1. 5 - 1 © 1997 Prentice-Hall, Inc. Importance of Normal Distribution n Describes many random processes or continuous phenomena n Can be used to approximate discrete probability distributions l Binomial l Poisson n Basis for classical statistical inference

2. 5 - 2 © 1997 Prentice-Hall, Inc. Normal Distribution n ‘Bell-shaped’ & symmetrical n Mean, median, mode are equal ‘Middle spread’ is 1.33  ‘Middle spread’ is 1.33  n Random variable has infinite range Mean Median Mode

3. 5 - 3 © 1997 Prentice-Hall, Inc. Standardize the Normal Distribution One table! Normal Distribution Standardized Normal Distribution

4. 5 - 4 © 1997 Prentice-Hall, Inc. Standardizing Example Normal Distribution

5. 5 - 5 © 1997 Prentice-Hall, Inc. Standardizing Example Normal Distribution Standardized Normal Distribution

6. 5 - 6 © 1997 Prentice-Hall, Inc. Obtaining the Probability.0478.0478.02 0.1.0478 Standardized Normal Probability Table (Portion) ProbabilitiesProbabilities Shaded area exaggerated

7. 5 - 7 © 1997 Prentice-Hall, Inc. Example P(3.8  X  5)

8. 5 - 8 © 1997 Prentice-Hall, Inc. Example P(3.8  X  5) Normal Distribution.0478 Standardized Normal Distribution Shaded area exaggerated

9. 5 - 9 © 1997 Prentice-Hall, Inc. Example P(2.9  X  7.1)

10. 5 - 10 © 1997 Prentice-Hall, Inc. Example P(2.9  X  7.1) Normal Distribution.1664.1664.0832.0832 Standardized Normal Distribution Shaded area exaggerated

11. 5 - 11 © 1997 Prentice-Hall, Inc. Example P(X  8)

12. 5 - 12 © 1997 Prentice-Hall, Inc. Example P(X  8) Normal Distribution Standardized Normal Distribution.1179.1179.5000.3821.3821 Shaded area exaggerated

13. 5 - 13 © 1997 Prentice-Hall, Inc. Central Limit Theorem As sample size gets large enough (  30)... sampling distribution becomes almost normal.

14. 5 - 14 © 1997 Prentice-Hall, Inc. Introduction to Estimation

15. 5 - 15 © 1997 Prentice-Hall, Inc. Statistical Methods

16. 5 - 16 © 1997 Prentice-Hall, Inc. Estimation Process Mean, , is unknown Population Random Sample I am 95% confident that  is between 40 & 60. Mean  X = 50 Sample

17. 5 - 17 © 1997 Prentice-Hall, Inc. Population Parameters Are Estimated

18. 5 - 18 © 1997 Prentice-Hall, Inc. Point Estimation n Provides single value l Based on observations from 1 sample n Gives no information about how close value is to the unknown population parameter Example: Sample mean  X = 3 is point estimate of unknown population mean Example: Sample mean  X = 3 is point estimate of unknown population mean

19. 5 - 19 © 1997 Prentice-Hall, Inc. Interval Estimation n Provides range of values l Based on observations from 1 sample n Gives information about closeness to unknown population parameter l Stated in terms of probability n Example: Unknown population mean lies between 50 & 70 with 95% confidence

20. 5 - 20 © 1997 Prentice-Hall, Inc. Key Elements of Interval Estimation Confidence interval Sample statistic (point estimate) Confidence limit (lower) Confidence limit (upper) A probability that the population parameter falls somewhere within the interval.

21. 5 - 21 © 1997 Prentice-Hall, Inc. Confidence Limits for Population Mean Parameter = Statistic ± Error © 1984-1994 T/Maker Co.

22. 5 - 22 © 1997 Prentice-Hall, Inc. Many Samples Have Same Interval 90% Samples  x_ XXXX   X  =  ± Z   x  +1.65   x  -1.65   x

23. 5 - 23 © 1997 Prentice-Hall, Inc. Many Samples Have Same Interval 90% Samples 95% Samples  +1.65   x  x_ XXXX  +1.96   x  -1.65   x  -1.96   x   X  =  ± Z   x

24. 5 - 24 © 1997 Prentice-Hall, Inc. Many Samples Have Same Interval 90% Samples 95% Samples 99% Samples  +1.65   x  +2.58   x  x_ XXXX  +1.96   x  -2.58   x  -1.65   x  -1.96   x   X  =  ± Z   x

25. 5 - 25 © 1997 Prentice-Hall, Inc. n Probability that the unknown population parameter falls within interval Denoted (1 -  Denoted (1 -   is probability that parameter is not within interval  is probability that parameter is not within interval n Typical values are 99%, 95%, 90% Level of Confidence

26. 5 - 26 © 1997 Prentice-Hall, Inc. Intervals & Level of Confidence Sampling Distribution of Mean Large number of intervals Intervals extend from  X - Z   X to  X + Z   X (1 -  ) % of intervals contain .  % do not.

27. 5 - 27 © 1997 Prentice-Hall, Inc. Factors Affecting Interval Width n Data dispersion Measured by  Measured by  n Sample size   X =  /  n   X =  /  n Level of confidence (1 -  ) Level of confidence (1 -  ) l Affects Z Intervals extend from  X - Z   X to  X + Z   X © 1984-1994 T/Maker Co.

