Appendix IV to Part 264 - Cochran's Approximation to the Behrens-Fisher Students' t-test
40:28.0.1.1.5.27.1.9.11 : Appendix IV
Appendix IV to Part 264 - Cochran's Approximation to the
Behrens-Fisher Students' t-test
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:
and the variance is calculated by:
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:
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:
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:
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 (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 |
[47 FR 32367, July 26, 1982]