Stat664 Homework #3 due April 2 Turn in problems marked with Note: Whenever appropriate/possible, inspect the possibilities for need of transformation, determine suitable power (considering only power transformation), analyze transformed data, and discuss the interpretability of the transformed scale. 58. Suppose the goal of an experimenter is to determine the settings of two factors, temperature T (in F ) and velocity V (in mph), which MINIMIZE a response Y. It is proposed to run a design with corners defined by T = 70 F, 80 F and V = 40 mph, 60 mph. The coded temperature and velocity are denoted by x and x 2. (a) Write out the formulae for the coded variables x and x 2 from which the levels can be coded as and +. (b) Should additional runs be made at the center point? Why or why not? (c) Suppose a satisfactory fit to the data is given by the model ŷ = 5 3x + 4x 2. In what direction should one proceed to MINIMIZE the response (or what is the steepest descent direction)? (d) After several iterations of traveling along the steepest descent direction, a central composite design were conducted centering at T = 90 F, V = 0 mph and the results were analyzed. Suppose an adequate model is obtained and after canonical reduction the model takes the form ŷ 0.8 = B X 2 + B 22 X, where (X, X 2 ) are the usual (x, x 2 ) variables after appropriate transformation. In order for this model to correspond to a simple MINIMUM, what must be true about B and B 22? What is the expected value of that MINIMUM response? 59. (2 points) In a resistance spot welding experiment, five factors were chosen to study their effects on the tensile strength, which is the maximum load a weld can sustain in a tensile test. The five factors are button diameter (A), welding time (B), holding time (C), electrode force (D), and machine type (E), each at two levels. The last factor is qualitative, while the others are quantitative. A 2 5 V design with I = ABCDE was used for the experiment. Each run has three replicates. The data are given below:
Factor Run A B C D E Tensile strength 330 330 65 2 + + 935 935 880 3 + + 770 770 770 4 + + 275 275 275 5 + + 880 935 880 6 + + 385 440 495 7 + + 220 65 440 8 + + + + 255 200 200 9 + + 75 75 660 0 + + 385 550 550 + + 000 65 495 2 + + + + 990 990 990 3 + + 275 660 550 4 + + + + 660 605 660 5 + + + + 880 935 935 6 + + + + 275 220 275 Source:Wu & Hamada, Experiments. Analyze the data. 60. (2 points) A food scientist has collected the data below. Factor Run A (temperature) B (mixing speed) C (time in sec.) Viscosity (25) 60 2 +(35) (low) 62 3 +(high) 82 4 + + (60) 86 5 +(90) 85 6 + + 85 7 + + 6 8 + + + 6 9 0(30) 0(medium) 0(75) 84 0 0 0 0 87 0 0 0 8 Source:Wu & Hamada, Experiments. (a) What observations among the runs can be used to estimate the underlying process variance σ 2? (b) Assume each run follows a normal distribution with the same variance. Estimate the standard error of the factorial effects. 2
(c) Suppose s 2 is the estimate of σ 2 obtained in (a). Calculate the factorial effects and determine which effects are significant. (d) Use half-normal plot of the effects to determine effects significance. Are the results the same as you reported in (c)? (e) Use Lenth method to decide effects significance and compare the results with those in (d). 6. To arrange a 2 6 design in 6 blocks, two blocking schemes are being considered. The first has the generators B =26, B 2 =36, B 3 =346, and B 4 =456; and the second is B =36, B 2 =234, B 3 =3456, and B 4 =23456. Which blocking scheme is better. Use the criteria as described in class. 62. An experimenter obtained eight yields for the design given below: 2 3 4 5 Yield 26 + + 28 + + + 23 + + + 22 + + + 9 + + + 8 + + 30 + + + + 28 Source:Wu & Hamada, Experiments. (a) Make two interaction plots for factors 3 and 4, one for 3 against 4, and the other for 4 against 3. (b) Calculate the main effect of 3 and 4 and the 3 4 interaction. (c) If it is known that the standard deviation of each observation is.5, what is the standard error of the estimates in (b)? (d) Based on the results in (a) (c), which of the three effects in (b) are significant? Comment on sinergistic and antagonistic interactions. 