Geoscience Reference
In-Depth Information
7.15
COMPARING TWO GROUPS BY THE
t
TEST
7.15.1
t
t
est
For
u
npaired
p
lots
An individual unit in a population may be characterized in a number of different ways. A single tree,
for example, can be described as alive or dead, hardwood or softwood, infected or not infected, and
so forth. When dealing with observations of this type, we usually want to estimate the proportion of
a population having a certain attribute. Or, if there are two or more different groups, we will often
be interested in testing whether or not the groups differ in the proportions of individuals having
the specified attribute. Some methods of handling these problems have been discussed in previous
sections.
Alternatively, we might describe a tree by a measurement of some characteristic such as its diam-
eter, height, or cubic volume. For this measurement type of observation we may wish to estimate the
mean for a group as discussed in the section on sampling for measurement variables. If there are two
or more groups we will frequently want to test whether or not the group means are different. Often
the groups will represent types of treatment that we wish to compare. Under certain conditions, the
t
or
F
tests may be used for this purpose.
Both of these tests have a wide variety of applications. For the present we will confine our
attention to tests of the hypothesis that there is no difference between treatment (or group) means.
The computational routine depends on how the observations have been selected or arranged. The
first illustration of a
t
test of the hypothesis that there is no difference between the means of two
treatments assumes that the treatments have been assigned to the experimental units completely at
random. Except for the fact that there are usually (but not necessarily) an equal number of units or
“plots” for each treatment, there is no restriction on the random assignment of treatments.
In this example the “treatments” were two races of white pine which were to be compared on the
basis of their volume production over a specified period of time. Twenty-two square, 1-acre plots
were staked out for the study; 11 of these were selected entirely at random and planted with seed-
lings of race A. The remaining 11 were planted with seedlings of race B. After the prescribed time
period the pulpwood volume (in cords—a stack of wood 4 ft wide by 4 ft high by 8 ft in length) was
determined for each plot:
Race A
Race B
11
5
9
9
6
9
8
10
11
9
13
8
10
8
11
6
5
6
8
8
10
7
Sum = 99
Sum = 88
Average = 9.0
Average = 8.0
To test the hypothesis that there is no difference between the race means (sometimes referred to as
a null hypothesis—general or default position) we compute
XX
sn n
nn
−
AB
t
=
2
( )
( ()
+
AB
AB
wh
er
e
X
A
and
X
B
=
Arithmetic means for groups A and B.
n
A
and
n
B
= Number of observations in groups A and B (
n
A
and
n
B
do not have to be the same).
s
2
= Pooled within-group variance (calculation shown below).
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