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# Solucionario Walpole 8 ED

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has changed as follows. Source of Sum of Degrees of Mean Computed Variation Squares Freedom Square f Paint Types Error 2.6308 3.2516 2 15 1.3154 0.2168 6.07 Total 5.8824 17 with P -value= 0.0117. Decision: Reject H0 at level 0.05; the average wearing quality differ significantly for three paints. The residual and normal probability plots are shown here: 1.0 1.5 2.0 2.5 3.0 \u2212 0. 8 \u2212 0. 6 \u2212 0. 4 \u2212 0. 2 0. 0 0. 2 0. 4 0. 6 Type R es id ua l \u22122 \u22121 0 1 2 \u2212 0. 8 \u2212 0. 6 \u2212 0. 4 \u2212 0. 2 0. 0 0. 2 0. 4 0. 6 Theoretical Quantiles Sa m pl e Qu an tile s While the homogeneity of the variances seem to be a little better, the normality assumption may still be invalid. 212 Chapter 13 One-Factor Experiments: General 13.67 (a) The hypotheses are H0 : \u3b11 = \u3b12 = \u3b13 = \u3b14 = 0, H1 : At least one of the \u3b1i\u2019s is not zero. Computation: Source of Sum of Degrees of Mean Computed Variation Squares Freedom Square f Locations Error 0.01594 0.00616 3 16 0.00531 0.00039 13.80 Total 0.02210 19 with P -value= 0.0001. Decision: Reject H0; the mean ozone levels differ significantly across the locations. (b) Using Tukey\u2019s test, the results are as follows. y¯4. y¯1. y¯3. y¯2. 0.078 0.092 0.096 0.152 Location 2 appears to have much higher ozone measurements than other locations. Chapter 14 Factorial Experiments (Two or More Factors) 14.1 The hypotheses of the three parts are, (a) for the main effects temperature, H \u2032 0 : \u3b11 = \u3b12 = \u3b13 = 0, H \u2032 1 : At least one of the \u3b1i\u2019s is not zero; (b) for the main effects ovens, H \u2032\u2032 0 : \u3b21 = \u3b22 = \u3b23 = \u3b24 = 0, H \u2032\u2032 1 : At least one of the \u3b2i\u2019s is not zero; (c) and for the interactions, H \u2032\u2032\u2032 0 : (\u3b1\u3b2)11 = (\u3b1\u3b2)12 = · · · = (\u3b1\u3b2)34 = 0, H \u2032\u2032\u2032 1 : At least one of the (\u3b1\u3b2)ij\u2019s is not zero. \u3b1 = 0.05. Critical regions: (a) f1 > 3.00; (b) f2 > 3.89; and (c) f3 > 3.49. Computations: From the computer printout we have Source of Sum of Degrees of Mean Computed Variation Squares Freedom Square f Temperatures Ovens Interaction Error 5194.08 4963.12 3126.26 3833.50 2 3 6 12 2597.0400 1654.3733 521.0433 319.4583 8.13 5.18 1.63 Total 17, 116.96 23 213 214 Chapter 14 Factorial Experiments (Two or More Factors) Decision: (a) Reject H \u2032 0; (b) Reject H \u2032\u2032 0 ; (c) Do not reject H \u2032\u2032\u2032 0 . 14.2 The hypotheses of the three parts are, (a) for the main effects brands, H \u2032 0 : \u3b11 = \u3b12 = \u3b13 = 0, H \u2032 1 : At least one of the \u3b1i\u2019s is not zero; (b) for the main effects times, H \u2032\u2032 0 : \u3b21 = \u3b22 = \u3b23 = 0, H \u2032\u2032 1 : At least one of the \u3b2i\u2019s is not zero; (c) and for the interactions, H \u2032\u2032\u2032 0 : (\u3b1\u3b2)11 = (\u3b1\u3b2)12 = · · · = (\u3b1\u3b2)33 = 0, H \u2032\u2032\u2032 1 : At least one of the (\u3b1\u3b2)ij\u2019s is not zero. \u3b1 = 0.05. Critical regions: (a) f1 > 3.35; (b) f2 > 3.35; and (c) f3 > 2.73. Computations: From the computer printout we have Source of Sum of Degrees of Mean Computed Variation Squares Freedom Square f Brands Times Interaction Error 32.7517 227.2117 17.3217 254.7025 2 2 4 27 16.3758 113.6058 4.3304 9.4334 1.74 12.04 0.46 Total 531.9875 35 Decision: (a) Do not reject H \u2032 0; (b) Reject H \u2032\u2032 0 ; (c) Do not reject H \u2032\u2032\u2032 0 . 