Chapter 5
OnePopulation Models
5.1
Study Suggestions
In the first four chapters of the text, the results of a
study literally were restricted to the subjects in the
study. Chapter 5 begins the investigation of methods
for extending what has been learned in a study. The
notion of a population is the fundamental concept in
volved in these extensions.
Throughout Chapter 5 and the remainder of the
text, I stress the existence of two types of popula
tions that correspond to the two types of subjects in
troduced in Chapter 1, distinct individuals or trials.
At first my approach may appear to be a wasteful ex
travagance because the mathematical techniques that
work for one type of population also work for the
other type. There are a number of advantages, how
ever, in having two types of populations; some of
these advantages are discussed below.
•
A finite population is a tangible entity. The pop
ulation of students registered at my university
this semester is very concrete. By contrast, an
infinite population is a bit more slippery for a
nonmathematician. The notion of “a mathemat
ical model for the process that generates the out
comes of my shooting free throws,” is a bit over
whelming for many of my students.
•
A census is possible for a finite population, but
not for an infinite population.
•
Obtaining a random sample presents a different
challenge for the two types of populations. For
a finite population the challenge lies in obtain
ing a listing of the population members, select
ing a sample of subjects at random, finding the
selected subjects, and convincing them to par
ticipate in the study.
For trials, the challenge
lies in verifying (see below) the assumptions of
Bernoulli trials.
•
The goals of inference can depend on the type
of population.
For example, in many applica
tions prediction is the most important method
of inference for trials, but it is rarely important
for a finite population.
For the material in Chapter 15, it is important to
realize that the population in Chapter 5 is a single
number
p
.
Pages 153–55 of the text provide a bruteforce jus
tification of the multiplication rule.
This argument
does not work well for my students and I no longer
use it in my class.
Instead, I appeal to the long
run relative frequency interpretation of probability
to motivate the multiplication rule. More precisely,
suppose
p
= 0
.
6
, and suppose interest lies in the
event
(
X
1
= 1
, X
2
= 0)
.
The condition for invoking the longrun relative fre
quency interpretation of probability is met: the ex
periment of selecting two cards at random with re
placement from the population box can certainly be
repeated a large number of times under identical con
ditions. Suppose that the experiment is, in fact, to be
repeated a large number of times.
Since
p
= 0
.
6
,
about 60 percent of the first selections will yield a
card marked 1. Regardless of what happens on the
first selection, because
q
= 0
.
4
, about 40 percent
of the second selections will yield a card marked
0.
Hence, of the 60 percent of first selections that
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 Fall '09
 ProfessorWardrop
 Statistics, Probability theory, Binomial distribution, Coin flipping, Bernoulli trial, binomial sampling distribution

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