North Dakota Administrative Code › Title 33.1 - Department of Environmental Quality › Article 33.1-24 - Hazardous Waste Management › Chapter 33.1-24-05 - Standards for Treatment, Storage, and Disposal Facilities and for the Management of Specific Hazardous Wastes and Specific Types of Hazardous Waste Management Facilities › STANDARDS FOR FACILITIES OPERATING UNDER A STANDARDIZED PERMIT › 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:
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					 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:
Cochran's Approximation to the Behrens-Fisher
					 Student's T-Test
Click
					 here to view image
And so the comparison t-statistic is:
The t-statistic (t*) is now compared with the comparison t
					 - statistic (tc) using the following decision
					 rule:
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					 here to view image
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 Statistic
					 (1st Edition, Section 5.8) by R. G. D. Steel and J.
					 H. Torrie.
Cochran's Approximation to the Behrens-Fisher
					 Student's T-Test
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
1.714
2.069
24
1.711
2.064
25
1.708
2.060
30
1.697
2.042
40
1.684
2.021