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STA 3032 Chap 6 Part 2, Page 1 of 20
Inferences from two samples
(Sections 9.3, 9.4, 10.3)
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 nonselection 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 make up a pair. Thus
we
have a random sample of pairs
.
These pairs either come naturally (e.g., twinsstudies, 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. In many experiments the same population unit is used as its pair (before – after
studies).
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Point Estimation:
The following table gives difference of two population parameters and their point estimators:
Difference of
Two Population Parameters
Estimators of the Difference of
Two Population Parameters
12
= Difference of two
population means
1
2
1
2
XX
= Difference of the two
sample means
p
1
– p
2
= Difference of two
population proportions
1
2
1
2
ˆˆ
p
p
p
p
= Difference of two
Sample proportions
We have the usual general formula for confidence intervals and the six steps of hypothesis
testing.
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This note was uploaded on 11/12/2011 for the course STA 3032 taught by Professor Kyung during the Summer '08 term at University of Florida.
 Summer '08
 Kyung
 Statistics

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