14.3 The hypotheses of the three parts are, (a) for the main effects environments, H \u2032 0 : \u3b11 = \u3b12 = 0, (no differences in the environment) H \u2032 1 : At least one of the \u3b1i\u2019s is not zero; (b) for the main effects strains, H \u2032\u2032 0 : \u3b21 = \u3b22 = \u3b23 = 0, (no differences in the strains) H \u2032\u2032 1 : At least one of the \u3b2i\u2019s is not zero; Solutions for Exercises in Chapter 14 215 (c) and for the interactions, H \u2032\u2032\u2032 0 : (\u3b1\u3b2)11 = (\u3b1\u3b2)12 = · · · = (\u3b1\u3b2)23 = 0, (environments and strains do not interact) H \u2032\u2032\u2032 1 : At least one of the (\u3b1\u3b2)ij\u2019s is not zero. \u3b1 = 0.01. Critical regions: (a) f1 > 7.29; (b) f2 > 5.16; and (c) f3 > 5.16. Computations: From the computer printout we have Source of Sum of Degrees of Mean Computed Variation Squares Freedom Square f Environments Strains Interaction Error 14, 875.521 18, 154.167 1, 235.167 42, 192.625 1 2 2 42 14, 875.521 9, 077.083 617.583 1004.586 14.81 9.04 0.61 Total 76, 457.479 47 Decision: (a) Reject H \u2032 0; (b) Reject H \u2032\u2032 0 ; (c) Do not reject H \u2032\u2032\u2032 0 . Interaction is not significant, while both main effects, environment and strain, are all significant. 14.4 (a) The hypotheses of the three parts are, H \u2032 0 : \u3b11 = \u3b12 = \u3b13 = 0 H \u2032 1 : At least one of the \u3b1i\u2019s is not zero; H \u2032\u2032 0 : \u3b21 = \u3b22 = \u3b23 = 0, H \u2032\u2032 1 : At least one of the \u3b2i\u2019s is not zero; H \u2032\u2032\u2032 0 : (\u3b1\u3b2)11 = (\u3b1\u3b2)12 = · · · = (\u3b1\u3b2)33 = 0, H \u2032\u2032\u2032 1 : At least one of the (\u3b1\u3b2)ij\u2019s is not zero. \u3b1 = 0.01. Critical regions: for H \u2032 0, f1 > 3.21; for H \u2032\u2032 0 , f2 > 3.21; and for H \u2032\u2032\u2032 0 , f3 > 2.59. Computations: From the computer printout we have Source of Sum of Degrees of Mean Computed Variation Squares Freedom Square f Coating Humidity Interaction Error 1, 535, 021.37 1, 020, 639.15 1, 089, 989.63 5, 028, 396.67 2 2 4 45 767, 510.69 510, 319.57 272, 497.41 111, 742.15 6.87 4.57 2.44 Total 76, 457.479 47 Decision: Reject H \u2032 0; Reject H \u2032\u2032 0 ; Do not reject H \u2032\u2032\u2032 0 . Coating and humidity do not interact, while both main effects are all significant. 216 Chapter 14 Factorial Experiments (Two or More Factors) (b) The three means for the humidity are y¯L = 733.78, y¯M = 406.39 and y¯H = 638.39. Using Duncan\u2019s test, the means can be grouped as y¯M y¯L y¯H 406.39 638.39 733.78 Therefore, corrosion damage is different for medium humidity than for low or high humidity. 14.5 The hypotheses of the three parts are, (a) for the main effects subjects, H \u2032 0 : \u3b11 = \u3b12 = \u3b13 = 0, H \u2032 1 : At least one of the \u3b1i\u2019s is not zero; (b) for the main effects muscles, H \u2032\u2032 0 : \u3b21 = \u3b22 = \u3b23 = \u3b24 = \u3b25 = 0, H \u2032\u2032 1 : At least one of the \u3b2i\u2019s is not zero; (c) and for the interactions, H \u2032\u2032\u2032 0 : (\u3b1\u3b2)11 = (\u3b1\u3b2)12 = · · · = (\u3b1\u3b2)35 = 0, H \u2032\u2032\u2032 1 : At least one of the (\u3b1\u3b2)ij\u2019s is not zero. \u3b1 = 0.01. Critical regions: (a) f1 > 5.39; (b) f2 > 4.02; and (c) f3 > 3.17. Computations: From the computer printout we have Source of Sum of Degrees of Mean Computed Variation Squares Freedom Square f Subjects Muscles Interaction Error 4, 814.74 7, 543.87 11, 362.20 2, 099.17 2 4 8 30 2, 407.37 1, 885.97 1, 420.28 69.97 34.40 26.95 20.30 Total 25, 819.98 44 Decision: (a) Reject H \u2032 0; (b) Reject H \u2032\u2032 0 ; (c) Reject H \u2032\u2032\u2032 0 . 14.6 The ANOVA table is shown as Source of Sum of Degrees of Mean Computed Variation Squares Freedom Square f P -value Additive Temperature Interaction Error 1.7578 0.8059 1.7934 1.8925 1 3 3 24 1.7578 0.2686 0.5978 0.0789 22.29 3.41 7.58 < 0.0001 0.0338 0.0010 Total 6.2497 32 Solutions for Exercises in Chapter 14 217 Decision: All main effects and interaction are significant. An interaction plot is given here. 1 1 3. 0 3. 2 3. 4 3. 6 3. 8 4. 0 Additive Ad he siv en es s 2 2 3 3 4 4 0 1 Temperature 1 2 3 4 50 60 70 80 14.7 The ANOVA table is Source of Sum of Degrees of Mean Computed Variation Squares Freedom Square f P -value Temperature Catalyst Interaction Error 430.475 2, 466.650 326.150 264.500 3 4 12 20 143.492 616.663 27.179 13.225 10.85 46.63 2.06 0.0003 < 0.0001 0.0745 Total 3, 487.775 39 Decision: All main effects are significant and the interaction is significant at level 0.0745. Hence, if 0.05 significance level