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Confidence Interval of a Proportion
FPP 20 - 21
Using the sample to learn about the box
 Box models and CLT assume we know the contents of the
box (the population).
 In real-world problems, we do not.
 In random samples, sample averages and percentages are
good estimates of population quantities, but are subject to
chance variation
 We need a method of accounting for chance variation when
trying to learn about the box.
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Major Assumption
 We did not cover chapter 20 much at all but in what follows
we are assuming that the data come from a SIMPLE
RANDOM SAMPLE.
 None of what follows is valid if data is not collected this way.
 If data not from a random sample there is little we can do
 If data is from a random sample whose sampling scheme is more
complicated than what we’ve learned in this class then computing
standard errors is more complex.
Confidence intervals
 Rather than a single estimate of a population quantity, we
desire a range of likely values that takes chance error into
account
 We call the range of plausible values a confidence interval
 The method of producing such intervals was developed by
Jerzy Neyman in the 1920s
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Confidence intervals
 We will motivate confidence intervals using a variable that
produces binary outcomes (categorical/qualitative)
 Thus the parameter of interest is a proportion or percent
 One of the tricky things about the rest of the semester will
be to identify the parameter of interest in a given problem.
 One good way of doing this is to identify the type of data being
considered
 Other parameters we will consider in the class mean, slope.
Confidence intervals for population
proportions/percentages
 Let p be some population proportion.
 Recall that the sample proportion has
 EV = p
and SE =
 In large samples, we can use the normal curve to make
probability statements about the sample proportion (CLT)
 Example: In 95% of random samples, the sample
proportion, p-hat, is within 1.96 SEs of p
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Mathematical derivation of the CI
picture
 According to the CLT for large sample sizes, in 95% of
random samples
Mathematical derivation
 We can put p in the middle of the inequality, so that in 95%
of random samples
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Confidence interval defined
 Using the sample proportion from the data in the SE, we get
 This is a 95% confidence interval for p
Application of CI
 In 1998, New York Times and CBS News polled 1048
randomly selected 13-17 year olds to ask them if they had a
TV in their room
 In sample, 692 had a television in their room
 Let p = proportion of 13-17 year olds in U.S. in 1998 who
had a TV in their room.
pˆ = 692/1048 = 0.660
€
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Application of CI cont
 Recall that SE of pˆ is
p(1- p)
n
 But we don’t know p. What do we do? Use p-hat
€ refer to this as the boot-strap method
 FPP
 Thus
SE of pˆ is 0.660(0.340)/1048 = 0.01463
 € A 95% CI for p is
 (0.660 – 1.96*0.01463, 0.660 + 1.96*0.01463 =
(0.632, 0.689)
General form of all CIs
 In what follows “est” means parameter estimate and “SE”
means standard error
 Lower limit = est. – (multiplier) * SE
 Upper limit = est. + (multiplier) * SE
CI equation for proportion : pˆ ± multiplier *
pˆ (1− pˆ )
n
 Question how do we find the multiplier?
€
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Determining multiplier
 For 95% confidence interval for p, the multiplier is the z-
score value such that 95% of area under the standard normal
curve falls between –z and +z
 One can choose any level of confidence for the interval
 95% is most common, with 99% and 90% distance seconds
 Example: multiplier for a 99% CI
Interpretation of CI
 The actual computation of confidence intervals is fairly
straight forward. There are subtle difficulties associated with
the interpretation
 The interpretation of CI intervals needs three things
 1. Statement of parameter in words (with reference to the
population)
 2. Statement of level of Confidence
 3. Statement of Interval
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Correct Interpretations
 “I am 90% confidence that the interval (0.5, 0.75) captures the
true proportion of Duke alumni that donate”
 “The interval (0.6, 0.99) gives a range of reasonable values for the
proportion of all patients having flu like symptoms actually have
the H1N1 virus. We are 95% confident of this.”
 “The proportion of all seventh-grade girls whose IQ is between
95.3 and 109.2 is somewhere between 0.75 and 0.9 with 99%
confidence.”
Incorrect Interpretations
 “99% of IQ’s are contained in the interval (95.3, 109.2).”
 “The probability that the interval (0.5, 0.75) captures the
true proportion of Duke alumni that donate is .90”
 “We are 95% confident that the interval (0.6, 0.99) contains
the sample proportion of patients that have swine flu.”
 “99% of the time, the proportion of seventh-grade girls with
an IQ larger than 109.2 is contained in the interval (0.75,
0.91).”
 “We are 90% confident that the interval (119.5, 128.1)
captures the yields in bushels per acre.”
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Statistical “Confidence”
 What do we mean when we say we are 95% confident?
 We are “confident” in the procedure that produced the interval
 That is, we know that 95% of all simple random samples will
produce a confidence interval that contains the value of the
parameter
 Note that there is NO PROBABILITY associated with CIs
 Statistical “confidence” describes what will happen in the long
run
Statistical “confidence” Cont.
 We don’t know if whether our one sample produces one of
the “unlucky” 5% CIs and doesn’t contain the value of the
parameter
 What does statistical “confidence” say about the likelihood of
one particular interval containing the value of the parameter?
 Nothing
 Nada
 Ziltz
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Confidence intervals
 Templates
Example of CI revisited
 In 1998, the New York Times and CBS News polled 1048 randomly
selected 13-17 year olds to ask them if they had a TV in their room.
 In sample, 692 had a television in their room.
 Let p = percentage of 13-17 year olds in U.S. in 1998 who had a TV in
their room
pˆ = 692 /1048 = 0.660. SE =
0.660(1− 0.660)
= 0.1463
1048
 A 95% CI for p is
 (0.660 – 1.95*0.1463, 0.660 + 1.96*0.1463) = (0.632, 0.689)
€
 We are 95% confident that the population percentage of 13-17 year olds
in the U.S. in 1998 who had a TV in their room is between 0.632 and
0.689
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Another example of CIs
 Opinion polls often use the phrasing,
 “85% of people think the economy is the number one issue.
The poll has a margin of error of plus and minus 3%.”
 This means that a 95% confidence interval stretches from
82% to 88%.
 The margin of error in the confidence interval formula is
 M.E. = multiplier*SE
Width of confidence interval
 Width of CI depends on two quantities:
 Multiplier
 SE
 Multiplier: determined by level of confidence
 More confidence requires a ______________er multiplier
and there for a ____________ CI
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Width of confidence interval
 Smaller SE implies a ________________ CI.
 SE ____________ as n increases.
 Therefore, increasing n ____________ width of CI
 More (randomly sampled) data means _________ accurate
inferences
True or false
 DSG sets up a table outside the Bryan Center. DSG
representatives at the table ask students to stop by and
fill out a survey on a proposed activities fee increase.
Out of the 100 people who complete the survey, 65 are
in favor of the increase. The Chronicle reports that the
percentage of Duke students who support an increase in
the activities fee is likely between 55.5% and 74.5%.
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Important caveat
 Once again a confidence interval will NOT remedy a poorly
designed study
 Bad data yield unreliable (worthless) intervals
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