Biology Reference
In-Depth Information
The value s
2
is sometimes called the empirical variance or sample variance.
The square root, s, is called the empirical standard deviation of the random
variable
x
.
x
If we need to measure two different random variables
and
having
x
a normal distribution, we record the data points for
and
as samples
A and B. Sample A contains the data points measured for
x
and
. In our earlier example,
samples A and B contained the yield data for corn varieties A and B,
presented in Tables 4-2 and 4-3. To distinguish between the
mathematical expressions using data from sample A from those using
sample B, we use the name of the sample as part of the notation. For
example, we use x
A
to denote the average of the data points from A,
and s
B
¼
sample B contains the points measured for
s
B
ð
to denote the maximum likelihood estimate of the
variance calculated by the formula in Eq. (4-4) with the data from
sample B. In the latter case, the value of N will correspond to the number
of data points in the sample B.
N
Þ
Recall that the sum of two independent, normally distributed variables
also has a normal distribution, with a mean equal to the sum of the
means and variance
e
qual to the sum of the v
ar
iances. It follows then,
the empirical mean x and the difference
w
ill
al
so have normal
distributions. The same holds for th
e
difference x
A
ð
x
i
x
Þ
when we
consider two samples. Particularly, x will have
a
mean equal to
x
B
;
m
and
2
/N. Further, because
a variance equal to
has normal
distribution, the estimate of the variance s
2
, which is a sum of the
squares of N such quantities, will have an approximately
s
ð
x
i
x
Þ
2
distribution
w
with N degrees of freedom.
When these considerations are paired with the definitions we gave for
w
2
distribution, t-distribution, and F-distribution, in general, the
following broad principles hold:
1. Any statistical test (such as the Z-test, as we shall see later) using
the difference x
A
x
B
, between the empirical means of two samples
A and B as their test statistic requires the use of a normal
distribution;
2. Any statistical test (such as the t-test, as we shall see later)
using a variant of the ratio x
s (empirical mean/empirical
standard deviation) as their test statistic, requires the use of a
t-distribution (as it is approximately normal/
=
p
Chi
square
);
and
3. Any statistical test (such as the F-test, as we shall see later) using
the ratio s
A
ð
s
B
ð
M
Þ=
N
Þ
(empirical variance of one sample with M
