Global Positioning System Reference
In-Depth Information
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2
0
2
H
1
:
σ
0
=σ
(4.265)
The zero hypothesis states that the a priori variance of unit weight statistically equals
the a posteriori variance of unit weight. Recall that the a posteriori variance of unit
weight is a random variable; the adjustment makes a sample value available for this
quantity on the basis of the observations (the samples). Both variances of unit weight
do not have to be numerically equal; but they should be statistically equal in the sense
of (4.67). If the zero hypothesis is accepted, the adjustment is judged to be correct. If
the numerical value
2
0
v
T
Pv
σ
σ
2
χ
=
(n
−
r)
=
(4.266)
σ
0
0
[13
is such that
2
<
2
n
−
r,
1
−α
/
2
χ
χ
(4.267)
Lin
—
9.2
——
Nor
PgE
2
>
2
n
−
r,
α
/
2
χ
χ
(4.268)
th
en the zero hypothesis is rejected. The significance level
, i.e., the probability of a
ty
pe-I error, or the probability of rejecting the zero hypothesis even though it is true,
is
generally fixed to 0.05. Here the significance level is the sum of the probabilities
in
both tails. Table 4.6 lists selected values from the chi-square distribution
α
n
−
r,
α
.
Re
jection of the zero hypothesis is taken to indicate that something is wrong with
th
e adjustment. The cause for rejection remains to be clarified. Figure 4.3 shows the
li
mits for the posteriori variance of unit weight as a function of the degree of freedom
gi
ven the significance level
χ
[13
α =
0
.
05.
of the type-II error, i.e., the probability of rejecting the alternative
hy
pothesis and accepting the zero hypothesis even though the alternative hypothesis
is
true, is generally not computed. Type-II errors are considered in Section 4.10.2
in
regards to reliability and in Section 7.8.3 in regards to discernibility of estimated
am
biguity sets.
The probability
β
TA
BLE 4.6
Selected Values for Chi-Square
Probability
α
Degree of
Fr
eedom (DF)
0.975
0.950
0.050
0.025
1
0.00
0.00
3.84
5.02
5
0.83
1.15
11.07
12.83
10
3.25
3.94
18.31
20.48
20
9.59
10.85
31.41
34.17
50
32.36
34.76
67.50
71.42
100
74.22
77.93
124.34
129.56
