GENERAL BACKGROUND - GPS Satellite Surveying - page 365

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the alternative hypothesis is accepted if the computed sample statistics t falls in that
region. Thus, reject H 0 if

t>t α (A.149)

The shape and location of the density function of the test statistics under the
alt ernative hypothesis depend on the specifications of the alternative hypothesis.
Th us, the probability of a type-II error,

, depends on the specifications of H 1 .A
de sirable approach in statistical testing would be to minimize the probability of
bo th types of errors. However, this is not practical, because all distributions of the
alt ernative hypotheses, which, in general, are of the noncentral type, would have to
be computed. Figure A.6 shows that the probability

β

β increases as α decreases. A

co mmon procedure is to fix the probability of a type-I error to, say, α = 0 . 05, and

no t compute .

The rule (A.149) is a one-tail test in the upper end of the distribution. Depending
on the situation, it might be desirable to employ a two-tail test. In that case the null
hy pothesis is rejected if

β

[36

Lin
—
0.8
——
No
PgE

| t | >t α / 2 (A.150)

and the distribution H 0 is symmetric. It is rejected if

t>t α / 2 (A.151)

t<t 1 −α / 2 (A.152)

[36

and the distribution is not symmetric. The critical regions are at both tails of the
distribution, with each tail covering a probability area of / 2.

However, much effort has gone into research as to how the magnitude of

α

can
be controlled (Baarda, 1968). After all, committing a type-II error implies accepting
the null hypothesis even though the alternative hypothesis is true. For example, it
could mean that it has been concluded that no deformation took place even though
actual deformations occurred. Such an error could be costly in many respects. In
Section 4.10.2 some consideration is given to the type-II error in regards to blunder
detection and internal and external reliability, again based on Baarda's work. Section
7.8.3 considers type-II errors in regard to ambiguity fixing.
The goodness-of-fit test is a simple and useful example of statistical testing. As-
sume we wish to test a series of observations to determine whether they come from a
certain population with a specified distribution. We subdivide the observation series
into n bins. Let n i denote the number of observations in bin i . The subdivision should
be such that n i

β

5. Compute for each bin the expected number d i of observations
based on the hypothetical distribution. It can be shown that

≥

n

d i ) 2

(n i −

2

χ = (A.153)

d i

i = 1