Confidence Intervals about a Population Proportion - MATH 130 by ghkgkyyt

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```									Conﬁdence Intervals about a Population
Proportion
MATH 130, Elements of Statistics I

J. Robert Buchanan

Department of Mathematics

Fall 2009

J. Robert Buchanan   Conﬁdence Intervals about a Population Proportion
Motivation

Example
PRINCETON, NJ – Gallup Poll Daily tracking ﬁnds
41% of Americans describing economic conditions as
"poor," down slightly from the 2008 high of 44%, but
still more than double the percentage who say the
economy is "excellent" or "good" (17%). The vast
majority, 85%, perceive the economy to getting worse.
– Jeff Jones
The results reported here are based on combined data from
1,544 interviews conducted March 19-21, 2008. For results
based on this sample, the maximum margin of sampling error is
±3 percentage points.

J. Robert Buchanan   Conﬁdence Intervals about a Population Proportion
Background

The point estimate for the population proportion p is the
ˆ
sample proportion p.
If 1544 are surveyed and 633 respond that the economy is
getting worse then

ˆ
p = 633/1544 ≈ 0.410.

J. Robert Buchanan   Conﬁdence Intervals about a Population Proportion
ˆ
Sampling Distribution of p

Theorem
For a simple random sample of size n such that n ≤ 0.05N (that
is, the sample is less than or equal to 5% of the population),
ˆ
the shape of the sampling distribution of p is approximately
normal provided np(1 − p) ≥ 10,
ˆ
the mean of the sampling distribution of p is µp = p,
ˆ
ˆ
the standard deviation of the sampling distribution of p is

p(1 − p)
σp =
ˆ                  .
n

J. Robert Buchanan   Conﬁdence Intervals about a Population Proportion
Conﬁdence Interval for a Sample Proportion

Suppose a simple random sample of size n is taken from a
population. A (1 − α) · 100% conﬁdence interval for p is given by

ˆ     ˆ
p(1 − p)
ˆ
Lower and Upper bounds: p ± zα/2 ·
n

ˆ     ˆ
Note: it must be the case that np(1 − p ) ≥ 10 and n ≤ 0.05N to
construct this interval.

J. Robert Buchanan   Conﬁdence Intervals about a Population Proportion
Example

Example
Of 1500 people surveyed, 850 had eaten pizza within the last
month. Construct the 95% conﬁdence interval estimate of the
population proportion of people who have eaten pizza within
the last month.

J. Robert Buchanan   Conﬁdence Intervals about a Population Proportion
Example

Example
In the city or area where you live, are you satisﬁed or
dissatisﬁed with the quality of water?
In the United States 1000 residents aged 15 or older were
surveyed and 870 replied they were satisﬁed with the water
quality. Construct the 99% conﬁdence interval estimate of all
US residents satisﬁed with their water quality.

J. Robert Buchanan   Conﬁdence Intervals about a Population Proportion
Example

Example
Have recent price increases in gasoline caused any
ﬁnancial hardship for you or your household?

In the United States 1025 residents aged 18 or older were
surveyed and 646 replied “yes”. Construct the 90% conﬁdence
interval estimate of all US residents who report that the price
increases in gasoline have caused some ﬁnancial hardship for
themselves or their household.

J. Robert Buchanan   Conﬁdence Intervals about a Population Proportion
Estimating the Sample Size

ˆ     ˆ
p(1 − p)
E     = zα/2 ·
n
zα/2 2
ˆ     ˆ
n = p(1 − p)
E

ˆ
Remark: to use this formula we need a prior estimate of p or
we must consider the worst case scenario.

J. Robert Buchanan   Conﬁdence Intervals about a Population Proportion
Worst Case Scenario

p1 p
0.25

0.20

0.15

0.10

0.05

p
0.2           0.4        0.6             0.8            1.0

J. Robert Buchanan   Conﬁdence Intervals about a Population Proportion
Estimating the Sample Size

The sample size required to obtain a (1 − α) · 100% conﬁdence
interval for p with a margin of error E is given by
2
zα/2
ˆ     ˆ
n = p(1 − p)
E

ˆ
(rounded up the next integer), where p is a prior estimate of p. If
a prior estimate of p is unavailable, the sample size required is
2
zα/2
n = 0.25
E

rounded up the next integer.

J. Robert Buchanan   Conﬁdence Intervals about a Population Proportion
Example

Example
Determine the sample size necessary to estimate the true
proportion of college students with blue eyes, if the estimate is
to have a margin of error of 0.02 with 90% conﬁdence.

J. Robert Buchanan   Conﬁdence Intervals about a Population Proportion
Example

Example
An automobile manufacturer purchases bolts from a supplier
who claim that the bolts are approximately 5% defective.
Determine the sample size necessary to estimate the true
proportion of defective bolts if the margin of error is to be 0.01
with 95% conﬁdence.

J. Robert Buchanan   Conﬁdence Intervals about a Population Proportion
Homework