May 17’09
SAMPLING DISTRIBUTION OF THE DIFFERENCE
#11
BETWEEN TWO SAMPLE MEANS.
CONSTRUCTION
OF THE CONFIDENCE INTERVAL
Frequently the interest in a statistical investigation is focused on
two
populations. For
example, an investigator may wish to know something about the
difference
between two
population means to find out, if it is reasonable to conclude that two population means
are different. If it was found that the population means are different, then the investigator
may wish to know by how much they differ. Thus, the
magnitude of the
difference
between two population means may be the subject of interest. In investigations of this
type knowledge of the properties of
sampling distribution
of the difference between two
means is important.
Example
. Suppose we have two populations (
1
and
2
) of individuals – the population
1
has experienced some condition thought to be associated with mental retardation, and
the other population
2
has
not
experienced the condition.
The distribution of intelligence
scores in each of the two populations is believed to be approximately normal with a
standard deviation of
20
.
Suppose, further, that we take a sample of
15
individuals from each population and
compute for each sample the mean intelligence score with the following results:
_
_
x
1
*
= x
mean_1
*
= 92
and
x
2
*
= x
mean_2
*
= 105
.
If there is no difference between the two populations, with respect to their
true
mean
intelligence scores, what is a probability of observing a difference between two sample
means that is less than or equal to the obtained value of
(x
mean_1
*
– x
mean_2
*
) = –13
?
Solution
. To answer this question we need to know the nature of the sampling distribution
of the relevant statistic,
the difference
between two sample means
,
x
mean_1
– x
mean_2
.
Although, in practice, we would not attempt to construct the desired sampling
distribution, we can conceptualize the manner in which it could be done when sampling
is from finite populations. We would begin by selecting from population
1
(of size
N
1
) all
possible samples of size
n
1
=
15
and computing the mean for each sample. Similarly, we
would select from population
2
(of size
N
2
) all possible samples of size
n
2
=
15
and
compute the mean for each of these samples. We would then take all possible
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 Spring '05
 Kzaer
 Normal Distribution

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