Chapter 6
Inference for a Population
6.1
Study Suggestions
Let
X
be a random variable with the binomial sam
pling distribution with parameters
n
and
p
. In Chap
ter 5 you learned that
X
can arise when taking a ran
dom sample from a finite population, or when ob
serving a sequence of Bernoulli trials. In Chapter 5
the focus was on using the values of
n
and
p
to com
pute the probability of an event involving
X
. Chap
ter 6 considers the more realistic situation in which
the value of
p
is unknown. In Chapter 6 you learn
how the observed value of
X
can be used to draw an
inference about the unknown value of
p
.
The first example of inference is the point esti
mate.
The proportion of successes
in the sample
,
ˆ
p
=
x/n
, is the point estimate of the proportion
of successes
in the population
. The main advantage
of the point estimate is that it is easy to understand
and to interpret, and its main disadvantage is that
it is usually incorrect.
The confidence interval es
timate sacrifices some of the simplicity of a point es
timate to achieve a high confidence of being correct.
(Remember that the terms correct and incorrect do
not refer to the estimate being
computed
without or
with error; they refer to whether the stated estimate
equals, in the case of a point estimate, or contains, in
the case of an interval estimate, the actual value of
p
.)
A confidence interval estimate has two parts to
it—the interval and the confidence level. The con
fidence level is selected by the researcher and the in
terval is computed from the data.
For example, if
a researcher selects 95 percent confidence, then the
interval is
ˆ
p
±
1
.
96
s
ˆ
p
ˆ
q
n
.
The “95 percent confidence” in the 95 percent confi
dence interval is interpreted as follows. Before col
lecting data, the probability is 95 percent that the
data to be obtained will yield an interval that in
cludes the actual value of
p
.
My students find Ta
ble 6.2 and Figure 6.1 on page 197 of the text to be
helpful in their achieving an understanding of what
“confidence” means. In particular, note that in a real
application in which the value of
p
is unknown, a
researcher will never know whether a confidence in
terval is correct.
If you find this uncertainty unac
ceptable, then you will probably never consider con
fidence intervals to be of much practical value. I view
a confidence interval as a reasonable middle posi
tion between the extremes of taking a census (which
is usually prohibitively expensive and time consum
ing), and collecting no data.
In the majority of applications I have seen, the
researcher simply wants to estimate the value of
p
.
Occasionally, the researcher will consider one of the
possible values of
p
to be of special interest. In this
latter case, the researcher can perform a hypothesis
test with the null hypothesis stating that
p
equals the
special value of interest,
p
0
.
With the solid back
ground of Fisher’s test, my students find the hypoth
esis testing in Chapter 6 to be fairly easy to under
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 Fall '09
 ProfessorWardrop
 Statistics, Binomial, Statistical hypothesis testing, researcher, percent confidence interval, percent prediction interval

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