561
Chapter 11
Inferences on Two Samples
11.1
Inferences about Two Means: Dependent Samples
1.
independent
2.
dependent
3.
Since the researcher claims the mean of population 1,
1
μ
, is less than the mean of
population 2,
2
μ
,
in matched pair data, the difference
1
2
μ
μ
−
should be negative.
Thus,
define
1
:
0
d
H
μ
<
with
i
i
i
d
X
Y
=
−
.
4.
To test a claim regarding the differences of two means with dependent sampling, (1) the
sample must be obtained using simple random sampling;
(2) the sample must be matched
pairs; and (3) the differences must either be normally distributed with no outliers or the
sample size must be large (i.e.,
30
n
≥
).
5.
Since the members of the two samples are married to each other, the sampling is
dependent.
6.
Because the 100 subjects are randomly allocated to one of two groups, the sampling is
independent.
7.
Because the 80 students are randomly allocated to one of two groups, the sampling is
independent.
8.
Because the samples are obtained by giving different treatments to the same subjects, the
sampling is dependent.
9.
Because the two sets of twins are chosen at random, the sampling is independent.
10.
Because the 30 subplots are randomly allocated to one of two groups, the sampling is
independent.
11.
(a)
7.6
7.6
7.4
5.7
8.3
6.6
5.6
8.1
6.6
10.7
9.4
7.8
9.0
8.5
0.5
1.0
3.3
3.7
0.5
2.4
2.9
i
i
i
i
i
X
Y
d
X
Y
=
−
−
−
−
−
−
Observation
1
2
3
4
5
6
7
(b)
Using technology,
1.614
d
≈ −
and
1.915
d
s
≈
.
(c)
The hypotheses are
0
:
0
d
H
μ
=
versus
1
:
0
d
H
μ
<
.
The level of significance is
0.05
α
=
.
The test statistic is
0
1.614
2.230
/
1.915/
7
d
d
t
s
n
−
=
=
≈ −
.
Classical approach
: Since this is a left-tailed test with 6 degrees of freedom, the critical
value is
0.05
1.943
t
−
= −
. Since the test statistic
0
2.230
t
≈ −
is less than the critical value
0.05
1.943
t
−
= −
(i.e., since the test statistic fall within the critical region), we reject
0
H
.

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