Geoscience Reference
In-Depth Information
7.15.1.1 Sample Size
If there is a real difference of
D
feet between the two races of white pine, how many replicates
(plots) would be needed to show that it is significant? To answer this, we first assume that the
number of replicates will be the same for each group
(n
A
= n
B
= n
). The equation for
t
can then be
written as
22
2
D
s
n
2
ts
D
t
=
or
n
=
2
2
Next we need an estimate of the within-group variance,
s
2
. As usual, this must be determined from
previous experiments, or by special study of the populations.
■
EXAMPLE 7.9
Problem:
Suppose that we plan to test at the 0.05 level and wish to detect a true difference of
D
= 1
cord if it exists. From previous tests, we estimate
s
2
= 5.0. Thus, we have
22
2
2
ts
D
50
10
.
.
2
n
=
=
2
t
Here we hit a snag. In order to estimate
n
we need a value for
t
, but the value of
t
depends on the
number of degrees of freedom, which depends on
n
. The situation calls for an iterative solution,
which is a mathematical procedure that generates a sequence of improving approximate solutions
for a class of problems—in other words, a fancy name for trial and error. We start with a guessed
value for
n
, say
n
0
= 20. As
t
has (
n
A
- 1) + (
n
B
- 1) = 2(
n
- 1) degrees of freedom, we'll use
t
= 2.025
(which is equal to
t
0.05
with 38 degrees of freedom) and compute
50
10
.
.
=
2
n
1
=
22025
(.
)
41
The proper value of
n
will be somewhere between
n
0
and
n
1
—much closer to
n
1
than to
n
0.
We can
now make a second guess at
n
and repeat the process. If we try
n
2
= 38,
t
will have 2(
n
- 1) = 74
degrees of freedom and
t
0.05
= 1.992. Thus,
50
10
.
.
=
2
n
3
=
21992
(.
)
39 7
.
Thus,
n
appears to be over 39 and we will use
n
= 40 plots for each group, or a total of 80 plots.
7.15.2
t
t
est
For
p
aired
p
lots
A second test was made of the two races of white pine. It also had 11 replicates of each race, but
instead of the two races being assigned completely at random over the 22 plots, the plots were
grouped into 11 pairs and a different race was randomly assigned to each member of a pair. The
cordwood volumes at the end of the growth period were as follows:
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