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Section 7.2Estimating a Population Proportion

ObjectiveFind the confidence interval for a population proportion p

Determine the sample size needed to estimate a population proportion p

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DefinitionsThe best point estimate for a population proportion p is the sample proportion p

Best point estimate : p

The margin of error E is the maximum likely difference between the observed value and true value of the population proportion p (with probability is 1–α)

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Margin of Error for Proportions

2

ˆ ˆpqE zn

E = margin of errorp = sample proportionq = 1 – p n = number sample values1 – α = Confidence Level

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Confidence Interval for a Population Proportion p

( p – E, p + E )ˆ ˆwhere

2

ˆ ˆpqE zn

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Finding the Point Estimate and E from a Confidence Interval

Margin of Error:

E = (upper confidence limit) — (lower confidence limit)

2

Point estimate of p:

p = (upper confidence limit) + (lower confidence limit)

2

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Round-Off Rule for Confidence Interval Estimates of p

Round the confidence interval limits for p to

three significant digits.

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Find the 95%confidence interval for the population proportion If a sample of size 100 has a proportion 0.67

Direct Computation

Example 1

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Using StatCrunch

Stat → Proportions → One Sample → with Summary

Example 1 Find the 95%confidence interval for the population proportion If a sample of size 100 has a proportion 0.67

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Using StatCrunch

Enter Values

Example 1 Find the 95%confidence interval for the population proportion If a sample of size 100 has a proportion 0.67

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Using StatCrunch

Click ‘Next’

Example 1 Find the 95%confidence interval for the population proportion If a sample of size 100 has a proportion 0.67

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Using StatCrunch

Select ‘Confidence Interval’

Example 1 Find the 95%confidence interval for the population proportion If a sample of size 100 has a proportion 0.67

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Using StatCrunch

Enter Confidence Level, then click ‘Calculate’

Example 1 Find the 95%confidence interval for the population proportion If a sample of size 100 has a proportion 0.67

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Using StatCrunch

From the output, we find the Confidence interval isCI = (0.578, 0.762)

Lower LimitUpper Limit

Standard Error

Example 1 Find the 95%confidence interval for the population proportion If a sample of size 100 has a proportion 0.67

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Sample Size

Suppose we want to collect sample data in order to estimate some population proportion. The question is how many sample items must be obtained?

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Determining Sample Size

(solve for n by algebra)

( )2 ˆp qZ n =

ˆ

E 2

zE =

p qˆ ˆn

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Sample Size for Estimating Proportion p

When an estimate of p is known: ˆˆ( )2 p qn =

ˆ

E 2z

When no estimate of p is known:use p = q = 0.5

( )2 0.25n = E 2

z

ˆˆ ˆ

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Round-Off Rule for Determining Sample Size

If the computed sample size n is not a whole number, round the value of n up to the next larger whole number.

Examples: n = 310.67 round up to 311 n = 295.23 round up to 296 n = 113.01 round up to 114

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A manager for E-Bay wants to determine the current percentage of U.S. adults who now use the Internet.

How many adults must be surveyed in order to be 95% confident that the sample percentage is in error by no more than three percentage points when…

(a) In 2006, 73% of adults used the Internet.

(b) No known possible value of the proportion.

Example 2

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(a) Given:

Given a sample has proportion of 0.73, To be 95% confident that our sample proportion is within three percentage points of the true proportion, we need at least 842 adults.

Example 2

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(b) Given:

For any sample, To be 95% confident that our sample proportion is within three percentage points of the true proportion, we need at least 1068 adults.

Example 2

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SummaryConfidence Interval of a Proportion

2

ˆ ˆpqE zn

( p – E, p + E )

E = margin of error

p = sample proportion

n = number sample values

1 – α = Confidence Level

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When an estimate of p is known:

ˆ( )2 p qn =

ˆE 2

z

When no estimate of p is known (use p = q = 0.5)

( )2 0.25n = E 2

z

SummarySample Size for Estimating a Proportion