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# Solucionario Walpole 8 ED

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flour and P -value = 0.6059 for \u201cCake\u201d flour. We also have the interaction plot which shows that sweetener at 100% concen- tration yielded a specific gravity significantly lower than the other concentrations for all-purpose flour. For cake flour, however, there were no big differences in the effect of sweetener concentration. 1 1 1 1 0. 80 0. 82 0. 84 0. 86 0. 88 Sweetener At tri bu te 2 2 2 2 0 50 75 100 Flour 1 2 all\u2212pupose cake 14.35 (a) The ANOVA table is given. Source of Sum of Degrees of Mean Computed Variation Squares Freedom Square f P -value Sauce Fish Sauce*Fish Error 1, 031.3603 16, 505.8640 724.6107 3, 381.1480 1 2 2 24 1, 031.3603 8, 252.9320 362.3053 140.8812 7.32 58.58 2.57 0.0123 < 0.0001 0.0973 Total 21, 642.9830 29 Interaction effect is not significant. (b) Both P -values of Sauce and Fish Type are all small enough to call significance. 14.36 (a) The ANOVA table is given here. Source of Sum of Degrees of Mean Computed Variation Squares Freedom Square f P -value Plastic Type Humidity Interaction Error 142.6533 143.7413 133.9400 50.5950 2 3 6 12 71.3267 47.9138 22.3233 4.2163 16.92 11.36 5.29 0.0003 0.0008 0.0070 Total 470.9296 23 The interaction is significant. (b) The SS for AB with only Plastic Type A and B is 24.8900 with 3 degrees of freedom. Hence f = 24.8900/3 4.2163 = 1.97 with P -value = 0.1727. Hence, there is no significant interaction when only A and B are considered. Solutions for Exercises in Chapter 14 233 (c) The SS for the single-degree-of-freedom contrast is 143.0868. Hence f = 33.94 with P -value < 0.0001. Therefore, the contrast is significant. (d) The SS for Humidity when only C is considered in Plastic Type is 2.10042. So, f = 0.50 with P -value = 0.4938. Hence, Humidity effect is insignificant when Type C is used. 14.37 (a) The ANOVA table is given here. Source of Sum of Degrees of Mean Computed Variation Squares Freedom Square f P -value Environment Stress Interaction Error 0.8624 40.8140 0.0326 0.6785 1 2 2 12 0.8624 20.4020 0.0163 0.0565 15.25 360.94 0.29 0.0021 < 0.0001 0.7547 Total 42.3875 17 The interaction is insignificant. (b) The mean fatigue life for the two main effects are all significant. 14.38 The ANOVA table is given here. Source of Sum of Degrees of Mean Computed Variation Squares Freedom Square f P -value Sweetener Flour Interaction Error 1.26893 1.77127 0.14647 2.55547 3 1 3 16 0.42298 1.77127 0.04882 0.15972 2.65 11.09 0.31 0.0843 0.0042 0.8209 Total 5.74213 23 The interaction effect is insignificant. The main effect of Sweetener is somewhat in- significant, since the P -value = 0.0843. The main effect of Flour is strongly significant. 14.39 The ANOVA table is given here. Source of Sum of Degrees of Mean Computed Variation Squares Freedom Square f P -value A B C AB AC Error 1133.5926 26896.2593 40.1482 216.5185 1.6296 2.2963 2.5926 1844.0000 2 2 2 4 4 4 8 27 566.7963 13448.1296 20.0741 54.1296 0.4074 0.5741 0.3241 68.2963 8.30 196.91 0.29 0.79 0.01 0.01 0.00 0.0016 < 0.0001 0.7477 0.5403 0.9999 0.9999 1.0000 Total 30137.0370 53 234 Chapter 14 Factorial Experiments (Two or More Factors) All the two-way and three-way interactions are insignificant. In the main effects, only A and B are significant. 14.40 (a) Treating Solvent as a class variable and Temperature and Time as continuous variable, only three terms in the ANOVA model show significance. They are (1) Intercept; (2) Coefficient for Temperature and (3) Coefficient for Time. (b) Due to the factor that none of the interactions are significant, we can claim that the models for ethanol and toluene are equivalent apart from the intercept. (c) The three-way interaction in Exercise 14.23 was significant. However, the general patterns of the gel generated are pretty similar for the two Solvent levels. 