Section 6.3

Confidence Intervals for Population Proportions

Larson/Farber 4th ed1

Section 6.3 Objectives

Larson/Farber 4th ed2

Find a point estimate for the population proportion

Construct a confidence interval for a population proportion

Determine the minimum sample size required when estimating a population proportion

Point Estimate for Population p

Larson/Farber 4th ed3

Population Proportion The probability of success in a single trial

of a binomial experiment. Denoted by pPoint Estimate for p The proportion of successes in a sample. Denoted by

read as “p hat” number of successes in sampleˆ number in sample

xpn

Point Estimate for Population p

Larson/Farber 4th ed4

Point Estimate for q, the proportion of failures Denoted by Read as “q hat”

1ˆ ˆq p

Estimate Population Parameter…

with Sample Statistic

Proportion: p p̂

Example: Point Estimate for p

Larson/Farber 4th ed5

In a survey of 1219 U.S. adults, 354 said that their favorite sport to watch is football. Find a point estimate for the population proportion of U.S. adults who say their favorite sport to watch is football. (Adapted from The Harris Poll)

Solution: n = 1219 and x = 354

354 0.29ˆ 0402 29.0%1219

xpn

Confidence Intervals for p

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ˆ ˆwhereˆ ˆ cpqp E p p E E zn

A c-confidence interval for the population proportion p •

•The probability that the confidence interval contains p is c.

Constructing Confidence Intervals for p

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1. Identify the sample statistics n and x.

2. Find the point estimate

3. Verify that the sampling distribution of can be approximated by the normal distribution.

4. Find the critical value zc that corresponds to the given level of confidence c.

ˆ xpn

Use the Standard Normal Table

.̂p

5, 5ˆ ˆnp nq p̂

In Words In Symbols

Constructing Confidence Intervals for p

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5. Find the margin of error E.

6. Find the left and right endpoints and form the confidence interval.

ˆ ˆcpqE zn

Left endpoint: Right endpoint: Interval:

p̂ Ep̂ E

ˆ ˆp E p p E

In Words In Symbols

Example: Confidence Interval for p

Larson/Farber 4th ed9

In a survey of 1219 U.S. adults, 354 said that their favorite sport to watch is football. Construct a 95% confidence interval for the proportion of adults in the United States who say that their favorite sport to watch is football.

Solution: Recall ˆ 0.290402p

1 0.290402ˆ ˆ 0.7095981q p

Solution: Confidence Interval for p

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Verify the sampling distribution of can be approximated by the normal distribution

p̂

1219 0.290402 354 5ˆnp

1219 0.709598 865 5ˆnq

• Margin of error:

(0.290402) (0.709598)1.96ˆ ˆ 0.0251219c

pqE zn

Solution: Confidence Interval for p

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Confidence interval:

ˆ

0.29 0.025

0.265

p E

Left Endpoint:

Right Endpoint:

0.265 < p < 0.315

ˆ

0.29 0.025

0.315

p E

Solution: Confidence Interval for p

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0.265 < p < 0.315

( )

• 0.290.26

50.315

With 95% confidence, you can say that the proportion of adults who say football is their favorite sport is between 26.5% and 31.5%.

Point estimate

p̂p̂ E p̂ E

Sample Size

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Given a c-confidence level and a margin of error E, the minimum sample size n needed to estimate p is

This formula assumes you have an estimate for and .

If not, use and

2

ˆ ˆ czn pqE

ˆ 0.5.qˆ 0.5p

p̂q̂

Example: Sample Size

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You are running a political campaign and wish to estimate, with 95% confidence, the proportion of registered voters who will vote for your candidate. Your estimate must be accurate within 3% of the true population. Find the minimum sample size needed if 1.no preliminary estimate is available.

Solution: Because you do not have a preliminary estimate for use andˆ 5.0.q ˆ 0.5p p̂

Solution: Sample Size

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c = 0.95 zc = 1.96 E = 0.03

2 21.96

(0.5)(0.5) 1067.110.

ˆ03

ˆ czn pqE

Round up to the nearest whole number.With no preliminary estimate, the minimum sample size should be at least 1068 voters.

Example: Sample Size

Larson/Farber 4th ed16

You are running a political campaign and wish to estimate, with 95% confidence, the proportion of registered voters who will vote for your candidate. Your estimate must be accurate within 3% of the true population. Find the minimum sample size needed if 2.a preliminary estimate gives .

ˆ 0.31p

Solution: Use the preliminary estimate

1 0.31 0. 9ˆ ˆ 61q p

ˆ 0.31p

Solution: Sample Size

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c = 0.95 zc = 1.96 E = 0.03

2 21.96

(0.31)(0.69) 913.020.

ˆ ˆ03

czn pqE

Round up to the nearest whole number.With a preliminary estimate of , the minimum sample size should be at least 914 voters.Need a larger sample size if no preliminary estimate is available.

ˆ 0.31p

Section 6.3 Summary

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Found a point estimate for the population proportion

Constructed a confidence interval for a population proportion

Determined the minimum sample size required when estimating a population proportion