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<regulation prefix="Regulation" num=".23" citation="COMAR 26.13.05.23" title_num="26" subtitle_num="13" chapter_num="05" heading="Cochran's Approximation to the Behrens-Fisher Student's t-test.">
  <paragraph num="A.">
    <text>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:</text>
  </paragraph>
  <paragraph num="B.">
    <text>The t-test uses these data summary measures to calculate a t-static (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.</text>
  </paragraph>
  <paragraph num="C.">
    <text>The t-statistic for all parameters except pH and similar monitoring parameters is:</text>
  </paragraph>
  <paragraph num="D.">
    <text>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.</text>
  </paragraph>
  <paragraph num="E.">
    <text>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 (tB-1) degrees of freedom, at the 0.05 level of significance.</text>
  </paragraph>
  <paragraph num="F.">
    <text>tm = t-tables with (nm-1)degrees of freedom, at the 0.05 level of significance.</text>
  </paragraph>
  <paragraph num="G.">
    <text>Finally, the special weighting WB and WM are defined as:</text>
  </paragraph>
  <paragraph num="H.">
    <text>The t-statistic (t*) is now compared with the comparison t-statistic (tc) using the following decision-rule:</text>
    <paragraph num="(1)">
      <text>If t* is equal to or larger than tc, then conclude that there most likely has been a significant increase in this specific parameter;</text>
    </paragraph>
    <paragraph num="(2)">
      <text>If t* is less than tc, then conclude that most likely there has not been a change in this specific parameter.</text>
    </paragraph>
  </paragraph>
  <paragraph num="I.">
    <text>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.</text>
  </paragraph>
  <paragraph num="J.">
    <text>If t* is equal 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.</text>
  </paragraph>
  <paragraph num="K.">
    <text>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.</text>
  </paragraph>
</regulation>