63. An experiment on a surface-finishing operation of an overhead cam block auxilliary drive shaft was reported by Sirvanci and Durmaz (993). The following is an adaption of the original experiment. The part had a target value of 75µm. The three experimental factors were type of insert (A), speed in rpm (B), and feed rate in millimeters per minute (C). A 2 3 design was used. The surface roughness of each part was measured by a Surtronic-3 device. The design and data for the drive shaft experiment are displayed below. The current setting for the surface-finishing process is insert type #5023, 780 rpm, and 60 mm/min. 3
Factor Run A B C Roughness (50) 54.6 73.0 39.2 55.4 52.6 2 (800) +(80) 86.2 66.2 79.2 86.0 82.6 3 +(000) 4.4 5.2 42.6 58.6 58.4 4 (#5023) + + 62.8 64.8 74.6 74.6 64.6 5 +(#5074) 59.6 52.8 55.2 6.0 6.0 6 + + 82.0 72.8 76.6 73.4 75.0 7 + + 43.4 49.0 48.6 49.6 55.2 8 + + + 65.6 65.0 64.2 60.8 77.4 Source:Wu & Hamada, Experiments. (a) Analyze the experiment for both location and dispersion effects. (b) Make recommendations on the optimal factor settings for location and dispersion effects. 64. (3 points) A company had received a contract to manufacture aluminum case with a plastic polymer coating on the case for transceivers. A research team was given the responsibility to develop a process to adhere the polymer coating to the aluminum. The team identified five factors in the process which they thought had any potential to affect adhesion of the polymer to the aluminum. They were the type of aluminum alloy (A), the type of solvent used to clean the aluminum (S), the molecular structure of the coating polymer (M), the percent catalyst used in the adhesion process (C), and the curing temperature for the process (T). A 2 5 V : I=ASMCT design was used and the results: A S M C T force + 4.5 + 39.6 + 43.9 + + + 38.8 + 48.7 + + + 52.0 + + + 55.8 + + + 43.2 + 39.5 + + + 42.6 + + + 44.0 + + + 33.8 + + + 53.6 + + + 48. + + + 5.3 + + + + + 48.7 4
Examine the effects using normal probability plot (or half normal probability plot) of effects. Also use Lenth method to examine the effects significance. Draw useful conclusion. State any assumption that your conclusion is based upon. 65. (2 points) (Montgomery) An article in Solid State Technology ( Orthogonal Design for Process Optimization and Its Application in Plasma Etching, May 987, pp. 27 32) describes the application of factorial design in developing a nitride etch process on a singlewafer plasma etcher. The process uses C 2 F 6 as the reactant gas. Four factors are of interest: A, anode-cathode gap (0.80 &.20cm); B, pressure in the reactor chamber (450 & 550mTorr); C, C 2 F 6 gass flow (25 & 200SCCM); D, powder applied to the cathode (275 & 325W). The response variable of interest is the etch rate (Å/min) for silicon nitride. A single replicate of a 2 4 design is run, and the data arranged in standard order in factors A, B, C, and D with corresponding run order recorded in parentheses are given: 550(3), 669(8), 604(2), 650(9), 633(4), 642(5), 60(6), 635(3), 037(), 749(4), 052(5), 868(0), 075(), 860(2), 063(7), 729(6). Analyze the data and draw appropriate conclusion. 66. (2 points) An animal scientist studied the relationship between metabolism of methionine, a sulfur amino acid, and carotene, vitamin A, as they affect the growth of chickens. The optimum levels of methionine and carotene were thought to be 0.9% methionine in the diet and 50 micrograms carotene per day. A central composite rotatable design was used for the experiment. Eight chicks were randomly assigned to each of the treatment diets and their weight gains were recorded after 38 days. The average weight gains for the treatments follow. 5
