Confidence Interval Estimation for a Population Proportion Lecture 33 Section 9.4 Mon, Nov 6, 2006.

Confidence Interval Confidence Interval Estimation for a Estimation for a

Population Population ProportionProportionLecture 33Lecture 33

Section 9.4Section 9.4

Mon, Nov 6, 2006Mon, Nov 6, 2006

Point EstimatesPoint Estimates

Point estimatePoint estimate – A single value of the – A single value of the statistic used to estimate the statistic used to estimate the parameter.parameter.

The problem with point estimates is The problem with point estimates is that we have no idea how close we that we have no idea how close we can expect them to be to the can expect them to be to the parameter.parameter.

That is, we have no idea of how That is, we have no idea of how large the error may be.large the error may be.

Interval EstimatesInterval Estimates

Interval estimateInterval estimate – an interval of – an interval of numbers that has a stated numbers that has a stated probability (often 95%) of containing probability (often 95%) of containing the parameter.the parameter.

An interval estimate is more An interval estimate is more informative than a point estimate.informative than a point estimate.

Interval EstimatesInterval Estimates

Confidence levelConfidence level – The probability – The probability that is associated with the interval.that is associated with the interval.

If the confidence level is 95%, then If the confidence level is 95%, then the interval is called a the interval is called a 95% 95% confidence intervalconfidence interval..

Approximate 95% Approximate 95% Confidence IntervalsConfidence Intervals

How do we find a 95% confidence How do we find a 95% confidence interval for interval for pp??

Begin with the sample size Begin with the sample size nn and the and the sampling distribution of sampling distribution of pp^̂..

We know that the sampling We know that the sampling distribution is normal with mean distribution is normal with mean pp and standard deviationand standard deviation

n

ppp

1ˆ

The Target AnalogyThe Target Analogy

Suppose a shooter hits within 4 Suppose a shooter hits within 4 rings (4 inches) of the bull’s eye 95% rings (4 inches) of the bull’s eye 95% of the time.of the time.

Then each individual shot has a 95% Then each individual shot has a 95% chance of hitting within 4 inches.chance of hitting within 4 inches.


Now suppose we are shown where Now suppose we are shown where the shot hit, but we are not shown the shot hit, but we are not shown where the bull’s eye is.where the bull’s eye is.

What is the probability that the What is the probability that the bull’s eye is within 4 inches of that bull’s eye is within 4 inches of that shot?shot?


Where is the bull’s eye?


4 inches


4 inches

95% chance that thebull’s eye is within this circle.

The Confidence IntervalThe Confidence Interval

In a similar way, 95% of the sample In a similar way, 95% of the sample proportions proportions pp^̂ should lie within 1.96 should lie within 1.96 standard deviations (standard deviations (pp^̂) of the ) of the parameter parameter pp..


p


p

1.96 p^


Therefore, if we compute a single Therefore, if we compute a single pp^̂, , then we expect that there is a 95% then we expect that there is a 95% chance that it lies within a distance chance that it lies within a distance 1.961.96pp^̂ of of pp..


Where is p?

p^


1.96 p^

p^


1.96 p^

95% chance that p is within this interval

p^


Thus, the confidence interval isThus, the confidence interval is

The trouble is, to know The trouble is, to know pp^̂, we must , we must know know pp. (See the formula for . (See the formula for pp^̂.).)

The best we can do is to use The best we can do is to use pp^̂ in in place of place of pp to estimate to estimate pp^̂..

pp ˆ96.1ˆ


That is,That is,

This is called the This is called the standard errorstandard error of of pp^̂ and is denoted SE( and is denoted SE(pp^̂).).

Now the 95% confidence interval Now the 95% confidence interval isis

n

ppp

ˆ1ˆˆ

pp ˆSE96.1ˆ

ExampleExample

Example 9.6, p. 585 – Study: Chronic Example 9.6, p. 585 – Study: Chronic Fatigue Common.Fatigue Common.

Rework the problem supposing that Rework the problem supposing that 350 out of 3066 people reported that 350 out of 3066 people reported that they suffer from chronic fatigue they suffer from chronic fatigue syndrome.syndrome.

