Geoscience Reference
In-Depth Information
Table 2.3 Chi-square test for normality, biotite ages, Grenville Province
Class i Limits
2
2 / f t
f o
z i
ʦ
(z i )
ʦ
(z i )
ʦ Z i 1
f t
ʔ ¼ | f o
f t |
ʔ
ʔ
ð
Þ
1
<
860
9
1.04
0.149
0.149
11.3
2.3
5.29
0.47
2
860-900
11
0.60
0.274
0.125
9.5
1.5
2.25
0.24
3
900-940
19
0.16
0.436
0.162
12.3
6.7
44.89
3.65
4
940-980
16
0.28
0.610
0.174
13.2
2.8
7.84
0.59
5
980-1020
5
0.72
0.764
0.154
11.7
6.7
44.89
3.84
6
1020-1060
7
1.16
0.877
0.113
8.6
1.6
2.56
0.30
7
> 1060
9
1
1.000
0.123
9.3
0.3
0.09 0.01
Sum ¼ 9.1
Source: Agterberg ( 1974 , Table XIV)
to one. For level of significance
2.36
representing the 97.5 % fractile of the cumulative F -distribution in statistical tables.
The 97.5 % fractile is used because this is a two-tailed significance test. Since F
(75, 17)
ʱ ¼
0.05, we should compare it F 0.975 ¼
F 0.975 the null hypothesis for equality of variance can be accepted.
The two sample variances of Table 2.2 can be combined with one another yielding
the overall variance s 2
<
8,064.
Next, the t -test can be used to test the hypothesis that the two population means
are equal. It consists of calculating the quantity:
¼
j
x 1
x 2
j
^ tf 1 þ
ð
f 2
Þ ¼
q
1
sx
ðÞ
n 1 þ
1
n 2
Because the t -test also is two-tailed, this statistic (
¼
0.89) for level of signifi-
cance
1.98. Because it is smaller than
this significance limit, the hypothesis of equality of means can be accepted. The two
means in Table 2.2 can be combined with one another yielding the overall mean age
of 958.5 Ma.
Strictly speaking, the preceding significance tests only can be applied if the age
dates are normally distributed. The chi-square test for goodness of fit can be used to
test for normality. Its usage is illustrated in Table 2.3 . First class limits are set and
the number of observations per class is counted. This gives the observed frequen-
cies ( f o ) to be compared with theoretical frequencies ( f e ) that are computed by
using the normal frequency distribution model. Normal fractiles are determined for
all class limits and successive differences between them are multiplied by total
number of observations (
ʱ ¼
0.05 can be compared with t 0.975 (92)
¼
¼
76). Next the following statistic is obtained:
X f 0
2
ð
f t
Þ
2
ˇ
f t
Two important points must be considered here: (1) The approximation is only
valid if all theoretical frequencies are at least 5; care must be taken that all classes
are sufficiently wide to allow for this (Cram ´ r 1947 ); and (2) the sum of the
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