Confidence Interval
Estimation for a
Population Proportion
Lecture 31
Section 9.4
Tue, Mar 27, 2007
Point Estimates
Point estimate
 The problem with point estimates is that
we have no idea how close we can expect
them to be to the parameter.
 That is, we have no idea of how large the
error may be.

Interval Estimates
Interval estimate
 An interval estimate is more informative
than a point estimate.

Interval Estimates
Confidence level
 If the confidence level is 95%, then the
interval is called a 95% confidence
interval.

Approximate 95% Confidence
Intervals
How do we find a 95% confidence interval
for p?
 Begin with the sample size n and the
sampling distribution of p^.
 We know that the sampling distribution is
normal with mean p and standard
deviation

 pˆ 
p1  p 
n
The Target Analogy
Suppose a shooter hits within 5 rings (5
inches) of the bull’s eye 95% of the time.
 Then each individual shot has a 95%
chance of hitting within 5 inches.

The Target Analogy
The Target Analogy
The Target Analogy
The Target Analogy
The Target Analogy
The Target Analogy
The Target Analogy
Now suppose we are shown where the
shot hit, but we are not shown where the
bull’s eye is.
 What is the probability that the bull’s eye is
within 5 inches of that shot?

The Target Analogy
The Target Analogy
The Target Analogy
Where is the bull’s eye?
The Target Analogy
5 inches
The Target Analogy
5 inches
95% chance that the
bull’s eye is within
this circle.
The Confidence Interval

In a similar way, 95% of the sample
proportions p^ should lie within 1.96
standard deviations (p^) of the parameter
p.
The Confidence Interval
p
The Confidence Interval
1.96 p^
p
The Confidence Interval
1.96 p^
p
The Confidence Interval
1.96 p^
p
The Confidence Interval
1.96 p^
p
The Confidence Interval
1.96 p^
p
The Confidence Interval
1.96 p^
p
The Confidence Interval

Therefore, if we compute a single p^, then
we expect that there is a 95% chance that
it lies within a distance 1.96p^ of p.
The Confidence Interval
The Confidence Interval
The Confidence Interval
p^
Where is p?
The Confidence Interval
1.96 p^
p^
The Confidence Interval
1.96 p^
p^
95% chance that p is
within this interval
Approximate 95% Confidence
Intervals

Thus, the 95% confidence interval would
be
pˆ  1.96 pˆ
The trouble is, to know p^, we must know
p. (See the formula for p^.)
 The best we can do is to use p^ in place of
p to estimate p^.

Approximate 95% Confidence
Intervals

That is,
 pˆ 

pˆ 1  pˆ 
n
This is called the standard error of p^ and
is denoted SE(p^).
ˆ) 
SE( p
ˆ 1  p
ˆ
p
n
Approximate 95% Confidence
Intervals

Therefore, the 95% confidence interval is
ˆ  1.96  SE p
ˆ
p
Example
Example 9.6, p. 585 – Study: Chronic
Fatigue Common.
 Rework the problem supposing that 350
out of 3066 people reported that they
suffer from chronic fatigue syndrome.
 How should we interpret the confidence
interval?

Standard Confidence Levels

The standard confidence levels are 90%,
95%, 99%, and 99.9%. (See p. 588 and
Table III, p. A-6.)
Confidence Level
90%
95%
z
1.645
1.960
99%
99.9%
2.576
3.291
The Confidence Interval

The confidence interval is given by the
formula
ˆ  z  SE p
ˆ
p
where z
 Is
given by the previous chart, or
 Is found in the normal table, or
 Is obtained using the invNorm function on the
TI-83.
Confidence Level

Rework Example 9.6, p. 585, by
computing a
 90%
confidence interval.
 99% confidence interval.
Which one is widest?
 In which one do we have the most
confidence?

TI-83 – Confidence Intervals
The TI-83 will compute a confidence
interval for a population proportion.
 Press STAT.
 Select TESTS.
 Select 1-PropZInt.

 (Note
that it is “Int,” not “Test.”)
TI-83 – Confidence Intervals
A display appears requesting information.
 Enter x, the numerator of the sample
proportion.
 Enter n, the sample size.
 Enter the confidence level, as a decimal.
 Select Calculate and press ENTER.

TI-83 – Confidence Intervals

A display appears with several items.
 The
title “1-PropZInt.”
 The confidence interval, in interval notation.
 The sample proportion p^.
 The sample size.

How would you find the margin of error?
TI-83 – Confidence Intervals

Rework Example 9.6, p. 585, using the TI83.
Probability of Error
We use the symbol  to represent the
probability that the confidence interval is in
error.
 That is,  is the probability that p is not in
the confidence interval.
 In a 95% confidence interval,  = 0.05.

Probability of Error

Thus, the area in each tail is /2.
Confidence
Level
90%
95%
99%
99.9%

invNorm(/2)
0.10
0.05
0.01
0.001
-1.645
-1.960
-2.576
-3.291
Which Confidence Interval is Best?

Which is better?
 A large
margin of error (wide interval), or
 A small margin of error (narrow interval).

Which is better?
 A low
level of confidence, or
 A high level of confidence.
Which Confidence Interval is Best?
Why not get a confidence interval that has
a small margin of error and has a high
level of confidence associated with it?
 Hey, why not a margin of error of 0 and a
confidence level of 100%?

Which Confidence Interval is Best?

Which is better?
 A smaller
sample size, or
 A larger sample size.
Which Confidence Interval is Best?
A larger sample size is better only up to
the point where its cost is not worth its
benefit.
 That is why we settle for a certain margin
of error and a confidence level of less than
100%.
