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# Solucionario Walpole 8 ED

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28 .0 28 .5 29 .0 AD Interaction D y 2 2 \u22121 1 A 1 2 \u22121 1 1 126 .0 27 .0 28 .0 29 .0 BC Interaction C y 2 2 \u22121 1 B 1 2 \u22121 1 15.4 The ANOVA table is displayed. Source of Degrees of Computed Variation Freedom f P -value A B AB C AC BC ABC D AD BD ABD CD ACD BCD ABCD Error 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 16 57.85 7.52 6.94 127.86 7.08 10.96 1.26 44.72 4.85 4.85 1.14 6.52 1.72 1.20 0.87 < 0.0001 0.0145 0.0180 < 0.0001 0.0171 0.0044 0.2787 < 0.0001 0.0427 0.0427 0.3017 0.0213 0.2085 0.2900 0.3651 Total 31 Solutions for Exercises in Chapter 15 239 All main effects and two-way interactions are significant, while all higher order inter- actions are insignificant. 15.5 The ANOVA table is displayed. Source of Degrees of Computed Variation Freedom f P -value A B C D AB AC AD BC BD CD Error 1 1 1 1 1 1 1 1 1 1 5 9.98 0.20 6.54 0.02 1.83 0.20 0.57 19.03 1.83 0.02 0.0251 0.6707 0.0508 0.8863 0.2338 0.6707 0.4859 0.0073 0.2338 0.8863 Total 15 One two-factor interaction BC, which is the interaction of Blade Speed and Condition of Nitrogen, is significant. As of the main effects, Mixing time (A) and Nitrogen Condition (C) are significant. Since BC is significant, the insignificant main effect B, the Blade Speed, cannot be declared insignificant. Interaction plots for BC at different levels of A are given here. 1 1 15 .9 16 .1 16 .3 16 .5 Speed and Nitrogen Interaction (Time=1) Nitrogen Condition y 2 2 1 2 Speed 1 2 1 1 15 .6 15 .7 15 .8 15 .9 16 .0 Speed and Nitrogen Interaction (Time=2) Nitrogen Condition y 2 2 1 2 Speed 1 2 1 2 15.6 (a) The three effects are given as wA = 301 + 304\u2212 269\u2212 292 4 = 11, wB = 301 + 269\u2212 304\u2212 292 4 = \u22126.5, wAB = 301\u2212 304\u2212 269 + 292 4 = 5. There are no clear interpretation at this time. (b) The ANOVA table is displayed. 240 Chapter 15 2k Factorial Experiments and Fractions Source of Degrees of Computed Variation Freedom f P -value Concentration Feed Rate Interaction Error 1 1 1 4 35.85 12.52 7.41 0.0039 0.0241 0.0529 Total 7 The interaction between the Feed Rate and Concentration is closed to be signifi- cant at 0.0529 level. An interaction plot is given here. 1 113 5 14 0 14 5 15 0 Feed Rate y 2 2 \u22121 1 Concentration 1 2 \u22121 1 The mean viscosity does not change much at high level of concentration, while it changes a lot at low concentration. (c) Both main effects are significant. Averaged across Feed Rate a high concentration of reagent yields significantly higher viscosity, and averaged across concentration a low level of Feed Rate yields a higher level of viscosity. 15.7 Both AD and BC interaction plots are shown in Exercise 15.3. Here is the interaction plot of AB. 1 1 26 .5 27 .0 27 .5 28 .0 28 .5 AB Interaction A y 2 2 \u22121 1 B 1 2 \u22121 1 For AD, at the high level of A, Factor D essentially has no effect, but at the low level of A, D has a strong positive effect. For BC, at the low level of B, Factor C has a strong negative effect, but at the high level of B, the negative effect of C is not as pronounced. For AB, at the high level of B, A clearly has no effect. At the low level of B, A has a strong negative effect. 15.8 The two interaction plots are displayed. Solutions for Exercises in Chapter 15 241 1 1 26 27 28 29 30 AD Interaction (B=\u22121) A y 2 2 \u22121 1 D 1 2 \u22121 1 1 1 25 .0 25 .5 26 .0 26 .5 27 .0 27 .5 AD Interaction (B=1) A y 2 2 \u22121 1 D 1 2 \u22121 1 It can be argued that when B = 1 that there is essentially no interaction between A and D. Clearly when B = \u22121, the presence of a high level of D produces a strong negative effect of Factor A on the response. 