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Statistics homework 5 An engineer is interested in the effects of cutting speed (A), tool geometry (B), and cutting angle (C) on the life (in hours) of a machine tool

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Statistics homework 5

An engineer is interested in the effects of cutting speed (A), tool geometry (B), and cutting angle (C) on the life (in hours) of a machine tool. Two levels of each factor are chosen, and three replicates of a 23 factorial design are run. The results are as follows:

A B C Treatment Combination Replicate I II III (1) 22 31 25 a 32 43 29 b 35 34 50 + ab 55 47 46 — + 44 45 38 ± — + ac 40 37 36 be 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) On the basis of an analysis of main effect and interaction plots, what coded factor levels of A, B, and C would you recommend using?

 

6.6. Reconsider the experiment described in Problem 6.1. 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.

(a) Estimate the factor effects. Which effects are large?

(b) Perform an analysis of variance, including a check for pure quadratic curvature. What are your conclusions?

(c) 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 in Problem 6.1, part (c)?

(d) Analyze the residuals.

(e) What conclusions would you draw about the appropriate operating conditions for this process?

6.21. I am always interested in improving my golf scores. Since a typical golfer uses the putter for about 35-45 percent of his or her strokes, it seems reasonable that improving one's putting is a logical and perhaps simple way to improve a golf score ("The man who can putt is a match for any man?' Willie Parks, 1864-1925, two time winner of the British Open). An experiment was conducted to study the effects of four factors on putting accuracy. The design factors are length of putt, type of putter, breaking putt versus straight putt, and level versus downhill putt. The response variable is distance from the ball to the center of the cup after the ball comes to rest. One golfer performs the experiment, a 2⁴ factorial design with seven replicates was used, and all putts are made in random order. The results are shown in Table P6.4.

? TABLE P6.4 The Putting Experiment from Problem 6.21

 

Design Factors Distance from Cup (replicates) Length of Break Slope pull (ft) Type of puller of putt of putt 1 2 3 4 5 6 7 10 Mallet Straight Level 10.0 18.0 14.0 12.5 19.0 16.0 18.5 30 Mallet Straight Level 0.0 16.5 4.5 17.5 20.5 17.5 33.0 10 Cavity back Straight Level 4.0 6.0 1.0 14.5 12.0 14.0 5.0 30 Cavity back Straight Level 0.0 10.0 34.0 11.0 25.5 21.5 0.0 10 Mallet Breaking Level 0.0 0.0 18.5 19.5 16.0 15.0 11.0 30 Mallet Breaking Level 5.0 20.5 18.0 20.0 29.5 19.0 10.0 10 Cavity back Breaking Level 6.5 18.5 7.5 6.0 0.0 10.0 0.0 30 Cavity back Breaking Level 16.5 4.5 0.0 23.5 8.0 8.0 8.0 10 Mallet Straight Downhill 4.5 18.0 14.5 10.0 0.0 17.5 6.0 30 Mallet Straight Downhill 19.5 18.0 16.0 5.5 10.0 7.0 36.0 10 Cavity back Straight Downhill 15.0 16.0 8.5 0.0 0.5 9.0 3.0 30 Cavity back Straight Downhill 41.5 39.0 6.5 3.5 7.0 8.5 36.0 10 Mallet Breaking Downhill 8.0 4.5 6.5 10.0 13.0 41.0 14.0 30 Mallet Breaking Downhill 21.5 10.5 6.5 0.0 15.5 24.0 16.0 10 Cavity back Breaking Downhill 0.0 0.0 0.0 4.5 1.0 4.0 6.5 30 Cavity back Breaking Downhill 18.0 5.0 7.0 10.0 32.5 18.5 8.0

(a) Analyze the data from this experiment. Which factors significantly affect putting performance?

(b) Analyze the residuals from this experiment. Are there any indications of model inadequacy?

6.28. In a process development study on yield, four factors were studied, each at two levels: time (A), concentration (B), pressure (C), and temperature (D). A single replicate of a 2⁴ design was run, and the resulting data are shown in Table P6.7.

