<br />16
<br />
<br />TECHNIQUES OF WATER-RESOURCES INVESTIGATIONS
<br />
<br />variance of b,=8'(0,,),
<br />variance of b,=8'(o,,),
<br />
<br />b,- tN_,8..;o;;<13,<b,+ IN -,8..;0;;,
<br />
<br />where tN_, is from the t distribution with N-4
<br />degrees of freedom at the selected a level.
<br />The regression coefficient, b" is significantly
<br />different from zero if the confidence l:im:its do
<br />not include zero.
<br />The regression computation using 14CIJ 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 C22J C2:h and C24:
<br />
<br />..
<br />
<br />Then
<br />
<br />and
<br />
<br />variance of b,=8'(0,,),
<br />
<br />and the standard errors are the square roots of
<br />the'variances.
<br />The confidence interval for 13" a population
<br />regression coefficient, lit a probability level a,
<br />may be expressed as
<br />
<br />I 2:;(""')0,, + 2:;("""")0,, + 2:; (""x,)o,,= 1
<br />II 2:;("""")c,, + 2:;(x,')o" + 2:;(x,x,)o,,=O
<br />III 2:;(",,,,)0,, + 2:;(",,,,)0,, + 2:;("/)0,,=0
<br />I 10.201820" + 6.381330" + 0.625540,,= 1
<br />l' -0" -0.6255080" -0.0613160,,= -0.0980216
<br />II 6.381330" + 6.90632"" - 0.069520,,=0
<br />(-0.625508) I -6.38133"" - 3.991570" - 0.39128",,=-0.625508
<br />2:;, 2.914750" - 0.460800,,= -0.625508
<br />II' -0" +0.1580920,,=0.214601
<br />III 0.625540" - 0.069520" + 0.33512",,=0
<br />(-0.061316) I 0.038360,,= -0.061316
<br />(0.158092) 2:;, 0.072850,,= -0.098888
<br />2:;, 0.223910,,= -0.160204
<br />0,,= -0.71548
<br />II' -0,,+0.158092(-0.71548) =0.214601
<br />0,,=-0.32771
<br />I' -""- (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 />.
<br />
<br />Solving for "", 0", and 0,,:
<br />I
<br />II
<br />III
<br />I
<br />l'
<br />II
<br />(-0.625508) I
<br />2:;,
<br />II'
<br />III
<br />(-0.061316) I
<br />(0.158092) 2:;,
<br />2:;,
<br />
<br />2:;(",')0" + 2:;(""",)0" + 2:;(""",)0,,=0
<br />2:;("":>:')0" + 2:;(",')0" + 2:;(",,,,)0,,=1
<br />2:;(""",)0" + 2:;(",,,,)0,, + 2:;("/)0,,=0
<br />10.201830" + 6.381330" + 0.62554c,,=0
<br />-0" -0.6255080" -0.0613160,,=0
<br />6.381330" + 6.906320" - 0.069520,,= 1
<br />-6.38133"" ~ 3.99157c" - 0.3912&,,= 0
<br />2.914750" - 0.460800,,-1
<br />-0" +0.1580920,,=-0.343082
<br />.625540" - .069520" + 0.335120,,=0
<br />0.03836",,=0
<br />0.072850" = 0.158092
<br />0.223910,,=0.158092
<br />0,,=0.70605
<br />-0,,+0.158092(0.70605) = -0.343082
<br />0,,=0.45470
<br />
<br />II'
<br />
<br />.
<br />
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