How should we interpret the How should we interpret the confidence interval?confidence interval?

Standard Confidence Standard Confidence LevelsLevels

The standard confidence levels are The standard confidence levels are 90%, 95%, 99%, and 99.9%. (See p. 90%, 95%, 99%, and 99.9%. (See p. 588 and Table III, p. A-6.)588 and Table III, p. A-6.)

Confidence Level

z

90%90% 1.6451.645

95%95% 1.9601.960

99%99% 2.5762.576

99.9%99.9% 3.2913.291


The confidence interval is given by the The confidence interval is given by the formulaformula

where where zz Is given by the previous chart, or Is given by the previous chart, or Is found in the normal table, or Is found in the normal table, or Is obtained using the invNorm function on Is obtained using the invNorm function on

the TI-83.the TI-83.

pzp ˆSEˆ

Confidence LevelConfidence Level

Rework Example 9.6, p. 585, by Rework Example 9.6, p. 585, by computing acomputing a 90% confidence interval.90% confidence interval. 99% confidence interval.99% confidence interval.

Which one is widest?Which one is widest? In which one do we have the most In which one do we have the most

confidence?confidence?

Probability of ErrorProbability of Error

We use the symbol We use the symbol to represent to represent the probability that the confidence the probability that the confidence interval is in error.interval is in error.

That is, That is, is the probability that is the probability that pp is is notnot in the confidence interval. in the confidence interval.

In a 95% confidence interval, In a 95% confidence interval, = = 0.05.0.05.

Probability of ErrorProbability of Error

Thus, the area in each tail is Thus, the area in each tail is /2./2.

ConfidenConfidencece

LevelLevel invNorm(invNorm(//

2)2)

90%90% 0.100.10 -1.645-1.645

95%95% 0.050.05 -1.960-1.960

99%99% 0.010.01 -2.576-2.576

99.9%99.9% 0.000.0011

-3.291-3.291

Which Confidence Interval Which Confidence Interval is Best?is Best?

Which is better?Which is better? A large margin of error (wide interval), A large margin of error (wide interval),

oror A small margin of error (narrow A small margin of error (narrow

interval).interval). Which is better?Which is better?

A low level of confidence, orA low level of confidence, or A high level of confidence.A high level of confidence.


Why not get a confidence interval Why not get a confidence interval that has a small margin of error and that has a small margin of error and has a high level of confidence has a high level of confidence associated with it?associated with it?

Hey, why not a margin of error of 0 Hey, why not a margin of error of 0 and a confidence level of 100%?and a confidence level of 100%?


Which is better?Which is better? A smaller sample size, orA smaller sample size, or A larger sample size.A larger sample size.


A larger sample size is better only A larger sample size is better only up to the point where its cost is not up to the point where its cost is not worth its benefit.worth its benefit.

That is why we settle for a certain That is why we settle for a certain margin of error and a confidence margin of error and a confidence level of less than 100%.level of less than 100%.

TI-83 – Confidence TI-83 – Confidence IntervalsIntervals

The TI-83 will compute a confidence The TI-83 will compute a confidence interval for a population proportion.interval for a population proportion.

Press STAT.Press STAT. Select TESTS.Select TESTS. Select 1-PropZInt.Select 1-PropZInt.


A display appears requesting A display appears requesting information.information.

Enter Enter xx, the numerator of the sample , the numerator of the sample proportion.proportion.

Enter Enter nn, the sample size., the sample size. Enter the confidence level, as a Enter the confidence level, as a

decimal.decimal. Select Calculate and press ENTER.Select Calculate and press ENTER.


A display appears with several items.A display appears with several items. The title “1-PropZInt.”The title “1-PropZInt.” The confidence interval, in interval The confidence interval, in interval

notation.notation. The sample proportion The sample proportion pp^̂.. The sample size.The sample size.

How would you find the margin of How would you find the margin of error?error?


Rework Example 9.6, p. 585, using Rework Example 9.6, p. 585, using the TI-83.the TI-83.

Confidence Interval Estimation for a Population Proportion Lecture 33 Section 9.4 Mon, Nov 6, 2006.

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