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14.11 A study was done to determine the impact of two factors, method of analysis and the laboratory do- ing the analysis, on the level of sulfur content in coal. Twenty-eight coal specimens were randomly assigned to 14 factor combinations, the structure of the experi- mental units represented by combinations of seven lab- oratories and two methods of analysis with two speci- mens per factor combination. The data, expressed in percent of sulfur, are as follows: Method Laboratory 1 2 0. 109 0.105 0.105 0. 108 0.129 0.122 0.127 0.124 0.115 0.112 0.109 0.111 0.108 0.108 0.117 0.118 0.097 0.096 0.110 0.097 0.114 0.119 0.116 0.122 0.155 0.145 0.164 0.160 (The data are taken from G. Taguchi, "Signal to Noise Ratio and Its Applications to Testing Material." Reports of Statistical Application Research, Union of Japanese Scientists and Engineers, Vol. 18, No. 4, 1971.) (a) Do an analysis of variance and show results in an analysis-of-variance table. (b) Is interaction significant? If so, discuss what it means to the scientist. Use a P-value in your con- clusion. (c) Are the individual main effects, laboratory, and method of analysis statistically significant? Discuss612 CH (d) The engineer felt confident that cutting speed and cutting angle should interact. If this interaction is significant, draw an interaction plot and discuss the engineering meaning of the interaction.15.19 (a) Obtain a } fraction of a 2* factorial design using BCD as the defining contrast. (b) Divide the } fraction into 2 blocks of 4 units each by confounding ABO.15.21 (a) Using the defining contrasts ABCE ABDF, obtain a fraction of a 2" design. (b) Show the analysis-of-variance table (sources of v ation and degrees of freedom) for all appropr tests, assuming that E and F do not interact all three-factor and higher interactions are nog be.15.25 In the study Durability of Rubber to Steel Ad- hesively Bonded Joints, conducted at the Department of Environmental Science and Mechanics and analyzed by the Statistics Consulting Center at Virginia Tech, an experimenter measured the number of breakdowns in an adhesive seal. It was postulated that concentra- tion of seawater A, temperature B, pl C, voltage D, and stress E influence the breakdown of an adhesive seal. A & fraction of a 2" factorial experiment was used, with the defining contrast being ABCDE. The data are as follows: A B C D E Response 462 -1 746 714 TELL 1070 474 832 764 LELELELELELELED 1087 522 LEELL. 854 773 1068 572 831 819 1104 Perform an analysis of variance on main effects and two factor interactions AD, AE, BD, BE, assuming that all three-factor and higher interactions are negligible. Use a = 0.05.(c) Show the analysis-of-variance table (sources of vari- ation and degrees of freedom ) for testing all uncon- founded main effects, assuming that all interaction effects are negligible