Chapter 7 Comparing Two Populations
7.1 Some new concepts
The framework:
Until now we had one population, one random sample from that population and just one
parameter (μ or π) with an unknown value and we made inference about the unknown value of
the parameter.
In this Chapter
•
We have
two populations,
a population of X’s [say, {X
1
, X
2
, …, X
N
} ] and a population
of Y’s, [say, {Y
1
, Y
2
, …, Y
M
}].
•
If X and Y are both quantitative variables,
the population means are μ
X
and μ
Y
, and
standard deviations are σ
X
and σ
Y
, respectively; we are interested in making inferences
about the difference of population means, μ
X
– μ
Y
.
•
If both X and Y are categorical variables,
each with two categories, then the parameter
of interest is the difference between the proportion of “Success”s, denoted by π
X
, in the
population of X’s and proportion of “Success”s, denoted by π
Y
in the population of Y’s.
•
We select (simple) random sample of size n
X
from the population of X’s and a (simple)
random sample of size n
Y
from the population of Y’s to make inferences about the
difference of the parameters of interest in the respective populations.
•
The samples can be either
independent
of each other or they may be
dependent
on each
other.
Definition:
Two random samples are said to be
independent samples
if the selection of a
unit from one population has no effect on the selection or non-selection of another unit
from the second population. Otherwise the samples are said to be
dependent samples.
Independent samples are used in most applications. However, in some applications the selection
of one unit will from one of the populations determines the selection of another one from the
second population. Such samples are said to be
dependent samples.
In such applications one
unit from each population come in pairs. Thus
we have a random sample of pairs
. These pairs
either come naturally (e.g., twins-studies, studies of married couples, observations on the same
person under two different conditions, etc.) or the pairs are created by the experimenter, being
matched on as many characteristics as possible, except the one characteristic of interest to the
researcher.
We have the usual general formulas for confidence intervals and the usual 6 steps of making
statistical tests.
STA 6126 Chap 7, Page 1 of 21