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In-Depth Information
Students studying for two to four months versus those studying six
to eight months.
There are some common steps involved in testing these three hypothe-
ses. The first step involves creating a difference score between the two
means being compared. We will refer to this value as
ˆ
(psi hat), which
statisticians use to refer to a difference between two means. The caret
or “hat” above the symbol indicates that this value is an estimate of the
difference between the population treatment means.
The second step is to compute a new comparison sum of squares, which
we will call
SS
A
comp
,andanewmeansquare(
MS
A
comp
), and form a new
F
ratio that will produce a new
F
A
comp
that can be evaluated to determine
if the difference between the two means (captured by
ψ
ˆ
ψ
)isstatistically
significant (see Keppel et al., 1992). The formulas for these three values
with their numerical examples for each hypothesis follow.
Sometimes researchers are interested in
simple
comparisons between
two treatment means. On other occasions, researchers examine more
complex
or sophisticated hypotheses that require the averaging of two or
more groups in order to compare them to another single group or an
average of two or more other groups. We will demonstrate the creation of
these simple and complex comparisons by weighting the treatment means
with special numbers or weights called “coefficients” (see Keppel et al.,
1992; Keppel & Wickens, 2004).
7.27.1 HYPOTHESIS 1
The coefficients used to assess the first hypothesis are as follows:
4
−
1
−
1
−
1
−
1
.
These coefficients indicate that the first treatment group (zero months
of study) is being compared to four other (combined) study groups.
This is accomplished by weighting or multiplying each treatment mean
by its respective coefficient to give that mean more or less input during
the computation of the comparison sum of squares. The formula that
generates this weighting (
ˆ
ψ
)isasfollows:
ˆ
ψ
1
=
(
c
1
)(
Y
1
)
+
(
c
2
)(
Y
2
)
+
(
c
3
)(
Y
3
)
+
(
c
4
)(
Y
4
)
+
(
c
5
)(
Y
5
)
=
(4)(412
.
86)
+
(
−
1)(474
.
29)
+
(
−
1)(552
.
86)
+
−
.
+
−
.
(
1)(614
29)
(
1)(623
86)
= −
613.86
.
(7.13)
ˆ
With
ψ
calculated, we are now ready to proceed with the formula for
the
SS
A
comp
.
n
ˆ
ψ
2
SS
A
comp
=
.
(7.14)
c
2
