1422TME100092305 Analyze the data from the experiment with \( a=0.05 \) and draw appropriate conclusions. 10 The yield of a chemical process is being studied. The two most important (7) variables are thought to be the pressure and the temperature. Three levels of each factor are selected, and a factorial experiment with two replicates is performed. \begin{tabular}{|c|c|c|c|} \hline \multirow{2}{*}{\begin{tabular}{c} Temperature \\ \( \left({ }^{\circ} \mathrm{C}\right) \) \end{tabular}} & \multicolumn{3}{|c|}{ Pressure (bar) } \\ \cline { 2 - 4 } & 100 & 150 & 200 \\ \hline \multirow{2}{*}{200} & 80.5 & 80.5 & 80.1 \\ \cline { 2 - 4 } & 80.2 & 80.6 & 80.3 \\ \hline \multirow{2}{*}{250} & 80.6 & 80.4 & 80.4 \\ \cline { 2 - 4 } & 80.4 & 80.7 & 80.6 \\ \hline \multirow{2}{*}{300} & 80.7 & 80.3 & 80.5 \\ \cline { 2 - 4 } & 80.1 & 80.2 & 80.2 \\ \hline \end{tabular} Analyze the yield data shown above and draw appropriate conclusions using a \( =0.05 \) 11 A mechanical engineer is studying the thrust force developed by a drill press. (7) He suspects that the drilling speed and the feed rate of the material are the most important factors. He selects four feed rates and uses a high and low drill speed chosen to represent the extreme operating conditions. He obtains the following results. \begin{tabular}{|c|c|c|c|c|} \hline \multirow{3}{*}{\begin{tabular}{c} Drill \\ speed \end{tabular}} & \multicolumn{4}{|c|}{ Feed rate } \\ \hline & & 0.02 & & 0.0 \\ \hline & 0.02 & 5 & 0.03 & 35 \\ \hline \multirow{2}{*}{250} & 2.6 & 2.8 & 2.9 & 2.7 \\ \hline & 2.8 & 2.5 & 2.7 & 2.6 \\ \hline \multirow{2}{*}{300} & 2.9 & 2.8 & 2.9 & 2.5 \\ \hline & 2.5 & 2.6 & 2.4 & 2.5 \\ \hline \end{tabular} Analyze the data and draw appropriate conclusions using \( a=0.05 \). 12 Construct a \( 2^{5-1} \) design. Show how the design may be run in two blocks of (7) eight observations each. Are any main effects or two-factor interactions confounded with blocks? Page 3of 3
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DOE Minitab. You are a consultant in the design of experiments for a chemical process company. One of the processes requires reaching and maintaining the temperature at which one of the company's processes is carried out, between 600 and 700 °F. To achieve this objective, the appropriate combination of five variables that affect temperature must be achieved: MgBr concentration (A); Oxygen rate (B); Nitrogen level (C); Ionization rate (D); and concentration of NaCl2 (E). To achieve the desired combination, you propose to carry out three runs of a 2^5 factorial design. Since your proposal would entail a high cost for the company, you are authorized to carry out a 2^5-1 design with two replicates. The results are the following: Factors Temperature A B C D E Replica 1 Replica 2 Treatment MgBr Oxygen Nitrogen Ionization NaCl2 Y1 Y2 1 20 1 200 5 0.9 765.9 768.7 2 20 1 200 10 0.1 243.6 234.1 3 20 1 500 5 0.1 495.1 593.5 4 20 1 500 10 0.9 518.3 456.4 5 20 10 200 5 0.1 505.4 522.9 6 20 10 200 10 0.9 487.9 498.6 7 20 10 500 5 0.9 768.0 702.8 8 20 10 500 10 0.1 296.7 271.7 9 80 1 200 5 0.1 635.3 622.8 10 80 1 200 10 0.9 387.0 380.9 11 80 1 500 5 0.9 676.3 680.0 12 80 1 500 10 0.1 366.1 325.4 13 80 10 200 5 0.9 640.1 650.7 14 80 10 200 10 0.1 357.3 347.2 15 80 10 500 5 0.1 526.8 527.4 16 80 10 500 10 0.9 419.9 411.0 a) Estimate the effects of the factors. What effects appear to be great? b) Perform the analysis of variance to confirm your conclusions for part (a). c) What is the equation to predict the temperature based on the results of this experiment? How good is this model? (hint: check model assumptions) d) What levels of the factors (both significant and non-significant, if any) included in your model would you recommend using to achieve the objective? (hint: perform optimization)
