The article Response Surface Methodology for Protein Extraction Optimization of Red Pepper Seed (Food Sci, and...

   

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The article "Response Surface Methodology for Protein Extraction Optimization of Red Pepper Seed" (Food Sci, and Tech., 2010: 226-231) gave data on the response variable y = protein yield (%) and the independent variables x1 = temperature (°C), x2 = pH, x3 = extraction time (min), and x 4 = solvent/meal ratio. a. Fitting the model with the four as predictors gave the following output: Predictor Coef SE Coef Constant 4.586 2.542 -1.80 O.084 O.01317 0.02707 0.49 0.631 x2 1.635O 0.2707 6.04 O.000 0.02883 0.01353 2.23 0-044 x4 O.0540O O.02707 I.99 0.058 Source DF MS 9721 Regression २esidua Brro 24 19.8882 10.5513 22.31 0.000 0.4396 Total 28 30.4395 Calculate and interpret the values of R2 and adjusted R2. Does the model appear to be useful? b. Fitting the complete second-order model gave the following results: b. Fitting the complete second-order model gave the following results: SE Coef 18.53 Predictor Constant Coef 119.49 45 0.37 Q.000 x1 0.1047 0.2839 0.718 G.000 28.678 3.625 0.1303 x2 7.91 3.13 x3 0.4074 O.007 X4 0.2711 0.2606 1.04 O.316 o.00 752 xlsqd x2sqd X38qd x4sqd O.002120 -O.36 O.727 -1.6452 0.2110 -7.80 O-000 O.0002121 0.000s275 0.40 O.694 0.002 110 O.02687 0.015152 7.28 0.000 O.02250 0.000550 xix2 O.80 0.437 xix3 O.001344 O.41 O.30 O.688 FO.000800 -0.o5900 0.0390O x1x4 0.002687 0.770 x2x3 O.01344 4.39 O.001 x2x4 O.02697 1.45 O.169 x3x4 0.002725 0.001344 2.03 O.062 S= 0.268703 R-Sq = 96.7% R-Sq(adj) =93.4% Source Regression Residual Total DF MS 29.4287 1.0108 14 2.1020 29.11 O.000 Error 14 O.0722 28 30.4395 Does at least one of the second-order predictors appear to be useful? Carry out an appropriate test of hypotheses. c. From the output in (b), a reasonable conjecture is that none of the predictors involving x 1 are providing useful information. When these predictors are eliminated, the value of SSE for the reduced regression model is 1.1887. Does this support the conjecture? d. Here is output from Minitab's best subsets option, with just the single best subset of each size identified. Which model(s) would you consider using (subject to checking model adequacy)? Minitab output for Exercise 58d 12 3 4x x x Xx x Si11 22 3 S. Mallows Cp 174.4 x x x X qqq gx x xx 1234 ddd d 2343 44 Vars R-Sq R-Sq (adj) 50.9 0.73030 0.61349 52.7 67.9 65.4 112.5 77.7 75.0 73.1 50.8 0.52124 83.4 80.7 0.45835 90.9 21.4 0.34731 ΧΧ 6 88 94.6 93.1 7.9 0.27422 X X X 95.8 0.24683 0.24137 0.23962 0.24132 0.24716 0.25328 0.26041 X x x X X X X X X X x X X X X X X X X XX 94.4 4.7 5.1 6.1 X X X X X X x x X X X X X X X X X X x x X X X x X X X X 96.2 94.6 96.4 96.6 94.7 94.6 94.4 94.1 93.8 93.4 x X X x x X X x x x x 10 7.5 11 96.6 9.4 96.6 11.2 13 13.1 96.7 96.7 14 15.0 0.26870 X X x x x X x x XX X x x x x sA2nyS6 789 O123m4 00 The article "Response Surface Methodology for Protein Extraction Optimization of Red Pepper Seed" (Food Sci, and Tech., 2010: 226-231) gave data on the response variable y = protein yield (%) and the independent variables x1 = temperature (°C), x2 = pH, x3 = extraction time (min), and x 4 = solvent/meal ratio. a. Fitting the model with the four as predictors gave the following output: Predictor Coef SE Coef Constant 4.586 2.542 -1.80 O.084 O.01317 0.02707 0.49 0.631 x2 1.635O 0.2707 6.04 O.000 0.02883 0.01353 2.23 0-044 x4 O.0540O O.02707 I.99 0.058 Source DF MS 9721 Regression २esidua Brro 24 19.8882 10.5513 22.31 0.000 0.4396 Total 28 30.4395 Calculate and interpret the values of R2 and adjusted R2. Does the model appear to be useful? b. Fitting the complete second-order model gave the following results: b. Fitting the complete second-order model gave the following results: SE Coef 18.53 Predictor Constant Coef 119.49 45 0.37 Q.000 x1 0.1047 0.2839 0.718 G.000 28.678 3.625 0.1303 x2 7.91 3.13 x3 0.4074 O.007 X4 0.2711 0.2606 1.04 O.316 o.00 752 xlsqd x2sqd X38qd x4sqd O.002120 -O.36 O.727 -1.6452 0.2110 -7.80 O-000 O.0002121 0.000s275 0.40 O.694 0.002 110 O.02687 0.015152 7.28 0.000 O.02250 0.000550 xix2 O.80 0.437 xix3 O.001344 O.41 O.30 O.688 FO.000800 -0.o5900 0.0390O x1x4 0.002687 0.770 x2x3 O.01344 4.39 O.001 x2x4 O.02697 1.45 O.169 x3x4 0.002725 0.001344 2.03 O.062 S= 0.268703 R-Sq = 96.7% R-Sq(adj) =93.4% Source Regression Residual Total DF MS 29.4287 1.0108 14 2.1020 29.11 O.000 Error 14 O.0722 28 30.4395 Does at least one of the second-order predictors appear to be useful? Carry out an appropriate test of hypotheses. c. From the output in (b), a reasonable conjecture is that none of the predictors involving x 1 are providing useful information. When these predictors are eliminated, the value of SSE for the reduced regression model is 1.1887. Does this support the conjecture? d. Here is output from Minitab's best subsets option, with just the single best subset of each size identified. Which model(s) would you consider using (subject to checking model adequacy)? Minitab output for Exercise 58d 12 3 4x x x Xx x Si11 22 3 S. Mallows Cp 174.4 x x x X qqq gx x xx 1234 ddd d 2343 44 Vars R-Sq R-Sq (adj) 50.9 0.73030 0.61349 52.7 67.9 65.4 112.5 77.7 75.0 73.1 50.8 0.52124 83.4 80.7 0.45835 90.9 21.4 0.34731 ΧΧ 6 88 94.6 93.1 7.9 0.27422 X X X 95.8 0.24683 0.24137 0.23962 0.24132 0.24716 0.25328 0.26041 X x x X X X X X X X x X X X X X X X X XX 94.4 4.7 5.1 6.1 X X X X X X x x X X X X X X X X X X x x X X X x X X X X 96.2 94.6 96.4 96.6 94.7 94.6 94.4 94.1 93.8 93.4 x X X x x X X x x x x 10 7.5 11 96.6 9.4 96.6 11.2 13 13.1 96.7 96.7 14 15.0 0.26870 X X x x x X x x XX X x x x x sA2nyS6 789 O123m4 00

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