South Carolina Code of Regulations › Chapter 61 - DEPARTMENT OF HEALTH AND ENVIRONMENTAL CONTROL › Subchapter 61-79 - Hazardous Waste Management Regulations › Part 61-79.264 - STANDARDS FOR OWNERS AND OPERATORS OF HAZARDOUS WASTE TREATMENT, STORAGE, AND DISPOSAL FACILITIES › Appendix 61-79.264.IV - COCHRAN'S APPROXIMATION TO THE BEHRENS-FISHER STUDENTS' T-TEST

Current through Register Vol. 48, No. 6, June 28, 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:
See Image
and the variance is calculated by:
See 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* 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:
See Image
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 t (this is supposed to be an inferior
				  roman B, but this entire section has been deleted in the supp, so I put this
				  here instead) and tM from standard (one-tailed) tables
				  where, tb = t-tables with (ng-1)
				  degrees of freedom, at the 0.05 level of significance and
				  tm = t-tables with (nm-1) degrees
				  of freedom, at the 0.05 level of significance.
Finally, the special weightings W and
				  Wm are defined as:
See
				  Image
and so the comparison t-statistic is:
See
				  Image
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, Section4.14) by G. W. Snedecor and W. G. Cochran, or
				  Principles and Procedures of Statistics (1st Edition, Section5.8) 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-ail)
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).