1
Inference about the Difference Between Two Means
The objective of many studies is to compare two means in addition
to estimating individual means.
Suppose there would are two populations, one with mean μ
1
and
the other with mean μ
2
, and we want to make inference about the
difference μ
1
 μ
2
.
We might want to test the null hypothesis
H
0
:μ
1
=μ
2
, or construct a confidence interval for the difference μ
1

μ
2
.
Which statistical method you use depends on how you obtained the
data.
Basically, there are two types of samples,
independent
and
paired
.
Independent Samples:
You have a sample of n
1
observations from
population 1 and a sample of n
2
observations from population 2.
The observations of the first sample are denoted y
11
,…,y
1n1
, and
the observations of the second sample are denoted y
21
,…,y
2n2
,
Paired Samples:
You have two samples, but each observation in
the first sample is related to an observation in the second sample.
The related observations make a pair.
The data are denoted
(y
11
,y
21
),…,(y
1n
,y
2n
), where y
1i
and y
2i
are two observations that are
related from samples 1 and 2, respectively.
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2
Inference about the Difference Between Two Means
Comparing means from two independent samples
0
1
2
1
2
:
:
a
H
H
µ
µ
µ
µ
=
>
Test Statistic:
(
)
(
)
2
2
1
1
2
2
2
1
2
1
2
2
1
2
1
2
1
1
,
2
1
1
2
p
p
n
s
n
s
y
y
t
s
n
n
s
n
n
d f
n
n
−
+
−
−
=
=
+
−
⎛
⎞
+
⎜
⎟
⎝
⎠
=
+
−
where
1
y
and
s
1
2
are the mean and variance from sample 1, and
2
y
and
s
2
2
are the mean and variance from sample 2. The sample
variance
s
1
2
and
s
2
2
have n
1
– 1 and n
2
– 1 degrees of freedom,
respectively.
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 Summer '08
 Staff
 Statistics, Normal Distribution, Statistical hypothesis testing, Means, hand span

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