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