Statistics for Environmental Engineers

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yi = (a0 + в0 Z1 + 70-Z 2) + a1 xi + ei


For category 1: yi = (a0 + во Z1) + a1 xi + ei For category 2: yi = (a0 + y0 Z 2) + a1 xi + eFor category 3: yi = a0 + a1 xi + ei

Case Study: Solution


The model under consideration allows a different slope and intercept for each storm. Two dummy variables are needed:


Zi= i for storm i and zero otherwise Z2 = i for storm 2 and zero otherwise


The model is:


pH = a0 + aiWA + Zi(e0 + eiWA) + Z2(y0 + yWA)


where the a’s, e’s, and y’s are estimated by regression. The model can be rewritten as:


pH = a0 + e0Zi + y0Z2 + aiWA + eiZiWA + yiZ2WA


The dummy variables are incorporated into the model by creating the new variables ZiWA and Z2WA. Table 40.1 shows how this is done.


Fitting the full six-parameter model gives:


Model A: pH = 5.77 — 0.00008WA + 0.998Zi + 1.65Z2 — 0.005Zi WA — 0.008Z2 WA (f-ratios)    (0.11)    (2.14)    (3.51)    (3.63)    (4.90)


which is also shown as Model A in Table 40.2 (top row). The numerical coefficients are the least squares estimates of the parameters. The small numbers in parentheses beneath the coefficients are the f-ratios for the parameter values. Terms with t < 2 are candidates for elimination from the model because they are almost certainly not significant.


The term WA appears insignificant. Dropping this term and refitting the simplified model gives Model B, in which all coefficients are significant:

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