Biology Reference
In-Depth Information
Because we are considering a one-sided hypothesis, we find a value
V
0
such that the area under the curve to the ri
gh
t of V
0
is exactly 0.05
[see Figure 4-4(B)]. Next, we plot the value
a ¼
x
B
x
A
on the horizontal
axis. If the value of
falls to the right of V
0
, this would mean that
if we reject the null hypothesis, the chance we are wrong (that is, the
chance we shall reject the null hypothesis when it is, in fact, true) would
be smaller than 5%.
a
If we have a two-sided alternative hypothesis H:
m
B
6¼ m
A
, we need to
find two numbers V
1
and V
2
[see Figure 4-4(C)] such that the area
to the left of V
1
and to the right of V
2
is 0.025, and thus the total area
under the curve outside the interval [V
1
,V
2
] is 0.05. If the value
a ¼
x
A
falls to the right of V
2
or to the left of V
1
, then the chance we
shall wrongly reject the null hypothesis will be less than 5%.
x
B
The p-value produced by statistical software such as MINITAB and
SPSS is equal (in the case of one-sided hypotheses) to the area under
the curve to the right of the value of our statistic
a ¼
x
B
x
A
:
Therefore,
if this p-value is less than 0.05, this means
x
A
is to the
right of the value V
0
, and therefore H
0
can be rejected with a chance
of type I error less than 5%. In the case of two-sided hypotheses,
the p-value repres
e
nts
th
e c
o
mbi
n
ed area under the curve outside
of the interval
a ¼
x
B
The important thing to
remember is that, in all cases, the p-value is exactly the probability for
type I error.
½j
x
B
x
A
j; j
x
B
x
A
j:
One question we have not addressed so far is how to determine the
actual type of the sampling distributions under the assumption the null
hypothesis is true. This choice is based on underlying assumptions for
the populations, as well as on the type of parameters and claims
referenced by the null hypothesis. As we shall see, the probability
distributions we introduced above play a fundamental role in this
process. The following cases will be quite common:
Case I. The null hypothesis deals with comparing mean values; and
Case II. The null hypothesis deals with comparing variances.
In the next two sections, we outline some basic statistical tests that allow
for hypothesis testing of the above cases. They represent essential
ideas that will be needed in later chapters to understand how to
interpret the results from other statistical analyses. As a very broad rule,
when the assumptions for normality are met and the sample size is large
enough, hypotheses of the type outlined in case I would use a
Z-test or t-test, while those outlined in case II would use an F-test.
In the next section, we complete the corn example, which represents
a special instance of case I. In Section C, we illustrate case II by
examining testing for heritability.
