Geoscience Reference
In-Depth Information
pd = makedist('normal',0,1)
y = icdf(pd,[.025,.975])
[p,h] = normspec(y,0,1,'outside');
and obtain a plot with two blue tails, one to the let and one to the right, with
the 2.5% areas shaded (Fig. 3.13 b).
Note that we cannot prove the null hypothesis, in other words
not guilty
is not the same as
innocent
. In practice, we design hypotheses based on our
data, test them, and then continue to work with those found to be true, those
we could not show to be false. h e inherent possibility to prove a hypothesis
to be false is therefore an important requirement for our hypotheses.
h e next sections introduce the most important hypothesis tests for earth
science applications: the two-sample
t
-test to compare the means of two data
sets, the two-sample
F
-test to compare the variances of two data sets, and the
ˇ
2
-test and Kolmogorov-Smirnov test to compare distributions (Sections 3.7
to 3.10). h e Mann-Whitney and Ansari-Bradley tests are alternatives to the
t
-test and
F
-test for comparing the medians and dispersions of two data sets
without requiring a normality assumption for the underlying population
(Sections 3.11 and 3.12). h e i nal section introduces methods that can be
used to i t distributions to our data sets (Section 3.13).
3.7 The t-Test
h e Student's
t
-test by William Sealy Gosset (Student 1908) compares
the means of two distributions. h e
one-sample t-test
is used to test the
hypothesis that the mean of a Gaussian-distributed population has a value
specii ed in the null hypothesis. h e
two-sample t-test
is employed to test the
hypothesis that the means of two Gaussian distributions are identical. In the
following text the two-sample
t
-test is introduced to demonstrate hypothesis
testing. Let us assume that two independent sets of
n
a
and
n
b
measurements
have been carried out on the same object, for instance measurements on two
sets of rock samples taken from two separate outcrops. h e
t
-test can be used
to determine whether both samples come from the same population, e.g.,
the same lithologic unit (
null hypothesis
) or from two dif erent populations
(
alternative hypothesis
). Both sample distributions must be Gaussian and the
variances for the two sets of measurements should be similar. h eappropriate
test statistic for the dif erence between the two means is then
