Use multiple linear regression to derive a predictive

relative error for your prediction. Explain possible causes for the discrepancy.

15.7 As compared with the models from Probs. 15.5 and

15.6, a somewhat more sophisticated model that accounts for the effect of both temperature and chloride on dis- solved oxygen saturation can be hypothesized as being of the form o = f 3 T + f 1 c That is, a third-order polynomial in temperature and a linear relationship in chloride is assumed to yield superior results. Use the general linear least-squares approach to fit this model to the data in Table P15.5. Use the resulting equation to estimate the dissolved oxygen concentration for a chloride concentration of 15 g L at T = 12 ◦ C. Note that the true value is 9.09 mg L. Compute the percent relative error for your prediction.

15.8 Use multiple linear regression to fit

x 1 1 1 2 2 3 3 4 4 x 2 1 2 1 2 1 2 1 2 y 15.1 17.9 12.7 25.6 20.5 35.1 29.7 45.4 40.2 Compute the coefficients, the standard error of the estimate, and the correlation coefficient.

15.9 The following data were collected for the steady flow

of water in a concrete circular pipe: Experiment Diameter, m Slope, mm Flow, m 3 s 1 0.3 0.001 0.04 2 0.6 0.001 0.24 3 0.9 0.001 0.69 4 0.3 0.01 0.13 5 0.6 0.01 0.82 6 0.9 0.01 2.38 7 0.3 0.05 0.31 8 0.6 0.05 1.95 9 0.9 0.05 5.66 Use multiple linear regression to fit the following model to this data: Q = α D α 1 S α 2 where Q = flow, D = diameter, and S = slope. 15.10 Three disease-carrying organisms decay exponen- tially in seawater according to the following model: pt = Ae −1.5t + Be −0.3t + Ce −0.05t Estimate the initial concentration of each organism A, B, and C given the following measurements: t 0.5 1 2 3 4 5 6 7 9 pt 6 4.4 3.2 2.7 2 1.9 1.7 1.4 1.1 15.11 The following model is used to represent the effect of solar radiation on the photosynthesis rate of aquatic plants: P = P m I I sat e − I Isat +1 where P = the photosynthesis rate mg m −3 d −1 , P m = the maximum photosynthesis rate mg m −3 d −1 , I = solar radiation μE m −2 s −1 , and I sat = optimal solar radiation μE m −2 s −1 . Use nonlinear regression to evaluate P m and I sat based on the following data: I 50 80 130 200 250 350 450 550 700 P 99 177 202 248 229 219 173 142 72 15.12 The following data are provided x 1 2 3 4 5 y 2.2 2.8 3.6 4.5 5.5 TABLE P15.5 Dissolved oxygen concentration in water as a function of temperature ◦ C and chloride concentration gL. Dissolved Oxygen mgL for Temperature ◦ C and Concentration of Chloride gL T , °C c = 0 gL c = 10 gL c = 20 gL 14.6 12.9 11.4 5 12.8 11.3 10.3 10 11.3 10.1 8.96 15 10.1 9.03 8.08 20 9.09 8.17 7.35 25 8.26 7.46 6.73 30 7.56 6.85 6.20 PROBLEMS 377 Fit the following model to this data using MATLAB and the general linear least-squares model y = a + bx + c x

15.13 In Prob. 14.8 we used transformations to linearize

and fit the following model: y = α 4 xe β 4 x Use nonlinear regression to estimate α 4 and β 4 based on the following data. Develop a plot of your fit along with the data. x 0.1 0.2 0.4 0.6 0.9 1.3 1.5 1.7 1.8 y 0.75 1.25 1.45 1.25 0.85 0.55 0.35 0.28 0.18

15.14 Enzymatic reactions are used extensively to charac-

terize biologically mediated reactions. The following is an example of a model that is used to fit such reactions: v = k m [S] 3 K + [S] 3 where v = the initial rate of the reaction Ms, [S] = the substrate concentration M, and k m and K are parameters. The following data can be fit with this model: [ S ], M v , Ms 0.01 6.078 × 10 −11 0.05 7.595 × 10 −9 0.1 6.063 × 10 −8 0.5 5.788 × 10 −6 1 1.737 × 10 −5 5 2.423 × 10 −5 10 2.430 × 10 −5 50 2.431 × 10 −5 100 2.431 × 10 −5 a Use a transformation to linearize the model and evaluate the parameters. Display the data and the model fit on a graph. b Perform the same evaluation as in a but use nonlinear regression. 15.15 Given the data x 5 10 15 20 25 30 35 40 45 50 y 17 24 31 33 37 37 40 40 42 41 use least-squares regression to fit a a straight line, b a power equation, c a saturation-growth-rate equation, and d a parabola. For b and c, employ transformations to linearize the data. Plot the data along with all the curves. Is any one of the curves superior? If so, justify.

15.16 The following data represent the bacterial growth in a

liquid culture over of number of days: Day 4 8 12 16 20 Amount × 10 6 67.38 74.67 82.74 91.69 101.60 112.58 Find a best-fit equation to the data trend. Try several possibilities—linear, quadratic, and exponential. Determine the best equation to predict the amount of bacteria after 30 days.

15.17 Dynamic viscosity of water μ10

–3 N · sm 2 is re- lated to temperature T ◦ C in the following manner: T 5 10 20 30 40 µ 1.787 1.519 1.307 1.002 0.7975 0.6529 a Plot this data. b Use linear interpolation to predict μ at T = 7.5 °C. c Use polynomial regression to fit a parabola to the data in order to make the same prediction.

15.18 Use the following set of pressure-volume data to find

the best possible virial constants A 1 and A 2 for the follow- ing equation of state. R = 82.05 mL atmgmol K, and T = 303 K. P V RT = 1 + A 1 V + A 2 V 2 P atm 0.985 1.108 1.363 1.631 V mL 25,000 22,200 18,000 15,000

15.19 Environmental scientists and engineers dealing with

the impacts of acid rain must determine the value of the ion product of water K w as a function of temperature. Scien- tists have suggested the following equation to model this relationship: − log 10 K w = a T a + b log 10 T a + cT a + d where T a = absolute temperature K, and a, b, c, and d are parameters. Employ the following data and regression to