Statistics for Environmental Engineers

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The value of k at the given values of the variables is:


00 = k =



(Sp-S)Q


SXV



(200-20)(1)


20(2000)(1)



0.0045


The other coefficients of the linearized model are:


0



0



0



S


0


0


S


X


Q


дк


))))Q)))))))


dS0


SXV


dk


Q


dS


SXV


dk


Co


I


о


CO


dX =


SX2V


dk


Co


0


I


Co


1


d))))Q)))


SXV

1


20(2000)(1



2.5 x 10-5



1= 20(2000)(1) =


200 — 20


20(2000)2 ( 1 )


200 — 20 = 20(2000)(1) =



-2.5 x 10-5 = -2.25 x 10-6 0.0045


The variance of k, assuming all covariances are zero, is:


3.11 x 10-7


ot = 0s„Oso + 02sG2s + 0WX + 02qoQ


which is computed from the separate contributions of the variables using the values and variances given earlier, and as shown in Table 49.1.


The estimated total variance is ok = 3.11 x 10 7 and the standard deviation is ok = 0.00056. These numbers are small, but the expected value of k = 0.0045 is also small. A better perspective is obtained by finding an approximate 95% confidence interval for the estimated value of k, based on the linear approximation of the true model:


k = 0.0045 ± 2(0.00056) = 0.0045 ± 0.0011

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