One-Population 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 brute-force 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 long-run 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