1. Confidence Intervals for Proportions
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 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 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%

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.

51. Sample Size
Solution – Use the most conservative value for _____________.
Solve for n

52. Sample size

53. Example #4
We would like to obtain a 95% confidence interval for the population proportion of U.S. registered voters that approve of President Bush’s handling of the war in Iraq. We would like this confidence interval to have a margin of error of no more than 3%. How many people should be in our sample?

54. Example #4 (cont.)

55. Example #4 (cont.)
What if we want a 95% confidence interval for the population proportion with a margin of error of no more than 1.5%?

56. Example #4 (cont.)

57. Example #5
The mayor of a small city has suggested that the state locate a new prison there, arguing that the construction project and resulting jobs will be good for the local economy. A total of 183 residents show up for a public hearing on the proposal and a show of hands finds only 31 in favor of the prison project. What can the city council conclude about public support for the mayor’s initiative?

58. Example #5 (cont.)
A random sample of 100 residents is taken from this small city. 38 of the 100 people are in favor of locating the state prison in their city. Find a 90% confidence interval for the population proportion of city residents that are in favor of locating the state prison in the city.

59. Example #5 (cont.)

60. Example #5 (cont.)

61. Example #5 (cont.)
Suppose the issue was placed on the election ballot. The city residents were asked to vote whether to allow the state to build a prison in their city. How do you think the city residents would vote? Would the issue pass or would the voters vote the prison down?

62. Example #5 (cont.)