STAT 496, Spring 2003Homework Assignmen t #51. An experiment is performed to determine the importance of four factors on the effi-ciency of a distillation column that is used to separate two chemical compounds. Thefour factors are: A - Relative Volatility that expresses a chemical relationship betweentwo items, B - Bottom Composition that indicates the ratio of one compound to thetotal volume of the two components at the bottom of the distillation column, C - TopComposition that indicates the ratio of one compound to the total volume of the twocomponents at the top of the distillation column, and D - Feed Composition which isthe ratio of the two components where they enter the system. The sixteen combina-tions of these four factors, each at two levels, is run in a completely randomized design.Below are the factor levels and the data.Factor Low (−1) High (+1)A: Relative Volatility 1.1 2.0B: Bottom Composition 0.005 0.200C: Top Composition 0.70 0.99D: Feed Composition 0.30 0.60Trmt. RunComb. XA XB XC XD Efficiency Order1 −1 −1 −1 −1 88 (11)a+1 −1 −1 −111(2)b −1+1−1 −131(6)ab +1 +1 −1 −14(4)c −1 −1+1−1 138 (10)ac +1 −1+1−1 18 (14)bc −1+1+1−1 80 (13)abc +1 +1 +1 −111(8)d −1 −1 −1 +1 84 (16)ad +1 −1 −1+1 12(5)bd −1+1−1 +1 31 (15)abd +1 +1 −1 +1 5 (12)cd −1 −1 +1 +1 140 ( 9)acd +1 −1+1+1 19(7)bcd −1+1+1+1 85(1)abcd +1 +1 +1 +1 11 ( 3)(a) For each factor, calculate the mean efficiency when that factor is at the high leveland at the low level. Display these means in four main effects plots, one for eachfactor. Comment on the apparent size of each main effect.(b) Construct interaction plots for the four factors. Pay particular attention to thecombinations of A - Relative Volatility with B - Bottom Composition, A - RelativeVolatility with C - Top Composition, B - Bottom Composition with C - TopComposition. Comment on the apparent presence or absence of interaction forthese three combinations.(c) Compute the overall grand sample mean and the 15 estimated effects. Be sure toindicate whether you are computing estimated full effects of half effects.(d) Construct a normal probability plot for the 15 estimated effects. Label on yourplot effects that appear to be significant.(e) You should be able to pick up some pseudo replication by eliminating a factorand all its interactions with the other three factors. This should give you a 23experiment with 2 replicates. With this reduced model, compute MSrepErrorandthe standard error of an estimate.(f) Which of the remaining estimated effects are significant? non-significant?(g) Give the most parsimonious (include only significant terms) prediction equation.Be sure to explicitly define each variable in the equation.(h) In order to maximize the efficiency of this distillation column, what settings ofthe four factors would you recommend?(i) Give a predicted value, and prediction interval, for the filtration rate at the choiceof factor levels you give in (h). Be sure to indicate what you are using for theestimate of error variation and why.(j) Using the prediction equation in (g) construct plots of residuals versus predictedvalues and residuals versus run order. Are there any discernible patterns in theseplots?Note: You may use JMP, or another computer program, to assist you with any of thegraphing or calculations needed to answer the questions