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Chapter 3
Estimation of
p
3.1
Point and Interval Estimates of
p
Suppose that we have Bernoulli Trials (BT). So far, in every example I have told you the (numer
ical) value of
p
.I
ns
c
i
e
n
c
e
,u
s
u
a
l
l
yt
h
ev
a
l
u
eo
f
p
is unknown to the researcher. In such cases,
scientists and statisticians use data from the BT to
estimate
the value of
p
.N
o
t
et
h
a
tt
h
ew
o
r
d
estimate
is a technical term that has a precise deFnition 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 distinguishbetweentwocreatures
. ±irst
,is
Nature
,whoknowsevery
th
ingand
,inpar
t
icu
lar
,knowstheva
lueof
p
.Secondi
sthere
sea
rche
r
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
.A
f
te
r
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
,inthe
n
BT. We
deFne
ˆ
p
=
x/n
,theproportionofsuccessesinthesample
,tobethe
point estimate
of
p
.
±or 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
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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 I at obtaining correct point estimates.
•
Reporter: Granted, but is anybody worse than you at obtainingcorrectpointestimates?
•
Researcher: (Mumbling) Well, no. You see, the problem is thatonlyNatureknowstheactual
value of
p
.Nomereresearcherw
i
l
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 Fall '11
 hanlon
 Bernoulli

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