Wisconsin Administrative Code › Department of Natural Resources › NR 600-699 - Environmental Protection - Hazardous Waste Management › Chapter NR 664 - Hazardous Waste Treatment, Storage And Disposal Facility Standards › Appendix IV - COCHRAN'S APPROXIMATION TO THE BEHRENS-FISHER STUDENTS' T-TEST

Current through June 24, 2024
Using all the available background data
				  (nb readings), calculate the background mean
				  (Xb) and background variance
				  (sb2). 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 ( 2-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 ( 2-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).