(a) Fit the second-order model. Original Factors Coded Variables Methionine Carotene x x 2 Weight Gain.83 85.36 + + 445.83 4.64 + 33 0.67 85.36 + 443 0.67 4.64 336.300 50.00 2 0 44 0.500 50.00 2 0 389 0.900 00.00 0 2 435 0.900 0.00 0 25 0.900 50.00 0 0 442 0.900 50.00 0 0 42 0.900 50.00 0 0 48 0.900 50.00 0 0 440 0.900 50.00 0 0 44 (b) Obtain an estimate for the experimental error σ 2. (c) Perform checks for exact quadratic in x direction and for exact quadratic in x 2 direction. (d) Perform canonical analysis and describe the response surface. 67. (Montgomery) An experiment was conducted to determine how three factors affect the scrumptiousness of brownies. The factors and the corresponding levels are Factor Low( ) High(+) P, pan material Glass Aluminum S, stirring method Spoon Mixer M, brand of mix Expensive Cheap The response was a subjective measure derived from a questionnaire given to the subjects who sampled each batch of brownies. (The questionnaire dealt with such issues as taste, appearance, consistency, aroma, and so forth). An eight person test panel sampled each batch and filled out the questionnaire. The design matrix and the response data are shown below: 6
Test Panel Results A B C 2 3 4 5 6 7 8 9 0 0 0 8 9 + 5 0 6 4 2 9 6 5 + 9 2 2 + + 6 7 5 2 3 3 + 0 5 8 6 8 9 4 + + 2 3 4 3 9 3 4 9 + + 0 2 3 0 7 7 7 3 + + + 5 2 5 3 9 4 Analyze the data and draw appropriate conclusion. 68. (2 points) (Montgomery) An experiment was conducted on a chemical process that produces a polymer. The four factors under study were T, temperature (00 & 20 C); C, catalyst concentration (4 & 8%); R, reaction time (20 & 30min); P, pressure (60 & 75psi). The design matrix and response data are shown below: run molecular T C R P order weight viscosity 8 2400 400 + 9 240 500 + 3 235 520 + + 8 250 630 + 3 265 380 + + 2625 525 + + 4 2400 500 + + + 7 2750 620 + 6 2400 400 + + 7 2390 525 + + 300 500 + + + 0 2520 500 + + 4 2625 420 + + + 9 2630 490 + + + 5 2500 500 + + + + 20 270 600 0 0 0 0 255 500 0 0 0 0 5 2500 460 0 0 0 0 6 2400 525 0 0 0 0 475 500 7
The center point is T = 0 C, C = 6%, R = 25min, and P = 67.5psi. Analyze the data. What operating conditions would you recommend if it was necessary to produce a product with molecular weight between 2400 and 2500, and the lowest possible viscosity? (Hint: Use superimposed contours of the fitted curves in these two responses.) 69. Three constituents polyethylene (x ), polystyrene (x 2 ), and polypropylene (x 3 ) were blended together and the resulting fiver material were spun to form yarn for draperies. The response is the observed elongation value (y) at each of the blends. Yarns with high elongation values is desirable. The data are given by Design Component Proportions Observed Average Point x x 2 x 3 Elongation (y) Elongation (y) 0 0.0, 2.4.7 2 0 5.0, 4.8, 6. 5.3 3 0 0 8.8, 0.0 9.4 4 0 2 2 0.0, 9.7,.8 0.5 5 0 0 6.8, 6.0 6.4 6 0 7.7, 6.4, 6.6 6.9 Source: Experiments with Mixtures by Cornell (a) What sort of design is this? (b) Produce a graphical representation of the design. (c) The second-order fitted response is given by 5 x = 3 4 6 x = 2 4 2 3 5 7 x = 3 i. Approximately what blend would you choose to maximize the elongation? ii. If a pure blend is desirable, which one would you choose? 70. A mixture experiment was conducted to evaluate the effect of three components on the octane ratings of gasoline. The components aklylate (x ), light straight run (x 2 ), and reformate (x 3 ) were used in a design with the following seven mixtures in two replicates. The octane ratings were obtained 8
Components mean octane rating x x 2 x 3 octane ratings, y y i 0 06.6, 05.0 05.80 0 0 83.3, 8.4 82.35 0 0 99.4, 9.4 95.40 0 94., 9.4 92.75 0 0.9, 98.0 99.95 0 92.3, 86.5 89.40 96.3, 9.7 94.00 3 3 3 Source: R.D. Snee (98). Developing blending models for gasoline and other mixtures. Technometrics 23, 9 30. A quadratic model was fitted to the data. (a) What sort of design is this? Draw a graphical representation of the design (with design points marked). (b) Determine the linear coefficients ˆβ i, i =, 2, 3. (c) The fitted response surface was described by the following contour map: x = 05 02.5 00 97.5 x 2 = 85 87.5 90 92.5 95 x 3 = What would you recommend if the objective is to find a mixture that maximize the mean octane rating? 9