28. 5 - 28 © 1997 Prentice-Hall, Inc. Confidence Interval Estimates

29. 5 - 29 © 1997 Prentice-Hall, Inc. Confidence Interval Estimate Mean (  Known)

30. 5 - 30 © 1997 Prentice-Hall, Inc. Confidence Interval Estimates

31. 5 - 31 © 1997 Prentice-Hall, Inc. Confidence Interval Mean (  Known) n Assumptions l Population standard deviation is known l Population is normally distributed If not normal, can be approximated by normal distribution (n  30) If not normal, can be approximated by normal distribution (n  30)

32. 5 - 32 © 1997 Prentice-Hall, Inc. Confidence Interval Mean (  Known) n Assumptions l Population standard deviation is known l Population is normally distributed If not normal, can be approximated by normal distribution (n  30) If not normal, can be approximated by normal distribution (n  30) n Confidence interval estimate

33. 5 - 33 © 1997 Prentice-Hall, Inc. Estimation Example Mean (  Known) The mean of a random sample of n = 25 is  X = 50. Set up a 95% confidence interval estimate for  if  = 10.

34. 5 - 34 © 1997 Prentice-Hall, Inc. Estimation Example Mean (  Known) The mean of a random sample of n = 25 is  X = 50. Set up a 95% confidence interval estimate for  if  = 10.

35. 5 - 35 © 1997 Prentice-Hall, Inc. Confidence Interval Solution*

36. 5 - 36 © 1997 Prentice-Hall, Inc. Confidence Interval Estimate Mean (  Unknown)

37. 5 - 37 © 1997 Prentice-Hall, Inc. Confidence Interval Estimates

38. 5 - 38 © 1997 Prentice-Hall, Inc. Confidence Interval Mean (  Unknown) n Assumptions l Population standard deviation is unknown l Population must be normally distributed n Use Student’s t distribution n Confidence interval estimate

39. 5 - 39 © 1997 Prentice-Hall, Inc. Student’s t Distribution 0 t (df = 5) Standard normal t (df = 13) Bell- shaped Symmetric ‘Fatter’ tails

40. 5 - 40 © 1997 Prentice-Hall, Inc. Student’s t Table t values  / 2 Assume: n = 3 df= n - 1 = 2  =.10  /2 =.05.05

41. 5 - 41 © 1997 Prentice-Hall, Inc. Student’s t Table Assume: n = 3 df= n - 1 = 2  =.10  /2 =.05 2.920 t values  / 2.05

42. 5 - 42 © 1997 Prentice-Hall, Inc. Estimation Example Mean (  Unknown) A random sample of n = 25 has  X = 50 & S = 8. Set up a 95% confidence interval estimate for .

43. 5 - 43 © 1997 Prentice-Hall, Inc. Thinking Challenge You’re a time study analyst in manufacturing. You’ve recorded the following task times (min.): 3.6, 4.2, 4.0, 3.5, 3.8, 3.1. What is the 90% confidence interval estimate of the population mean task time? AloneGroupClass

44. 5 - 44 © 1997 Prentice-Hall, Inc. Confidence Interval Solution*  X = 3.7 S = 3.8987 S = 3.8987 n = 6, df = n - 1 = 6 - 1 = 5 n = 6, df = n - 1 = 6 - 1 = 5 S /  n = 3.8987 /  6 = 1.592 S /  n = 3.8987 /  6 = 1.592 t.05,5 = 2.0150 t.05,5 = 2.0150 3.7 - (2.015)(1.592)  3.7 + (2.015)(1.592) 3.7 - (2.015)(1.592)  3.7 + (2.015)(1.592) 0.492  6.908 0.492  6.908

45. 5 - 45 © 1997 Prentice-Hall, Inc. Estimation of Mean for Finite Populations

46. 5 - 46 © 1997 Prentice-Hall, Inc. Confidence Interval Estimates

47. 5 - 47 © 1997 Prentice-Hall, Inc. Estimation for Finite Populations n Assumptions l Sample is large relative to population s n / N >.05 n Use finite population correction factor Confidence interval (mean,  unknown) Confidence interval (mean,  unknown)

48. 5 - 48 © 1997 Prentice-Hall, Inc. Confidence Interval Estimate of Proportion

49. 5 - 49 © 1997 Prentice-Hall, Inc. Confidence Interval Estimates

50. 5 - 50 © 1997 Prentice-Hall, Inc. Confidence Interval Proportion n Assumptions l Two categorical outcomes l Population follows binomial distribution l Normal approximation can be used  n·p  5 & n·(1 - p)  5 n Confidence interval estimate

51. 5 - 51 © 1997 Prentice-Hall, Inc. Estimation Example Proportion A random sample of 400 graduates showed 32 went to grad school. Set up a 95% confidence interval estimate for p.

52. 5 - 52 © 1997 Prentice-Hall, Inc. Estimation Example Proportion A random sample of 400 graduates showed 32 went to grad school. Set up a 95% confidence interval estimate for p.

53. 5 - 53 © 1997 Prentice-Hall, Inc. Thinking Challenge You’re a production manager for a newspaper. You want to find the % defective. Of 200 newspapers, 35 had defects. What is the 90% confidence interval estimate of the population proportion defective? AloneGroupClass

54. 5 - 54 © 1997 Prentice-Hall, Inc. Confidence Interval Solution* n·p  5 n·(1 - p)  5

55. 5 - 55 © 1997 Prentice-Hall, Inc. This Class... n What was the most important thing you learned in class today? n What do you still have questions about? n How can today’s class be improved? Please take a moment to answer the following questions in writing: