter 6

2. 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
As a result of this class, you should be able to ...

3. Inferential Statistics9

4. Statistical Methods

5. Inferential Statistics
1. Involves:
Estimation
Hypothesis Testing
2. Purpose
Make Decisions about Population Characteristics
Population?

6. Inference Process

7. Inference Process
Population

8. Inference Process
Population
Sample

9. Inference Process
Population
Sample statistic (X)
Sample

10. Inference Process Estimates & tests Population Sample statistic (X)

11. Estimators 1. Random Variables Used to Estimate a Population Parameter
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 
3. Theoretical Basis Is Sampling Distribution

12. Sampling Distributions9

13. Sampling Distribution
1. Theoretical Probability Distribution
2. Random Variable is Sample Statistic
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

14. 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
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15. Population Characteristics
Summary Measures
Population Distribution
Have students verify these numbers.

16. All Possible Samples of Size n = 2
Sample with replacement

17. All Possible Samples of Size n = 2
16 Sample Means
Sample with replacement

18. Sampling Distribution of All Sample Means

19. Summary Measures of All Sample Means
Have students verify these numbers.

20. Sampling Distribution
Comparison
Population
Sampling Distribution

21. Standard Error of Mean
1. Standard Deviation of All Possible Sample Means,X
Measures Scatter in All Sample Means,X
2. Less Than Pop. Standard Deviation

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

23. Properties of Sampling Distribution of Mean9

24. Properties of Sampling Distribution of Mean
1. Unbiasedness
Mean of Sampling Distribution Equals Population Mean
2. Efficiency
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
An estimator is a random variable used to estimate a population parameter (characteristic).
Unbiasedness
An estimator is unbiased if the mean of its sampling distribution is equal to the population parameter.
Efficiency
The efficiency of an unbiased estimator is measured by the variance of its sampling distribution.
If two estimators, with the same sample size, are both unbiased, then the one with the smaller variance has greater relative efficiency.
Consistency
An estimator is a consistent estimator of a population parameter if the larger the sample size, the more likely it is that the estimate will come close to the parameter.

25. Unbiasedness
Unbiased
Biased 

26. Sampling distribution of mean Sampling distribution of median
Efficiency
Sampling distribution of mean
Sampling distribution of median 

27. Consistency
Larger sample size
Smaller sample size 

28. Sampling from Normal Populations9

29. Sampling from Normal Populations
Central Tendency
Dispersion
Sampling with replacement
Population Distribution
Sampling Distribution
n = 4 X = 5
n =16 X = 2.5

30. Standardizing Sampling Distribution of Mean
Standardized Normal Distribution

31. 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?
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32. Sampling Distribution Solution*
Standardized Normal Distribution
.3830
.1915
.1915

33. Sampling from Non-Normal Populations9

34. Sampling from Non-Normal Populations
Central Tendency
Dispersion
Sampling with replacement
Population Distribution
Sampling Distribution
n = 4 X = 5
n =30 X = 1.8

35. Central Limit Theorem 9

36. Central Limit Theorem

37. Central Limit Theorem
As sample size gets large enough (n  30) ...

38. Central Limit Theorem As sample size gets large enough (n  30) ...
sampling distribution becomes almost normal.

39. Central Limit Theorem As sample size gets large enough (n  30) ...
sampling distribution becomes almost normal.

40. 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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End of Chapter
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