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Confidence Intervals for Proportions Review

Understand the concept of confidence intervals for proportions and learn how to calculate and interpret them. Explore the relationship between confidence level, margin of error, and sample size.

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Confidence Intervals for Proportions Review

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  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.

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