Chapter 7

Confidence Intervals and Sample Size for Proportions

1

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Do you think people are happy because they are successful or successful because they are happy?

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7.3 Confidence Intervals and Sample Size for Proportions

p = population proportion

(read p “hat”) = sample proportion

For a sample proportion,

where X = number of sample units that possess the characteristics of interest and n = sample size.

p̂

ˆ ˆ ˆ ˆand or 1X n X

p q q pn n

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In a recent survey of 150 households, 54 had central air conditioning. Find and , where is the proportion of households that have central air conditioning.

Since X = 54 and n = 150,

Example 7-8: Air Conditioned Households

54ˆ 0.36 36%

150

Xp

n

ˆ ˆ1 1 0.36 0.64 64% q p

p̂ q̂ p̂

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Example: Happiness

Find , where is the proportion of people who believe they are successful because they are happy.

Bluman, Chapter 7 5

p̂ p̂

when np 5 and nq 5.

Formula for a Specific Confidence Interval for a Proportion

2 2

ˆ ˆ ˆ ˆˆ ˆ

pq pqp z p p z

n n

Rounding Rule: Round off to three decimal places.

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A survey conducted by Sallie Mae and Gallup of 1404 respondents found that 323 students paid for their education by student loans.

Find the 90% confidence of the true proportion of students who paid for their education by student loans.

Example 7-9: Covering College Costs

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Example 7-9: Covering College CostsSince α = 1 – 0.90 = 0.10, zα/2 = 1.65.

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Example 7-9: Covering College CostsYou can be 90% confident that the percentage of students who pay for their college education by student loans is between 21.1 and 24.9%.

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A survey of 1721 people found that 15.9% of individuals purchase religious books at a Christian bookstore. Find the 95% confidence interval of the true proportion of people who purchase their religious books at a Christian bookstore.

Example 7-10: Religious Books

0.159 0.841 0.159 0.8410.159 1.96 0.159 1.96

1721 1721 p

0.142 0.176 p

You can say with 95% confidence that the true percentage is between 14.2% and 17.6%.

2 2

ˆ ˆ ˆ ˆˆ ˆ

pq pqp z p p z

n n

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Example: Happiness

Find the 90% confidence of the true proportion of people who believe that that people are successful because they are happy.

Bluman, Chapter 7 11

If necessary, round up to the next whole number.

Formula for Minimum Sample Size Needed for Interval Estimate of a Population Proportion

2

2ˆ ˆ

zn pq

E

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A researcher wishes to estimate, with 95% confidence, the proportion of people who own a home computer. A previous study shows that 40% of those interviewed had a computer at home. The researcher wishes to be accurate within 2% of the true proportion. Find the minimum sample size necessary.

Example 7-11: Home Computers

2

1.960.40 0.60

0.02

2304.96

The researcher should interview a sample of at least 2305 people.

2

2ˆ ˆ

zn pq

E

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Chapter 7Confidence Intervals and Sample Size

Section 7-3Example 7-12

Page #380

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A researcher wishes to estimate the percentage of M&M’s that are brown. He wants to be 95% confident and be accurate within 3% of the true proportion.

How large a sample size would be necessary?

Since no prior knowledge of is known, assign a value of 0.5 and then = 1 – 0.5 = 0.5. Substitute in the formula, using E = 0.03.

Example 7-12: M&M Colors

p̂q̂

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Example 7-12: M&M Colors

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Example: Happiness

If we wished to be accurate within 2.5% on our happiness question, how large of a sample would we need to take?

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Bluman, Chapter 7 18

Internet Access

Construct a 95% confidence interval for the true proportion who frequent it at least once a day.

How large of a sample would be need to be accurate within 2%?

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