Hawaii Administrative Rules › Title 11 - DEPARTMENT OF HEALTH › Subtitle 1 - GENERAL DEPARTMENTAL PROVISIONS › Chapter 264 - HAZADOUS WASTE MANAGEMENT STANDARDS FOR OWNERS AND OPERATERS OF HAZADOUS WASTE TREATMENT, STORAGE, AND DISPOSAL FACILITIES › Appendix IV to Part 264 - Cochran's Approximation to the Behrens- Fisher Students' T-Test

Current through June, 2024
Using all the available background data
				  (
nb
readings), calculate the background
				  mean (
Xb
) and background variance
				  (
sb3
). For
				  the single monitoring well under investigation (
nm
reading), calculate the monitoring mean
				  (
Xm
) and monitoring variance
				  (
sm2
).
For any set of data (
X1 ,
				  X2
,...,
Xn
) the mean is calculated by:
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and the variance is calculated by:
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where "n" denotes the number of observations in the set of
				  data.
The t-test uses these data summary measures to calculate a
				  t-statistic (
t*
) and a comparison t-statistic
				  (
tc)
. The
t*
value is
				  compared to the
tc
value and a
				  conclusion reached as to whether there has been a statistically significant
				  change in any indicator parameter.
The t-statistic for all parameters except pH and similar
				  monitoring parameters is:
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If the value of this t-statistic is negative then there is
				  no significant difference between the monitoring data and background data. It
				  should be noted that significantly small negative values may be indicative of a
				  failure of the assumption made for test validity or errors have been made in
				  collecting the background data.
The t-statistic (
tc
)
				  , against which
t*
will be compared, necessitates finding
tb
and
tm
from standard (one-tailed)
				  tables
where,
tb
= t-tables with
				  (
nb
-1) degrees of freedom, at the 0.05
				  level of significance.
tm
= t-tables with
				  (
nm
-1) degrees of freedom, at the 0.05
				  level of significance.
Finally, the special weightings
Wb
and
Wm
are defined as:
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and so the comparison t-statistic is:
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The t-statistic (t*) is now compared with the comparison
				  t-statistic (
tc
) using the following
				  decision-rule:
If
t*
is equal to or larger than
tc
, then conclude that there most
				  likely has been a significant increase in this specific parameter.
If
t*
is less than
tc
, then conclude that most likely
				  there has not been a change in this specific parameter.
The t-statistic for testing pH and similar monitoring
				  parameters is constructed in the same manner as previously described except the
				  negative sign (if any) is discarded and the caveat concerning the negative
				  value is ignored. The standard (two-tailed) tables are used in the construction
tc
for pH and similar monitoring
				  parameters.
If
t*
is equal to or larger than
tc
, then conclude that there most
				  likely has been a significant increase (if the initial
t*
had
				  been negative, this would imply a significant decrease). If
t*
is less than
tc
, then conclude that
				  there most likely has been no change.
A further discussion of the test may be found in
				  "Statistical Methods" (6th Edition, Section 4.14) by G. W. Snedecor and W. G.
				  Cochran, or "Principles and Procedures of Statistics" (1st Edition, Section
				  5.8) by R. G. D. Steel and J. H. Torrie.
Standard T-Tables 0.05 Level of Significance
Degrees of freedom
t-values (one-tail)
t-values (two-tail)
1
6.314
12.706
2
2.920
4.303
3
2.353
3.182
4
2.132
2.776
5
2.015
2.571
6
1.943
2.447
7
1.895
2.365
8
1.860
2.306
9
1.833
2.262
10
1.812
2.228
11
1.796
2.201
12
1.782
2.179
13
1.771
2.160
14
1.761
2.145
15
1.753
2.131
16
1.746
2.120
17
1.740
2.110
18
1.734
2.101
19
1.729
2.093
20
1.725
2.086
21
1.721
2.080
22
1.717
2.074
23
1.714
2.069
24
1.711
2.064
25
1.708
2.060
30
1.697
2.042
40
1.684
2.021
Adopted from Table III of "Statistical Tables for
				  Biological, Agricultural, and Medical Research" (1947, R. A. Fisher and F.
				  Yates).