Code of Massachusetts Regulations › 310 CMR - DEPARTMENT OF ENVIRONMENTAL PROTECTION › Title 310 CMR 30.000 - Hazardous waste › GENERAL PROVISIONS › Section 30.673 - Cochran's Approximation to the Behrens-Fisher Students' t-Test

Current through Register 1525, July 5, 2024
(1) Using all the available background data
				(nb readings), calculate the background mean
				(XB) and background variance (sb
									2). For the single monitoring well under
				investigation (nm reading), calculate the monitoring
				mean (Xm) and monitoring variance
				(sm
									2).
(2) For any set of data
				(X1, X2 . . .
				Xn) the mean is calculated by:
X1 + X2 . . .
				  +Xn
X = n and the variance is calculated by:
				  s2 = (X1-
				  X)2 +
				  (X2-X)2> . . . +
				  (Xn - X)2 n-1 where "n"
				  denotes the number of observations in the set of data.
(3) 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.
(4) The t-statistic for
				all parameters except pH and similar monitoring parameters is:
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				  view image
If the value of this t-statistic is negative, 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.
(5) 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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				  view image
(6)
				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 change
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.
(7) 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.
(8)
				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.
(9) 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. (10) 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. Y ates).