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In an experiment conducted at the Department of Mechanical Engineering and analyzed by the Statistics Consulting Center at the Virginia Polytechnic Institute and State University, a sensor detects an electrical charge each time a turbine blade makes one; rotation. The sensor then measures the amplitude of the electrical current. Six factors arc rprn $A$, temperature $B$, gap between blades $C$, gap between blade and casing $D$, location of input $E$, and location of detection $F .$ A $\frac{1}{4}$ fraction of a $2^{\circ}$ factorial experiment is used, with defining contrasts being $A B C E$ and $B C D F .$ The data are as follows: $$ \begin{array}{rrrrrrr} A & B & C & D & E & F & \text { Response } \\ \hline-1 & -1 & -1 & -1 & -1 & -1 & 3.89 \\ 1 & -1 & -1 & -\mathrm{i} & 1 & -1 & 10.46 \\ -1 & 1 & -1 & -1 & 1 & 1 & 25.98 \\ 1 & 1 & -1 & -1 & -1 & 1 & 39.88 \\ -1 & -1 & 1 & -1 & 1 & 1 & 61.88 \\ 1 & -1 & 1 & -1 & -1 & 1 & 3.22 \\ -1 & 1 & 1 & -1 & -1 & -1 & 8.94 \\ 1 & 1 & 1 & -1 & 1 & -1 & 20.29 \\ -1 & -1 & -1 & 1 & -1 & 1 & 32.07 \\ 1 & -1 & -1 & 1 & 1 & 1 & 50.76 \\ -1 & 1 & -1 & 1 & 1 & -1 & 2.80 \\ 1 & 1 & -1 & 1 & -1 & -1 & 8.15 \\ -1 & -1 & 1 & 1 & 1 & -1 & 16.80 \\ 1 & -1 & 1 & 1 & -1 & -1 & 25.47 \\ -1 & 1 & 1 & 1 & -1 & 1 & 44.44 \\ 1 & 1 & 1 & 1 & 1 & 1 & 2.45 \end{array} $$ Perform an analysis of variance on main effects, and two-factor interactions. assuming that all three-factor and higher interactions are negligible. Use $\alpha=0.05$.
$2^{k}$ Factorial Experiments and Fractions
Analysis of Fractional Factorial Experiments
In a metallurgy experiment, it is desired to test the effect of four factors and their interactions on the concentration (percent by weight) of a particular phosphorus compound in casting material. The variables are $A,$ percent phosphorus in the refinement; $B,$ percent remelted material; $C$, fluxing time; and $D$, holding time. The four factors are varied in a $2^{4}$ factorial experiment with two castings taken at each factor combination. The 32 castings were made in random order. The following table shows the data and an ANOVA table is given in Figure 15.8 on page $630 .$ Discuss the effects of the factors and their interactions on the concentration of the phosphorus compound. $$ \begin{array}{cccc} & \multicolumn{3}{c} {\text { Weight }} \\ \text { Treatment } & \multicolumn{2}{c} {\% \text { of Phosphorus Compound }} \\ \cline { 2 - 4 } \text { Combination } & \text { Rep 1 } & \text { Rep 2 } & \text { Total } \\ \hline(1) & 30.3 & 28.6 & 58.9 \\ a & 28.5 & 31.4 & 59.9 \\ b & 24.5 & 25.6 & 50.1 \\ a b & 25.9 & 27.2 & 53.1 \\ c & 24.8 & 23.4 & 48.2 \\ a c & 26.9 & 23.8 & 50.7 \\ b c & 24.8 & 27.8 & 52.6 \\ a b c & 22.2 & 24.9 & 47.1 \\ d & 31.7 & 33.5 & 65.2 \\ a d & 24.6 & 26.2 & 50.8 \\ b d & 27.6 & 30.6 & 58.2 \\ a b d & 26.3 & 27.8 & 54.1 \\ c d & 29.9 & 27.7 & 57.6 \\ a c d & 26.8 & 24.2 & 51.0 \\ b c d & 26.4 & 24.9 & 51.3 \\ a b c d & 26.9 & 29.3 & 56.2 \\ \hline \text { Total } & 428.1 & 436.9 & 865.0 \end{array} $$
Nonreplicated $2^{k}$ Factorial Experiment
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