15.9 (a) The parameter estimates for x1, x2 and x1x2 are given as follows. Variable Degrees of Freedom Estimate f P -value x1 x2 x1x2 1 1 1 5.50 \u22123.25 2.50 5.99 \u22123.54 2.72 0.0039 0.0241 0.0529 (b) The coefficients of b1, b2, and b12 are wA/2, wB/2, and wAB/2, respectively. (c) The P -values are matched exactly. 15.10 The effects are given here. A B C D AB AC AD BC \u22120.2625 \u22120.0375 0.2125 0.0125 \u22120.1125 0.0375 \u22120.0625 0.3625 BD CD ABC ABD ACD BCD ABCD 0.1125 0.0125 \u22120.1125 0.0375 \u22120.0625 0.1125 \u22120.0625 The normal probability plot of the effects is displayed. \u22121 0 1 \u2212 0. 2 0. 0 0. 1 0. 2 0. 3 Normal Q\u2212Q Plot Theoretical Quantiles Sa m pl e Qu an tile s A AB ABC AD ACD ABCD B D CD AC ABD BD BCD C BC (a) It appears that all three- and four-factor interactions are not significant. (b) From the plot, it appears that A and BC are significant and C is somewhat significant. 242 Chapter 15 2k Factorial Experiments and Fractions 15.11 (a) The effects are given here and it appears that B, C, and AC are all important. A B C AB AC BC ABC \u22120.875 5.875 9.625 \u22123.375 \u22129.625 0.125 \u22121.125 (b) The 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 0.11 4.79 12.86 1.58 12.86 0.00 0.18 0.7528 0.0600 0.0071 0.2440 0.0071 0.9640 0.6861 Total 15 1 1 30 35 40 45 50 AC Interaction A y 2 2 \u22121 1 C 1 2 \u22121 1 (c) Yes, they do agree. (d) For the low level of Cutting Angle, C, Cutting Speed, A, has a positive effect on the life of a machine tool. When the Cutting Angle is large, Cutting Speed has a negative effect. 15.12 A is not orthogonal to BC, B is not orthogonal to AC, and C is not orthogonal to AB. If we assume that interactions are negligible, we may use this experiment to estimate the main effects. Using the data, the effects can be obtained as A: 1.5; B: \u22126.5; C: 2.5. Hence Factor B, Tool Geometry, seems more significant than the other two factors. 15.13 Here is the block arrangement. Block Block Block 1 2 1 2 1 2 (1) c ab abc a b ac bc (1) c ab abc a b ac bc (1) c ab abc a b ac bc Replicate 1 Replicate 2 Replicate 3 AB Confounded AB Confounded AB Confounded Solutions for Exercises in Chapter 15 243 Analysis of Variance Source of Variation Degrees of Freedom Blocks A B C AC BC ABC Error 5 1 1 1 1 1 1 12 Total 23 15.14 (a) ABC is confounded with blocks in the first replication and ABCD is confounded with blocks in second replication. (b) Computing the sums of squares by the contrast method yields the following ANOVA table. Source of Degrees of Mean Computed Variation Freedom Square f P -value Blocks A B C D AB AC BC AD BD CD ABC ABD ACD BCD ABCD Error 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 13 2.32 2.00 0.50 4.50 8.00 0.50 0.32 0.50 0.72 0.32 0.18 1.16 0.32 0.02 0.18 0.53 0.60 3.34 0.83 7.51 13.36 0.83 0.53 0.83 1.20 0.53 0.30 1.93 0.53 0.03 0.30 0.88 0.0907 0.3775 0.0168 0.0029 0.3775 0.4778 0.3775 0.2928 0.4778 0.5928 0.1882 0.4778 0.8578 0.5928 0.3659 Total 31 Only the main effects C and D are significant. 15.15 L1 = \u3b31 + \u3b32 + \u3b33 and L2 = \u3b31 + \u3b32 + \u3b34. For treatment combination (1) we find L1 (mod 2) = 0. For treatment combination a we find L1 (mod 2) = 1 and L2 (mod 2) = 1. After evaluating L1 and L2 for all 16 treatment combinations we obtain the following blocking scheme: 244 Chapter 15 2k Factorial Experiments and Fractions Block 1 Block 2 Block 3 Block 4 (1) ab acd bcd c abc