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Chapter 19

Confidence Intervalsfor Proportions

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A Confidence Interval

Recall that the sampling distribution model of iscentered at p, with standard deviation .

Since we don’t know p, we can’t find the truestandard deviation of the sampling distributionmodel, so we need to find the standard error:

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A Confidence Interval (cont.)

By the 68-95-99.7% Rule, we know

about 68% of all samples will have ’s within 1SE of p

about 95% of all samples will have ’s within 2SEs of p

about 99.7% of all samples will have ’s within3 SEs of p

We can look at this from ’s point of view…

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A Confidence Interval (cont.)

Consider the 95% level:

There’s a 95% chance that p is no more than 2SEs away from .

So, if we reach out 2 SEs, we are 95% surethat p will be in that interval. In other words, ifwe reach out 2 SEs in either direction of , wecan be 95% confident that this interval containsthe true proportion.

This is called a 95% confidence interval.

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A Confidence Interval (cont.)

Example 1

54 out of 104 sea corals are infected with adisease. What is the 95% confidence interval?

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What Does “95% Confidence” Really Mean?

Each confidence interval uses a sample statisticto estimate a population parameter.

But, since samples vary, the statistics we use,and thus the confidence intervals we construct,vary as well.

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What Does “95% Confidence” Really Mean?(cont.)

The figure to theright shows thatsome of ourconfidenceintervals capturethe true proportion(the greenhorizontal line),while others donot:

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What Does “95% Confidence” Really Mean?(cont.)

Our confidence is in the process of constructingthe interval, not in any one interval itself.

Thus, we expect 95% of all 95% confidenceintervals to contain the true parameter that theyare estimating.

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Margin of Error: Certainty vs. Precision

We can claim, with 95% confidence, that theinterval contains the true populationproportion.

The extent of the interval on either side of iscalled the margin of error (ME).

In general, confidence intervals have the formestimate ± ME.

The more confident we want to be, the larger ourME needs to be.

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Margin of Error: Certainty vs. Precision (cont.)

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Margin of Error: Certainty vs. Precision (cont.)

To be more confident, we wind up being less precise.

We need more values in our confidence interval to bemore certain.

Because of this, every confidence interval is a balancebetween certainty and precision.

The tension between certainty and precision is alwaysthere.

Fortunately, in most cases we can be both sufficientlycertain and sufficiently precise to make usefulstatements.

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Margin of Error: Certainty vs. Precision (cont.)

The choice of confidence level is somewhatarbitrary, but keep in mind this tension betweencertainty and precision when selecting yourconfidence level.

The most commonly chosen confidence levelsare 90%, 95%, and 99% (but any percentage canbe used).

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Margin of Error: Certainty vs. Precision (cont.)

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Critical Values

The ‘2’ in (our 95% confidenceinterval) came from the 68-95-99.7% Rule.

Using a table or technology, we find that a moreexact value for our 95% confidence interval is1.96 instead of 2.

We call 1.96 the critical value and denote it z*.

For any confidence level, we can find thecorresponding critical value.

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Critical Values (cont.)

Example: For a 90% confidence interval, thecritical value is 1.645:

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Assumptions and Conditions

All statistical models depend upon assumptions.

Different models depend upon different assumptions.

If those assumptions are not true, the model might beinappropriate and our conclusions based on it may bewrong.

You can never be sure that an assumption is true, but youcan often decide whether an assumption is plausible bychecking a related condition.

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Assumptions and Conditions (cont.)

Here are the assumptions and the correspondingconditions you must check before creating a confidenceinterval for a proportion:

Independence Assumption: The data values are assumedto be independent from each other. We check threeconditions to decide whether independence is reasonable.

Plausible Independence Condition: Is there any reasonto believe that the data values somehow affect eachother? This condition depends on your knowledge ofthe situation—you can’t check it with data.

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Assumptions and Conditions (cont.)

Randomization Condition: Were the data sampled atrandom or generated from a properly randomizedexperiment? Proper randomization can help ensureindependence.

10% Condition: Is the sample size no more than 10%of the population?

Sample Size Assumption: The sample needs to be largeenough for us to be able to use the CLT.

Success/Failure Condition: We must expect at least 10“successes” and at least 10 “failures.”

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One-Proportion z-Interval

When the conditions are met, we are ready to find theconfidence interval for the population proportion, p.

The confidence interval is

where

The critical value, z*, depends on the particularconfidence level, C, that you specify.

Example 2

The Gallup Poll asked 537 randomly sampledadults the question “do you believe the deathpenalty is fair or unfair in this country today?” Ofthese, 53% answered fairly and 7% said theydidn’t know. What can we conclude from thissurvey?

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Calculator

Stat – Tests – A:1-PropZInt

X – # of successes

N – total in sample

C-level – 95%

Calculate

Try for 54 of 104 sea corals

Try for 53% who said yes to the death penalty

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What Can Go Wrong?

Don’t Misstate What the Interval Means:

Don’t suggest that the parameter varies.

Don’t claim that other samples will agree withyours.

Don’t be certain about the parameter.

Don’t forget: It’s the parameter (not the statistic).

Don’t claim to know too much.

Do take responsibility (for the uncertainty).

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What Can Go Wrong? (cont.)

Margin of Error Too Large to Be Useful:

We can’t be exact, but how precise do we need tobe?

One way to make the margin of error smaller is toreduce your level of confidence. (That may not bea useful solution.)

You need to think about your margin of errorwhen you design your study.

To get a narrower interval without giving upconfidence, you need to have less variability.

You can do this with a larger sample…

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What Can Go Wrong? (cont.)

Choosing Your Sample Size:

In general, the sample size needed to produce aconfidence interval with a given margin of error ata given confidence level is:

where z* is the critical value for your confidencelevel.

To be safe, round up the sample size you obtain.

Example 3

Suppose a candidate is planning a poll and wantsto estimate voter support within 3% with 95%confidence. How large a sample does she need?

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