3.4. A computer ANOVA output is shown below. Fill in the blanks. You may give bounds on the P-value. One-way ANOVA Source DF SS MS F P Factor 3 36.15??? Error??? Total 19 196.04 3.11. A pharmaceutical manufacturer wants to investigate the bioactivity of a new drug. A completely randomized single-factor experiment was conducted with three dosage levels, and the following results were obtained. Dosage Observations 20g 24 28 37 30 30g 37 44 39 35 40g 42 47 52 38 (a) Is there evidence to indicate that dosage level affects bioactivity? Use a = 0.05. (b) Analyze the residuals from this experiment and comment on the model adequacy.
5.1. The following output was obtained from a computer program that performed a two-factor ANOVA on a factorial experiment. Two-way ANOVA: y versus A, B Source DF SS MS F P A 1? 0.0002?? B? 180.378??? Interaction 3 8.479?? 0.932 Error 8 158.797? Total 15 347.653 Fill in the blanks in the ANOVA table. You can use bounds on the P-values. (a) How many levels were used for factor B? (b) How many replicates of the experiment were performed? (c) What conclusions would you draw about this experiment? 5.4. A mechanical engineer is studying the thrust force developed by a drill press. 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. Analyze the data and draw conclusions. Use a = 0.05. (A) Feed Rate (B) Drill Speed 0.015 0.030 0.045 0.060 125 2.70 2.45 2.60 2.75 2.78 2.49 2.74 2.86 200 2.83 2.85 2.86 2.94 2.86 2.80 2.87 2.88
6.2. An engineer is interested in the effects of cutting speed (A), tool geometry (B), and cutting angle on the life (in hours) of a machine tool. Two levels of each factor are chosen, and three replicates of a 2 3 factorial design are run. The results are as follows: Treatment Replicate A B C Combination I II III - - - (1) 22 31 25 + - - a 32 43 29 - + - b 35 34 50 + + - ab 55 47 46 - - + c 44 45 38 + - + ac 40 37 36 - + + bc 60 50 54 + + + abc 39 41 47 (a) Estimate the factor effects. Which effects appear to be large? (b) Use the analysis of variance to confirm your conclusions for part (a). (c) Write down a regression model for predicting tool life (in hours) based on the results of this experiment. (d) Analyze the residuals. Are there any obvious problems? (e) Based on the analysis of main effects and interaction plots, what levels of A, B, and C would you recommend using? (f) Suppose that the experimenter only performed the eight trials from replicate I. In addition, he ran four center points and obtained the following response values: 36, 40, 43, 45. Estimate the factor effects. Which effects are large? (g) Write down an appropriate model for predicting tool life, based on the results of this experiment. Does this model differ in any substantial way from the model (h) What conclusions would you draw about the appropriate operating conditions for this process? 6.12. A nickel-titanium alloy is used to make components for jet turbine aircraft engines. Cracking is a potentially serious problem in the final part, as it can lead to non-recoverable failure. A test is run at the parts producer to determine the effects of four factors on cracks. The four factors are pouring temperature (A), titanium content (B), heat treatment method (C), and the amount of grain refiner used (D). Two replicated of a 2 4 design are run, and the length of crack (in mm) induced in a sample coupon subjected to a standard test is measured. The data are shown below: Treatment Replicate Replicate A B C D Combination I II - - - - (1) 7.037 6.376 + - - - a 14.707 15.219 - + - - b 11.635 12.089 + + - - ab 17.273 17.815 - - + - c 10.403 10.151 + - + - ac 4.368 4.098 - + + - bc 9.360 9.253 + + + - abc 13.440 12.923 - - - + d 8.561 8.951 + - - + ad 16.867 17.052 - + - + bd 13.876 13.658 + + - + abd 19.824 19.639 - - + + cd 11.846 12.337
+ - + + acd 6.125 5.904 - + + + bcd 11.190 10.935 + + + + abcd 15.653 15.053 (a) Estimate the factor effects. Which factors appear to be large? (b) Conduct an analysis of variance. Do any of the factors affect cracking? Use a=0.05. (c) Write down a regression model that can be used to predict crack length as a function of the significant main effects and interactions you have identified in part (b). (d) Analyze the residuals from this experiment. (e) Is there an indication that any of the factors affect the variability in cracking? (f) What recommendations would you make regarding process operations? Use interaction and/or main effect plots to assist in drawing conclusions
7.2 Consider the experiment described in Problem 6.2. Analyze this experiment assuming that each replicate represents a block of a single production shift. 7.3. Consider the data from the first replicate of Problem 6.2. Suppose that these observations could not all be run using the same bar stock. Set up a design to run these observations in two blocks of four observations each with ABC confounded. Analyze the data.
