Confidence Intervals for Proportions Review

1. Chapter 19 Confidence Intervals for Proportions

2. Review • Categorical Variable • One Label or Category • _____ = proportion of population members belonging in this one category • _____ = proportion of sample members belonging in this one category

3. Review • These ___________________ are random events. • Long term behavior of _______ • Called Sampling distribution • Mean = ___________________ • Standard deviation = ____________________ • As long as • ___________________________________ • ___________________________________ • Shape = _____________________________

4. Example • What proportion of the U.S. adult population believe in the existence of ghosts? • Population – __________________________ • Parameter (p) – proportion of ________________ that ________________________

5. Problem • ________ is unknown. • We want to know ____________.

6. (Partial) Solution • Sample the population (n = 1000) • Statistic _____ – proportion of ____________ _______________ that _________________. • Out of 1000 people 388 of them __________________.

7. Estimating p. • How good is our estimate for p? • Sampling variability says • __________ is never the same as p. • Whenever you take a sample, you will _____________________________________.

8. Estimating p. • So why do we calculate __________ if it’s always wrong? • We know the long-term behavior of ____________.

9. Estimating p • I know ___________ is different from ________. • I also know how much ____________ is likely to be away from _____________.

10. Problem – I don’t know p. • Formula includes value of ______________. • Replace p with ____________. • This is called a _____________________.

11. Example • 38.8% of sample of 1000 U.S. adults believe in ghosts. • How much is this likely to be off by?

12. Example • My value of _________ is likely to be off by 1.5%. • 38.8% - 1.5% = 37.3% • 38.8% +1.5% = 40.3% • ____________________________________.

13. Confidence • We don’t know that for sure. • Our value for ______________ could be farther away from p. • How confident am I that p is between 37.3% and 40.3%? • ___________________________________

14. Review of 68-95-99.7 Rule • Approx. 68% of all samples have a _______ value within ______________ of p. • Approx. 95% of all samples have a ________ value within ______________ of p. • Approx. 99.7% of all samples have a _________ value within ___________ of p.

15. Example of 68-95-99.7 Rule • Approx. 68% of all samples have a _______ value between _________ and ____________. • Approx. 95% of all samples have a ________ value between _________ and ____________. • Approx. 99.7% of all samples have a _________ value between _____________ and ___________.

16. My sample information • ________ = 0.388 • Where does this value belong in the sampling distribution? • Answer: _________________ • Why: _____________________

17. Confidence • I am approx. _____________ confident that my _______ value is within _________ of p.

18. Confidence • I am approx. _____________ confident that my _______ value is within _________ of p.

19. Confidence • I am approx. _____________ confident that my _______ value is within _________ of p.

20. Confidence Interval for p

21. Confidence Interval for p • Gives interval of most likely values of p given the information from the sample. • Confidence level tells how confident we are parameter is in interval.

22. Confidence Levels • Common Confidence levels 80%, 90%, 95%, 98%, 99% • 100% confidence?

23. Confidence Interval for p.

24. Values for z* • z* - based on Confidence Level (C%). • Find z* from N(0,1) table • Middle C% of dist. between –z* and z*

25. Confidence Level = 90%

26. Confidence Level = 95%

27. Confidence Level = 98%

28. Confidence Level = 99%

29. Summary of values for z*

30. Example #1 • In a sample of 1000 U.S. adults, 38.8% stated they believed in the existence of ghosts. Find a 95% confidence interval for the population proportion of all U.S. adults who believe in the existence of ghosts.

31. Example #1 – Conditions

32. Example #1 – CI

33. Example #1 – Interpretation of CI

34. Example #2 • An insurance company checks police records on 582 accidents selected at random and notes that teenagers were at the wheel in 91 of them. Find the 90% confidence interval for the population proportion of all accidents that involve teenage drivers.

35. Example #2 – Conditions

36. Example #2 – CI

37. Example #2 – Interpretation of CI

38. Example #3 • 344 out of a sample of 1,010 U.S. adults rated the economy as good or excellent in a recent (October 4-7, 2007) Gallup Poll. Find a 98% confidence interval for the proportion of all U.S. adults who believe the economy is good or excellent.

39. Example #3 – Conditions

40. Example #3 – CI

41. Example #3 – Interpretation of CI

42. Meaning of Confidence Level • Capture Rate

43. Properties of CIs • Margin of Error = ______________________ • Width of CI = ________________________

44. For a fixed sample size (n) • Effect of Confidence Level on Margin of Error.

45. For a fixed sample size (n) • Smaller confidence level means smaller ME. • Larger confidence level means larger ME. • Idea:

46. For a fixed Confidence Level C% • Effect of sample size on Margin of Error

47. For a fixed Confidence Level C% • Smaller samples mean larger ME. • Larger samples mean smaller ME. • Idea:

48. Trade-Off • Goal #1: • Goal #2:

49. Trade-Off • Goal #1 and #2 conflict. • Solution?:

50. Sample Size • Before taking sample, determine sample size so that for a specified confidence level, we get a certain margin of error. • Problem – we don’t know _______ because we haven’t taken sample.