14.41 The ANOVA table is displayed. Source of Sum of Degrees of Mean Computed Variation Squares Freedom Square f P -value Surface Pressure Interaction Error 2.22111 39.10778 112.62222 565.72000 2 2 4 9 1.11056 19.55389 28.15556 62.85778 0.02 0.31 0.45 0.9825 0.7402 0.7718 Total 719.67111 17 All effects are insignificant. 14.42 (a) This is a two-factor fixed-effects model with interaction. yijk = µ+ \u3b1i + \u3b2j+)\u3b1\u3b2)ij + \u1ebijk,\u2211 i \u3b1i = 0, \u2211 j \u3b2j = 0, \u2211 i (\u3b1\u3b2)ij = \u2211 j (\u3b1\u3b2)ij = 0, \u1ebijk \u223c n(x; 0, \u3c3) (b) The ANOVA table is displayed. Source of Sum of Degrees of Mean Computed Variation Squares Freedom Square f P -value Time Temperature Interaction Error 0.16668 0.27151 0.03209 0.01370 3 2 6 12 0.05556 0.13575 0.00535 0.00114 48.67 118.91 4.68 < 0.0001 < 0.0001 0.0111 Total 0.48398 23 The interaction is insignificant, while two main effects are significant. (c) It appears that using a temperature of \u221220\u25e6C with drying time of 2 hours would speed up the process and still yield a flavorful coffee. It might be useful to try some additional runs at this combination. Solutions for Exercises in Chapter 14 235 14.43 (a) Since it is more reasonable to assume the data come from Poisson distribution, it would be dangerous to use standard analysis of variance because the normality assumption would be violated. It would be better to transform the data to get at least stable variance. (b) The ANOVA table is displayed. Source of Sum of Degrees of Mean Computed Variation Squares Freedom Square f P -value Teller Time Interaction Error 40.45833 97.33333 8.66667 25.50000 3 2 6 12 13.48611 48.66667 1.44444 2.12500 6.35 22.90 0.68 0.0080 < 0.0001 0.6694 Total 171.95833 23 The interaction effect is insignificant. Two main effects, Teller and Time, are all significant. (c) The ANOVA table using a squared-root transformation on the response is given. Source of Sum of Degrees of Mean Computed Variation Squares Freedom Square f P -value Teller Time Interaction Error 0.05254 1.32190 0.11502 0.35876 3 2 6 12 0.17514 0.66095 0.01917 0.02990 5.86 22.11 0.64 0.0106 < 0.0001 0.6965 Total 2.32110 23 Same conclusions as in (b) can be reached. To check on whether the assumption of the standard analysis of variance is violated, residual analysis may used to do diagnostics. Chapter 15 2k Factorial Experiments and Fractions 15.1 Either using Table 15.5 (e.g., SSA = (\u221241+51\u221257\u221263+67+54\u221276+73) 2 24 = 2.6667) or running an analysis of variance, we can get the Sums of Squares for all the factorial effects. SSA = 2.6667, SSB = 170.6667, SSC = 104.1667, SS(AB) = 1.500. SS(AC) = 42.6667, SS(BC) = 0.0000, SS(ABC) = 1.5000. 15.2 A simplified ANOVA table is given. Source of Degrees of Computed Variation Freedom f P -value A B AB C AC BC ABC Error 1 1 1 1 1 1 1 8 1294.65 43.56 20.88 116.49 16.21 0.00 289.23 < 0.0001 0.0002 0.0018 < 0.0001 0.0038 0.9668 < 0.0001 Total 15 All the main and interaction effects are significant, other than BC effect. However, due to the significance of the 3-way interaction, the insignificance of BC effect cannot be counted. Interaction plots are given. 237 238 Chapter 15 2k Factorial Experiments and Fractions 1 1 5 10 15 20 C=\u22121 A y 2 2 \u22121 1 B 1 2 \u22121 1 1 1 6 8 10 12 14 16 18 C=1 A y 2 2 \u22121 1 B 1 2 \u22121 1 15.3 The AD and BC interaction plots are printed here. The AD plot varies with levels of C since the ACD interaction is significant, or with levels of B since ABD interaction is significant. 1 1 26 .5 27 .0 27 .5