1. 6 - 1 © 1998 Prentice-Hall, Inc. Chapter 6 Sampling Distributions

2. 6 - 2 © 1998 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 a probability problem involving sampling distributions

3. 6 - 3 © 1998 Prentice-Hall, Inc. Inferential Statistics

4. 6 - 4 © 1998 Prentice-Hall, Inc. Types of Statistical Applications

5. 6 - 5 © 1998 Prentice-Hall, Inc. Inferential Statistics 1.Involves Estimation Estimation Hypothesis testing Hypothesis testing

6. 6 - 6 © 1998 Prentice-Hall, Inc. Inferential Statistics 1.Involves Estimation Estimation Hypothesis testing Hypothesis testing Population?

7. 6 - 7 © 1998 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?

8. 6 - 8 © 1998 Prentice-Hall, Inc. Inference Process

9. 6 - 9 © 1998 Prentice-Hall, Inc. Inference Process Population

10. 6 - 10 © 1998 Prentice-Hall, Inc. Inference Process Population Sample

11. 6 - 11 © 1998 Prentice-Hall, Inc. Inference Process Population Sample Sample statistic (X)

12. 6 - 12 © 1998 Prentice-Hall, Inc. Inference Process Population Sample Sample statistic (X) Estimate & test population parameter

13. 6 - 13 © 1998 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

14. 6 - 14 © 1998 Prentice-Hall, Inc. Sampling Distributions

15. 6 - 15 © 1998 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

16. 6 - 16 © 1998 Prentice-Hall, Inc. Developing Sampling Distributions Population size, N = 4 Random variable, x, is # televisions owned Values of x: 1, 2, 3, 4 Equally distributed (p=1/4) © 1984-1994 T/Maker Co. Suppose there’s a population...

17. 6 - 17 © 1998 Prentice-Hall, Inc. Population Characteristics

18. 6 - 18 © 1998 Prentice-Hall, Inc. Population Characteristics Summary Measures

19. 6 - 19 © 1998 Prentice-Hall, Inc. Population Characteristics Summary Measures

20. 6 - 20 © 1998 Prentice-Hall, Inc. Population Characteristics Population Distribution Summary Measures

21. 6 - 21 © 1998 Prentice-Hall, Inc. Let’s Draw All Possible Samples of Size n = 2

22. 6 - 22 © 1998 Prentice-Hall, Inc. Let’s Draw All Possible Samples of Size n = 2 16 Samples Sample with replacement

23. 6 - 23 © 1998 Prentice-Hall, Inc. Let’s Draw All Possible Samples of Size n=2 16 Samples 16 Sample Means Sample with replacement

24. 6 - 24 © 1998 Prentice-Hall, Inc. Sampling Distribution of All Sample Means 16 Sample Means Sampling Distribution

25. 6 - 25 © 1998 Prentice-Hall, Inc. Summary Measures of All Sample Means (n=16)

26. 6 - 26 © 1998 Prentice-Hall, Inc. Comparison of Population & Sampling Distribution

27. 6 - 27 © 1998 Prentice-Hall, Inc. Comparison of Population & Sampling Distribution Population

28. 6 - 28 © 1998 Prentice-Hall, Inc. Comparison of Population & Sampling Distribution Population Sampling Distribution

29. 6 - 29 © 1998 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

30. 6 - 30 © 1998 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)

31. 6 - 31 © 1998 Prentice-Hall, Inc. Properties of Sampling Distribution of Mean

32. 6 - 32 © 1998 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

33. 6 - 33 © 1998 Prentice-Hall, Inc. Unbiasedness  UnbiasedBiased

34. 6 - 34 © 1998 Prentice-Hall, Inc. Efficiency  Sampling distribution of median Sampling distribution of mean

35. 6 - 35 © 1998 Prentice-Hall, Inc. Consistency Smaller sample size Larger sample size 

36. 6 - 36 © 1998 Prentice-Hall, Inc. Sampling from Normal Populations

37. 6 - 37 © 1998 Prentice-Hall, Inc. Sampling from Normal Populations Central Tendency

38. 6 - 38 © 1998 Prentice-Hall, Inc. Sampling from Normal Populations Central Tendency

39. 6 - 39 © 1998 Prentice-Hall, Inc. Sampling from Normal Populations Central Tendency Dispersion Sampling with replacement Central Tendency Dispersion Sampling with replacement

40. 6 - 40 © 1998 Prentice-Hall, Inc. Sampling from Normal Populations Central Tendency Dispersion Sampling with replacement Central Tendency Dispersion Sampling with replacement Population Distribution

41. 6 - 41 © 1998 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

42. 6 - 42 © 1998 Prentice-Hall, Inc. Standardizing Sampling Distribution of Mean Suppose you want to make probability statements about the sampling distribution...

43. 6 - 43 © 1998 Prentice-Hall, Inc. Standardizing Sampling Distribution of Mean Sampling Distribution

44. 6 - 44 © 1998 Prentice-Hall, Inc. Standardizing Sampling Distribution of Mean Sampling Distribution Standardized Normal Distribution

45. 6 - 45 © 1998 Prentice-Hall, Inc. Standardizing Sampling Distribution of Mean Sampling Distribution Standardized Normal Distribution

46. 6 - 46 © 1998 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? © 1984-1994 T/Maker Co. AloneGroupClass

47. 6 - 47 © 1998 Prentice-Hall, Inc. Sampling Distribution Solution* Sampling Distribution.3830.3830.1915.1915 Standardized Normal Distribution

48. 6 - 48 © 1998 Prentice-Hall, Inc. Sampling from Non-Normal Populations

49. 6 - 49 © 1998 Prentice-Hall, Inc. Sampling from Non-Normal Populations Central Tendency

50. 6 - 50 © 1998 Prentice-Hall, Inc. Sampling from Non-Normal Populations Central Tendency

51. 6 - 51 © 1998 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

52. 6 - 52 © 1998 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

53. 6 - 53 © 1998 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

54. 6 - 54 © 1998 Prentice-Hall, Inc. Central Limit Theorem

55. 6 - 55 © 1998 Prentice-Hall, Inc. Central Limit Theorem

56. 6 - 56 © 1998 Prentice-Hall, Inc. Central Limit Theorem As sample size gets large enough (n  30)...

57. 6 - 57 © 1998 Prentice-Hall, Inc. Central Limit Theorem As sample size gets large enough (n  30)... sampling distribution becomes almost normal.

58. 6 - 58 © 1998 Prentice-Hall, Inc. Central Limit Theorem As sample size gets large enough (n  30)... sampling distribution becomes almost normal.

59. 6 - 59 © 1998 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 a probability problem involving sampling distributions

60. 6 - 60 © 1998 Prentice-Hall, Inc. This Class... 1.What was the most important thing you learned in class today? 2.What do you still have questions about? 3.How can today’s class be improved? Please take a moment to answer the following questions in writing:

61. End of Chapter Any blank slides that follow are blank intentionally.