Information Technology Reference
In-Depth Information
18.2.2
Statistical Reasoning
In statistical analysis we use much probability concepts and many of these are im-
precise by nature. According to the Zadeh's
FLe
, the idea on the fuzzy probability
generally means approximate probability variables and approximate values of these
variables [3, 25]. For example,
•
The probability that John's age is 20 is approximately 0.95.
•
The probability that John's age is 20 is very high.
•
The probability that John's age is approximately 20 is very high.
•
The probability that John is young is very high.
In particular, statistical tests apply much probability and random distributions when
we consider the acceptance of the null and alternative hypotheses according to the
tests of significance [6],[11],[12],[28]. We accept the null hypothesis if the value
of our test variable does not deviate too much from the “usual” case, otherwise
we reject it and accept the alternative hypothesis. In practice we operate with the
p-values (level of significance, (0
1) in a computer environment in which case
we consider the rejection of the null hypotheses if the obtained p-value is sufficiently
small. In other words, the p-value is our risk to draw an erroneous conclusion if we
reject the null hypothesis.
Traditionally the statistical hypothesis verification is based on bivalent reasoning,
and thus the rejection of one hypothesis automatically means the acceptance of the
other. Hence, formally we reason that if the p-value is greater than a given limit
point, we accept the null hypothesis, otherwise we reject it.
≤
p
≤
The usual threshold
.
.
values for
p
are
01 (5 % and 1 % levels of significance, respectively). For
example, when the t-test is applied to two independent samples of data (the two-
tailed case), we have the null hypothesis that there is no difference in the means
between the groups, whereas the alternative hypothesis asserts that this difference
prevails.
In practice we nevertheless take into account the borderline cases when the ac-
ceptance of the null hypothesis is considered. For example, we may pay special
attention to the p-values which are in the close neighborhood of
p
05 or
05 in order to
avoid erroneous conclusions. Hence, we actually apply approximate reasoning and
probability.
In approximate reasoning we may operate with the degrees of acceptance and
rejection in this context and thus acquire more informative outcomes. For example,
we can establish the meta-rule that the smaller the p-value, the lower the risk of error
for rejecting the null hypothesis, in other words, the higher the degree of rejection
for the null hypothesis. Simultaneously, then it also holds that the higher the degree
of accepting the alternative hypothesis. On Osgood scale it would mean such values
as
high degree of acceptance
,
fairly high degree of acceptance
,
borderline case
,
fairly high degree of rejection
and
high degree of rejection
. We may even construct
a fuzzy inference engine including this type of fuzzy rule base for this task.
=
.
