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6-1) 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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6-1) 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 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
-	-	+	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) On the basis of an analysis of main effects 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 2* 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

putt (ft)             Type of putter        of putt           of putt       1 2 3 4 5 6 7

10 Mallet Straight Level 10.0 18.0 140 125 190 16.0 18.5

30 Mallet Straight Level 00 165 45 175 205 17.5 33.0

10 Cavity back Straight Level 40 60 10 145 120 140 5.0

30 Cavity back Straight Level 0.0 10.0 340 110 255 215 0.0

10 Mallet Breaking Level 00 O00 185 195 160 15.0 11.0

30 Mallet Breaking Level 5.0 205 180 200 29.5 19.0 10.0

10 Cavity back Breaking Level 65 185 75 60 00 100 0.0

30 Cavity back Breaking Level 165 45 00 235 80 80 8.0

10 Mallet Straight Downhill 45 180 145 100 00 175 6.0

30 Mallet Straight Downhill 19.5 180 160 5.5 100 7.0 36.0

10 Cavity back Straight Downhill 150 160 85 00 O05 90 3.0

30 Cavity back Straight Downhill 415 390 65 35 70 85 36.0

10 Mallet Breaking Downhill 80 45 65 100 13.0 41.0 14.0

30 Mallet Breaking Downhill 21.5 105 65 0.0 155 240 16.0

10 Cavity back Breaking Downhill 0.00 0.0 00 4.5 10 40 £65

30 Cavity back Breaking Downhill 180 5.0 70 100 32.5 18.5 8.0

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

(b) Analysis 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 (Ibs)	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
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 -100 -1.00 -1.00 -1.00 811

1.00 -1.00 -1.00 -100 -100 5.56

-100 100 -1.00 —-100 -1.00 5.77

1.00 1.00 -1.00 -100  -100 5.82

-100 -100 1.00 -1.00 -100 9.17

1.00 -1.00 100 -100 -100 78

-1.00 100 100 -1.00 -100 3.23

1.00 1.00 1.00 -1.00 -100 5.69

-1.00 -1.00 —1.00 1.00 -1.00 8.82

1.00 -1.00 —1.00 1.00 -100 14.23

-1.00 1.00 —1.00 100 -100 92

1.00 1.00 —1.00 1.00 -1.00 8.94

-1,00 -1.00 1.00 1.00 -100 8.68

1.00 -1.00 1.00 1.00 -100 11.49

-1.00 1.00 1.00 1.00 -100 6.25

1.00 1.00 ‘1.00 1.00 -100 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

1.00 100 —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

—100 —100 -—1.00 1.00 1.00 12.43

100 -1.00 —-1.00 1.00 1.00 17.55

— 1.00 100 —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

0. 0 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.