Chapter 19 Confidence Intervals for Proportions Review o

Chapter 19 Confidence Intervals for Proportions

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

Review o o These __________ are random events. Long term behavior of _______ n n Called Sampling distribution Mean = __________ Standard deviation = __________ As long as o o o ___________________________________ Shape = _______________

Example o What proportion of the U. S. adult population believe in the existence of ghosts? n n Population – _____________ Parameter (p) – proportion of ________ that ____________

Problem o o ____ is unknown. We want to know ______.

(Partial) Solution o o o Sample the population (n = 1000) Statistic _____ – proportion of _______________ that _________. Out of 1000 people 388 of them _________.

Estimating p. o o How good is our estimate for p? Sampling variability says n n _____ is never the same as p. Whenever you take a sample, you will ___________________.

Estimating p. o o So why do we calculate _____ if it’s always wrong? We know the long-term behavior of ______.

Estimating p o o I know ______ is different from ____. I also know how much ______ is likely to be away from _______.

Problem – I don’t know p. o o o Formula includes value of _______. Replace p with ______. This is called a ___________.

Example o o 38. 8% of sample of 1000 U. S. adults believe in ghosts. How much is this likely to be off by?

Example o o My value of _____ is likely to be off by 1. 5%. 38. 8% - 1. 5% = 37. 3% 38. 8% +1. 5% = 40. 3% __________________.

Confidence o o 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%? __________________

Review of 68 -95 -99. 7 Rule o o o 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.

Example of 68 -95 -99. 7 Rule o o o 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 ______.

My sample information o o ____ = 0. 388 Where does this value belong in the sampling distribution? Answer: _________ Why: ___________

Confidence o I am approx. _______ confident that my _______ value is within _____ of p.

Confidence o I am approx. _______ confident that my _______ value is within _____ of p.

Confidence o I am approx. _______ confident that my _______ value is within _____ of p.

Confidence Interval for p

Confidence Interval for p o o 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.

Confidence Levels o o Common Confidence levels 80%, 95%, 98%, 99% 100% confidence?

Confidence Interval for p.

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

Confidence Level = 90%

Confidence Level = 95%

Confidence Level = 98%

Confidence Level = 99%

Summary of values for z*

Example #1 o 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.

Example #1 – Conditions

Example #1 – CI

Example #1 – Interpretation of CI

Example #2 o 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.

Example #2 – Conditions

Example #2 – CI

Example #2 – Interpretation of CI

Example #3 o 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.

Example #3 – Conditions

Example #3 – CI

Example #3 – Interpretation of CI

Meaning of Confidence Level o Capture Rate

Properties of CIs o o Margin of Error = ___________ Width of CI = ____________

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

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

For a fixed Confidence Level C% o Effect of sample size on Margin of Error

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

Trade-Off o Goal #1: o Goal #2:

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

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

Sample Size o Solution – Use the most conservative value for _______. o Solve for n

Sample size

Example #4 o 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?

Example #4 (cont. )

Example #4 (cont. ) o What if we want a 95% confidence interval for the population proportion with a margin of error of no more than 1. 5%?

Example #4 (cont. )

Example #5 o 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?

Example #5 (cont. ) o 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.

Example #5 (cont. )

Example #5 (cont. )

Example #5 (cont. ) o 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?

Example #5 (cont. )