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% 100% confidence?
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.)