North Dakota Administrative Code › Title 33 - State Department of Health › Article 33-24 - Hazardous Waste Management › Chapter 33-24-05 - Standards for Treatment, Storage, and Disposal Facilities and for the Management of Specific Hazardous Wastes and Specific Types of Hazardous Waste Management Facilities › Appendix II - Cochran's Approximation to the Behrens-Fisher Student's T-Test

Current through Supplement No. 393, July, 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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				  here to view image
.
And the variance is calculated by:
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				  here to view image
.
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* 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:
Click
				  here to view image
.
If the value of this T-statistic is negative, then there is
				  no significant difference between the monitoring data and the 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
				  (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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				  here to view image
.
And so the comparison T-statistic is:
Click
				  here to view image
.
The T-statistic (T*) is now compared with the comparison T -
				  statistic (T c) 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 (1 - tailed)
T - Values (2 - tailed)
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
011.714
2.069
24
1.711
2.064
25
1.708
2.060
30
1.697
2.042
40
1.684
2.021
Taken from 40 CFR, Part 264, Appendix IV, 47FR34329, July 26,
				  1982.