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<br />16
<br />
<br />TECHNIQUES OF WATER-RESOURCES INVESTIGATIONS
<br />
<br />variance of b,=8'(c,.),
<br />variance of b,=S'(c,,),
<br />
<br />b.-tN_,S.JC,;<fJ,<b.+tN.,S .JC,;,
<br />
<br />where tN-, is from the t distribution with N-4
<br />degrees of freedom at the selected a level.
<br />The regression coeflicient, b" is significantly
<br />different from zero if the confidence limits do
<br />not include zero.
<br />The regression computation using "c" multi-
<br />pliers and the data of table 3 are given below.
<br />Solutions of the normal equations follow the
<br />same pattern as previously described. Solving
<br />for c,., c", and "":
<br />
<br />.
<br />
<br />Then
<br />
<br />and
<br />
<br />variance of b,=8'(o.J,
<br />
<br />and the standard errors are the square roots of
<br />the.variances.
<br />The confidence interval for P., a population
<br />regression coeflicient, at a probability level a,
<br />may be expressed as
<br />
<br />I L;(x,')c,. + L;(x,z,)c", + L;(x,x.}",,= 1
<br />II L;(x,x,)c,. + L;(z,')c" + L;(z,x.)",,=O
<br />III L;(z,x.)c,. + L;(z,x.)c" + L;(x,')o..=O
<br />I 10.201820" + 6.3813&,. + 0.62554c,.= 1
<br />I' -c,. -O,625508c" -0.06131&,.= -0.0980216
<br />II 6,381330" +6.90632c" - 0.06952",,=0
<br />(-0.625508) I -6.38133c" - 3.99157c" - 0.39128c,.=-0.625508
<br />L;. 2.91475c" 0.4608Oc..- 0,625508
<br />11/ -c" +0.158092",,=0.214601
<br />III O,62554c" 0.069520" + 0,335120..=0
<br />(-0.061316) I - 0,038360..= -0.061316
<br />(0.158092) L;. - 0.07285c,.=-O,098888
<br />L;, 0.22391",,=-0.160204
<br />",,=-0,71548 .
<br />II' -0,,+0.158092('-0,71548)=0.214601
<br />0,,=-0.32771
<br />I' -0,,- (0,625508)( -0.32771) - (0.061316) (-0,71548)= -0.0980216
<br />-0,,+0,204985+0,043870= -0.0980216
<br />0,,=0.34688
<br />III (0,62554)(0,34688)- (0.06952) (-0,32771) + (0.33512) (-0.71548) =0
<br />o~o Check (to five places)
<br />
<br />Solving for c,., 0", and 0..:
<br />I L;(z,')o" + L;(z,z,)c" + L;(x.x.)o,,=O
<br />II L;(x,z,)o" + L;(x.')o" + L;(z,x,)o..= 1
<br />III L;(x.x,)o" + L;(z,x.}c" + L;(x,')",,=O
<br />I 10,201830" + 6.38133<:" + 0.62554c,.=0
<br />1/ -c" -0.6255080" -0.061316",,=0
<br />II 6.381330" + 6,906320" - 0.069520..= 1
<br />(-0.625508) I -6.38133<:" _ 3.99157c" - 0.39128c,.=0
<br />L;. 2.91475c" - 0.4608Oc,,= 1
<br />II' -c", +0,1580920,,=-0.343082
<br />III .62554c" - .069520" + 0.335120,,=0
<br />(-0,061316) I - 0.03836",,=0
<br />(0,158092) L;. - 0.07285c,.=0.158092
<br />L;, 0,22391",,=0.158092
<br />"" = O. 70605
<br />11/ -0,,+0.158092(0,70605)= -0.343082
<br />0,,=0.45470
<br />
<br />.
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