Confidence Intervals for Proportions Chapter 19 Confidence Intervals for Proportions

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

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

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

Problem ________ is unknown. We want to know ____________.

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

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 _____________________________________.

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

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

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

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

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

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%? ___________________________________

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.

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 ___________.

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

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

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

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

Confidence Interval for p

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.

Confidence Levels Common Confidence levels 80%, 90%, 95%, 98%, 99%

Confidence Interval for p.

Values for z* 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 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 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 #3 344 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 Capture Rate

Properties of CIs Margin of Error = ______________________ Width of CI = ________________________

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

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

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

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

Trade-Off Goal #1: Goal #2:

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

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.

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

Sample size

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?

Example #4 (cont.)

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%?

Example #4 (cont.)

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?

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.

Example #5 (cont.)

Example #5 (cont.)

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?

Example #5 (cont.)