? TABLE P6.7 Process Development Experiment from Problem 6.28

Run Number Actual Run Order A B C D Yield (lbs) Factor Levels Low (—) High (+) 1 5 — 12 A (h) 2.5 3 2 9 + — — 18 B (%) 14 18 3 8 + — 13 C (psi) 60 80 4 13 + + — — 16 D (°C) 225 250 5 3 — — + — 17 6 7 + + 15 7 14 — + + — 20 8 1 + + + — 15 9 6 — — — + 10 10 11 + — + 25 11 2 + — + 13 12 15 + + — + 24 13 4 + + 19 14 16 + + + 21 15 10 + + + 17 16 12 + + + + 23

(a) Construct a normal probability plot of the effect estimates. Which factors appear to have large effects?

(b) Conduct an analysis of variance using the normal probability plot in part (a) for guidance in forming an error term. What are your conclusions?

(c) Write down a regression model relating yield to the important process variables.

(d) Analyze the residuals from this experiment. Does your analysis indicate any potential problems?

(e) Can this design be collapsed into a 23 design with two replicates? If so, sketch the design with the average and range of yield shown at each point in the cube. Interpret the results.

 

6.39. An article in Quality and Reliability Engineering International (2010, Vol. 26, pp. 223-233) presents a 25 factorial design. The experiment is shown in Table P6.12.

? TABLE P6.12 The 25 Design in Problem 6.39

A B C D E y -1.00 -1.00 -1.00 -1.00 -1.00 8.11 1.00 -1.00 -1.00 -1.00 -1.00 5.56 -1.00 1.00 -1.00 -1.00 -1.00 5.77 1.00 1.00 -1.00 -1.00 -1.00 5.82 -1.00 -1.00 1.00 -1.00 -1.00 9.17 1.00 -1.00 1.00 -1.00 -1.00 7.8 -1.00 1.00 1.00 -1.00 -1.00 3.23 1.00 1.00 1.00 -1.00 -1.00 5.69 -1.00 -1.00 -1.00 1.00 -1.00 8.82 1.00 -1.00 -1.00 1.00 -1.00 14.23 -1.00 1.00 -1.00 1.00 -1.00 9.2 1.00 1.00 -1.00 1.00 -1.00 8.94 -1.00 -1.00 1.00 1.00 -1.00 8.68 1.00 -1.00 1.00 1.00 -1.00 11.49 -1.00 1.00 1.00 1.00 -1.00 6.25 1.00 1.00 1.00 1.00 -1.00 9.12 -1.00 -1.00 -1.00 -1.00 1.00 7.93 1.00 -1.00 -1.00 -1.00 1.00 5 -1.00 1.00 -1.00 -1.00 1.00 7.47 1.00 1.00 -1.00 -1.00 1.00 12 -1.00 -1.00 1.00 -1.00 1.00 9.86 1.00 -1.00 1.00 -1.00 1.00 3.65 -1.00 1.00 1.00 -1.00 1.00 6.4 1.00 1.00 1.00 -1.00 1.00 11.61 -1.00 -1.00 -1.00 1.00 1.00 12.43 1.00 -1.00 -1.00 1.00 1.00 17.55 -1.00 1.00 -1.00 1.00 1.00 8.87 1.00 1.00 -1.00 1.00 1.00 25.38 -1.00 -1.00 1.00 1.00 1.00 13.06 1.00 -1.00 1.00 1.00 1.00 18.85 -1.00 1.00 1.00 1.00 1.00 11.78 1.00 1.00 1.00 1.00 1.00 26.05

(a) Analyze the data from this experiment. Identify the significant factors and interactions.

(b) Analyze the residuals from this experiment. Are there any indications of model inadequacy or violations of the assumptions?

(c) One of the factors from this experiment does not seem to be important. If you drop this factor, what type of design remains? Analyze the data using the full factorial model for only the four active factors. Compare your results with those obtained in part (a).

(d) Find settings of the active factors that maximize the predicted response.