Chapter 19Confidence intervals forproportions
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Sample proportion
•A good estimate of the population proportion
•Natural sampling variability
•How does the sample proportion vary fromsample to sample?
–Approximately normal
–About 95% of all samples have within 2 sd of p
–Population proportion, p, is unknown
–Standard deviation is unknown
We can estimate!
•Standard error estimates standard deviation
•If estimates p accurately, by the 68-95-99.7% Rule, we know
–about 68% of all samples will have ’s within 1 SE ofp
–about 95% of all samples will have ’s within 2 SEsof p
–about 99.7% of all samples will have ’s within 3 SEsof p
What can we say about p?
•The true value of p is . (Wrong!)
•The true value of p is close to . (OK!)
•We are 95% confident that the true value of p isbetween
and
•This is a 95% confidence interval of p
•In this specific context, we can also call it one-proportion z-interval
Margin of error
•The extent of the confidence interval on eitherside of is called the margin of error
–The margin of error for our 95% confidence interval is2se.
–For the 99.7% confidence interval, the margin of erroris 3se.
–The more confident we want to be, the larger themargin of error must be.
–Certainty versus precision
•Confidence interval: estimate ± margin of error
What Does “95% Confidence”Really Mean?
•Each confidence interval uses a samplestatistic to estimate a populationparameter.
•But, since samples vary, the statistics weuse, and thus the confidence intervals weconstruct, vary as well.
What Does “95% Confidence”Really Mean? (cont.)
•The figure to the rightshows that some ofour confidenceintervals capture thetrue proportion (thegreen horizontal line),while others do not.
What Does “95% Confidence”Really Mean? (cont.)
•Our confidence is in the process ofconstructing the interval, not in any oneinterval itself.
•Thus, we expect 95% of all “95%-confidence intervals” to contain the trueparameter that they are estimating.
Margin of Error: Certainty vs.Precision
•We can claim, with 95% confidence, thatthe interval contains the truepopulation proportion.
–The extent of the interval on either side of iscalled the margin of error (ME).
•In general, confidence intervals have theform estimate ± ME.
•The more confident we want to be, thelarger our ME needs to be.
Margin of Error: Certainty vs.Precision (cont.)
Margin of Error: Certainty vs.Precision (cont.)
•To be more confident, we wind up being lessprecise.
–We need more values in our confidence interval to bemore certain.
•Because of this, every confidence interval is abalance between certainty and precision.
•The tension between certainty and precision isalways there.
–Fortunately, in most cases we can be both sufficientlycertain and sufficiently precise to make usefulstatements.
Margin of Error: Certainty vs. Precision
•The choice of confidence level issomewhat arbitrary, but keep in mind thistension between certainty and precisionwhen selecting your confidence level.
•The most commonly chosen confidencelevels are 90%, 95%, and 99% (but anypercentage can be used).
Critical value
•The number of SE’s affects the margin oferror
•This number is called the critical value,denoted by z*
•For a 95% confidence interval, the precisecritical value is 1.96
•For a 90% confidence interval, the precisecritical value is 1.645
Critical Values (cont.)
•Example: For a 90% confidence interval,the critical value is 1.645:
Margin of error
•A confidence interval too wide is not very useful
•How large a margin of error can we tolerate?
–Reduce level of confidence makes margin of errorsmaller (not a good idea, though)
–You should think about this ahead of time, when youdesign your study. Choose a larger sample!
–Generally, a margin of error of 5% or less isacceptable
Sample size
•Suppose a candidate is planning a polland wants to estimate voter support within3% with 95% confidence. How large asample does she need?
•
•But we do not know before we conductthe poll!
•However,
Sample size
•To be conservative, we set
•Solve for n, we have n = 1067.1
•So, we will need at least 1068 respondents to keep themargin of error as small as 3% with a confidence level of95%
–In practice, you may need more since many people do not respond
–If the response rate is too low, then the study might be a voluntaryresponse study, which can be biased
•To cut the se in half, we must quadruple the sample size n
Assumptions
•Independence
–Plausible independence condition
–Randomization condition
•Sampled at random?
•From a properly randomized experiment?
–10% condition
•Sampling without replacement can be viewedroughly the same as sampling with replacement
Assumptions
•Sample size assumption
–Normal approximation comes from the CLT
–Sample size must be large enough
–Check the success/failure condition
•
Example
•In May 2002, the Gallup Poll asked 537randomly sampled adults “generallyspeaking, do you believe the death penaltyis applied fairly or unfairly in this countrytoday?”
•53% answered “fairly”
•7% said “don’t know”
•What can we conclude from this survey?
Example
•Plausible independence? (Yes.)
•Randomization condition? (Yes.)
•10% condition? (Yes.)
•Success/failure condition (Yes.)
•Let’s find a 95% confidence interval
–Standard error:
–Margin of error:
–95% confidence interval
A deeper understanding ofconfidence interval
•How should we interpret the 95% confidence?
–Randomness is from sample to sample
–This does not say p is random
•We are the one who is uncertain, not the parameter.
–The interval itself is random (varies from sample tosample)
–This interval is about p, not about
•The sample proportion varies
–Report confidence interval or margin or error
•Always pay attention to the requiredassumptions
–Independence
–Sample size
•In practice, watch out for biased samples
What have we learned?
•Finally we have learned to use a sample to saysomething about the world at large.
•This process (statistical inference) is based onour understanding of sampling models, and willbe our focus for the rest of the book.
•In this chapter we learned how to construct aconfidence interval for a population proportion.
•And, we learned that interpretation of ourconfidence interval is key—we can’t be certain,but we can be confident.