8.3. Suppose that in Problem 6.12, only a one-half fraction of the 2 4 design could be run. Construct the design and perform the analysis, using the data from replicate I. 8.4. An article in Industrial and Engineering Chemistry ( More on Planning Experiments to Increase Research Efficiency, 1970, pp. 60-65) uses a 2 5-2 design to investigate the effect of A = condensation, B = amount of material 1, C = solvent volume, D = condensation time, and E = amount of material 2 on yield. The results obtained are as follows: e = 23.2 ad = 16.9 cd = 23.8 bde = 16.8 ab = 15.5 bc = 16.2 ace = 23.4 abcde = 18.1 (a) Verify that the design generators used were I = ACE and I = BDE. (b) Proof following complete defining relation and the aliases for this design.. (c) Estimate the main effects. A=ABDE=CE=BCD B=DE=ABCE=ACD C=BCDE=AE=ABD D=BE=ACDE=ABC E=BD=AC=ABCDE AB=ADE=BCE=CD AD=ABE=CDE=BC (d) Prepare an analysis of variance table. Verify that the AB and AD interactions are available to use as error. (e) Plot the residuals versus the fitted values. Also construct a normal probability plot of the residuals. Comment on the results. 6 3 2-8.15. Construct a III design. Determine the effects that may be estimated if a second fraction of this design is run with all signs reversed.
11.2. The region of experimentation for three factors are time ( 40 T1 80 min), temperature ( 200 T 2 300 C), and pressure ( 20 P 50 psig). A first-order model in coded variables has been fit to yield data from a 2 3 design. The model is ŷ = 30 + 5x + x 1 + 2. 5x2 3. 5 Is the point T 1 = 85, T 2 = 325, P=60 on the path of steepest ascent? 3 11.4. A chemical plant produces oxygen by liquefying air and separating it into its component gases by fractional distillation. The purity of the oxygen is a function of the main condenser temperature and the pressure ratio between the upper and lower columns. Current operating conditions are temperature x ) = - 220 C and pressure ratio x ) = 1.2. Using the following data find the path of steepest ascent. ( 2 Temperature (x 1 ) Pressure Ratio (x 2 ) Purity -225 1.1 82.8-225 1.3 83.5-215 1.1 84.7-215 1.3 85.0-220 1.2 84.3-220 1.2 84.5-220 1.2 83.9-220 1.2 84.3 11.9. The data shown in Table P11.3 were collected in an experiment to optimize crystal growth as a function of three variables x 1, x 2, and x 3. Large values of y (yield in grams) are desirable. Fit a second order model and analyze the fitted surface. Under what set of conditions is maximum growth achieved? Table P11.3 x 1 x 2 x 3 y -1-1 -1 66-1 -1 1 70-1 1-1 78-1 1 1 64 1-1 -1 80 1-1 1 70 1 1-1 100 1 1 1 75-1.682 0 0 100 1.682 0 0 80 0-1.682 0 68 0 1.682 0 63 0 0-1.682 65 0 0 1.682 82 0 0 0 113 0 0 0 100 0 0 0 118 0 0 0 88 0 0 0 100 0 0 0 85 ( 1
11.13. A central composite design is run in a chemical vapor deposition process, resulting in the experimental data shown in Table P11.7. Four experimental units were processed simultaneously on each run of the design, and the responses are the mean and variance of thickness, computed across the four units. (USE MINITAB) Table P11.7 x 1 x 2 y 2 s -1-1 360.6 6.689 1-1 445.2 14.230-1 1 412.1 7.088 1 1 601.7 8.586 1.414 0 518.0 13.130-1.414 0 411.4 6.644 0 1.414 497.6 7.649 0-1.414 397.6 11.740 0 0 530.6 7.836 0 0 495.4 9.306 0 0 510.2 7.956 0 0 487.3 9.127 (a) Fit a model to the mean response. Analyze the residuals. (b) Fit a model to the variance response. Analyze the residuals. (c) Fit a model to the ln(s 2 ). Is this model superior to the one you found in part (b)? (d) Suppose you want the mean thickness to be in the interval 450±25. Find a set of operating conditions that achieve the objective and simultaneously minimize the variance. (e) Discuss the variance minimization aspects of part (d). Have you minimized total process variance?