Chapter Eight Part 2 (Sections 8.4 & 8.5) Estimation

1. Understandable StatisticsSeventh EditionBy Brase and BrasePrepared by: Lynn SmithGloucester County College Chapter Eight Part 2 (Sections 8.4 & 8.5) Estimation

2. When estimating the mean, how large a sample must be used in order to assure a given level of confidence? Use the formula:

3. How do we determine the value of the population standard deviation, ? Use the standard deviation, s, of a preliminary sample of size 30 or larger to estimate .

4. Determine the sample size necessary to determine (with 99% confidence) the mean time it takes to drive from Philadelphia to Boston. We wish to be within 15 minutes of the true time. Assume that a preliminary sample of 45 trips had a standard deviation of 0.8 hours.

5. ... determine with 99% confidence... z0.99 = 2.58

6. ... We wish to be within 15 minutes of the true time. ... E = 15 minutes = 0.25 hours

7. ...a preliminary sample of 45 trips had a standard deviation of 0.8 hours. Since the preliminary sample is large enough, we can assume that the population standard deviation is approximately equal to 0.8 hours.

8. Minimum Sample Size =

9. Rounding Sample Size Any fractional value of n is always rounded to the next higher whole number.

10. Minimum Sample Size • n  68.16 • Round to the next higher whole number. • To be 99% confident in our results, the minimum sample size = 69.

11. Formula for Minimum Sample Size for Estimating p for the Binomial Distribution If p is an estimate of the true population proportion,

12. Formula for Minimum Sample Size for Estimating p for the Binomial Distribution If we have no preliminary estimate for p, the probability is at least c that the point estimate r/n for p will be in error by less than the quantity E if n is at least:

13. Formula for Minimum Sample Size for Estimating p for the Binomial Distribution If we have no preliminary estimate for p, use the following formula to determine minimum sample size:

14. The manager of a furniture store wishes to estimate the proportion of orders delivered by the manufacturer in less than three weeks. She wishes to be 95% sure that her point estimate is in error either way by less than 0.05. Assume no preliminary study is done to estimate p.

15. She wishes to be 95% sure ... z0.95 = 1.96

16. ... that her point estimate is in error either way by less than 0.05. E = 0.05

17. ... no preliminary study is done to estimate p. The minimum required sample size (if no preliminary study is done to estimate p) is 385.

18. If a preliminary estimate indicated that p was approximately equal to 0.75: The minimum required sample size (if this preliminary study is done to estimate p) is 289.

19. How can we tell if two populations are the same? • Compare the difference in population means. • Compare the difference in population proportions.

20. Paired Data: Dependent Samples Members of each pair have a natural matching of characteristics. Example: weight before and weight after a diet.

21. Independent Samples Take a sample from one population and an unrelated random sample from the other population.

22. Testing the Differences of Means for Large Independent Samples • Let x1 and x2 have normal distributions with means 1 and 2 and standard deviations 1 and 2 respectively. • Take independent random samples of size n1 and n2 from each distribution.

27. If both n1 and n2 are 30 or larger The Central Limit Theorem can be applied even if the original distributions are not normal.

28. If both n1 and n2 are 30 or larger The sample standard deviations (s1 and s2) are good approximations of the population standard deviations (1 and 2.)

29. Confidence Intervals for the Differences in Means for Large Independent Samples

30. A c Confidence Interval for 1 – 2 for Large Samples (n1 and n2 30)

31. Symbols Used

32. Symbols Used

33. Some Levels of Confidence and Their Critical Values

34. Determine a 95% Confidence Interval for the Difference in Population Mean Exam results:

35. A 95% Confidence Interval for 1 – 2 for Large Samples (n1 and n2 30)

36. A c% Confidence Interval for 1 – 2 for Large Samples (n1 and n2 30)

37. We conclude (with 95% confidence) that the difference in the mean exam results from the two populations falls between 1.13 and 4.87 points.

38. Confidence Intervals for the Differences of Two Means of Small Independent Samples

39. Assumptions • Independent random samples are drawn from two populations with means 1 and 2. • The parent populations have normal (or approximately normal) distributions. • The standard deviations for the populations (1 and 2) are approximately equal.

40. A c% Confidence Interval for 1 – 2 for Small Samples where the Standard Deviations are Approximately Equal

41. Best Estimate of the Common or Pooled Standard Deviation for Two Populations

42. Other Symbols Used

43. Symbols Used

44. Determine a 99% Confidence Interval for the Difference in Population Means:

45. Find s = Pooled Standard Deviation

46. To Find a 99% Confidence Interval • d.f. = 14 + 17 – 2 = 29. • Use the column headed by c = 0.990 in Table 6 Appendix II to find t0.99 for d.f. = 29. • t0.99 = 2.756.

47. Find E, when tc = 2.756 and s = 2.192

48. The 99% Confidence Interval for 1 – 2 for Small Samples

49. Conclusion We are 99% confident that the differences in speeds (m.p.h. for Group 1 minus m.p.h. for Group 2) between the two groups ranges from negative 0.18 to positive 4.18 m.p.h.

50. Confidence Intervals for the difference of two proportions from binomial distributionsp1 – p2