Chapter 3
Estimation of
p
3.1
Point and Interval Estimates of
p
Suppose that we have BT. So far, in every example I have told you the (numerical) value of
p
.
In science, usually the value of
p
is unknown to the researcher.
In such cases, scientists and
statisticians use data from BT to
estimate
the value of
p
. Note that the word
estimate
is a technical
term that has a precise definition in this course. I don’t particularly like the choice of the word
estimate
for what we do, but I am not the tsar of the Statistics world!
It will be very convenient for your learning if we distinguish between two creatures. First, is
Nature
, who knows everything and in particular knows the value of
p
. Second is the researcher
who is ignorant of the value of
p
.
Here is the idea. A researcher plans to observe
n
BT, but does not know the value of
p
. After
the BT have been observed the researcher will use the information obtained to make a statement
about what
p
might be.
After observing the BT, the researcher counts the number of successes,
x
, in the
n
BT. We
define
ˆ
p
=
x/n
, the proportion of successes in the sample, to be the
point estimate
of
p
.
For example, if I observe
n
= 20
BT and count
x
= 13
successes, then my point estimate of
p
is
ˆ
p
= 13
/
20 = 0
.
65
.
It is trivially easy to calculate
ˆ
p
=
x/n
; thus, based on your experiences in previous math
courses, you might expect that we will move along to the next topic. But we won’t.
What we do in a Statistics course is
evaluate the behavior
of our procedure. What does that
mean?
Well, recall that in first lecture I stated that statisticians evaluate procedures by seeing how they
perform
in the long run
.
We say that the point estimate
ˆ
p
is
correct
if, and only if,
ˆ
p
=
p
.
Obviously, any honest
researcher wants the point estimate to be correct. Let’s go back to the example of a researcher who
observes 13 successes in 20 BT and calculates
ˆ
p
= 13
/
20 = 0
.
65
.
The researcher schedules a press conference and the following exchange is recorded.
•
Researcher: I know that all Americans are curious about the value of
p
. I am here today to
announce that based on my incredible effort, wisdom and brilliance, I estimate
p
to be 0.65.
23
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•
Reporter: Great, but what is the actual value of
p
? Are you saying that
p
= 0
.
65
?
•
Researcher: Well, I don’t actually know what
p
is, but I certainly hope that it equals 0.65.
As I have stated many times, nobody is better than me at obtaining correct point estimates.
•
Reporter: Granted, but is anybody worse than you at obtaining correct point estimates?
•
Researcher: (Mumbling) Well, no. You see, the problem is that only Nature knows the actual
value of
p
. No mere researcher will ever know it.
•
Reporter: Then why are we here?
Before we follow the reporter’s suggestion and give up, let’s see what we can learn.
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 Fall '09
 ProfessorWardrop
 Statistics, Estimation theory, researcher, interval estimate

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