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<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 />