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Chapter 7
Confidence Intervals and Sample Size for Proportions
1
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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
3Bluman Chapter 7
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
4Bluman Chapter 7
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.
6Bluman Chapter 7
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
12Bluman Chapter 7
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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