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Design Of Experiments (IE 5342)

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Solutions from Montgomery, D. C. (2017) Design and Analysis of Experiments, Wiley, NY Chapter 7 Blocking and Confounding in the 2k Factorial Design Solutions 7 Consider the experiment described in Problem 6. Analyze this experiment assuming that each replicate represents a block of a single production shift. Source of Variation Cutting Speed (A) Tool Geometry (B) Cutting Angle (C) AB AC BC ABC Blocks Error Total Sum of Squares 0 770 280 16 468 48 28 0 482 2095 Design Expert Output Response: Life in hours ANOVA for Selected Factorial Model Analysis of variance table sum of Sum of Mean Source Squares DF Square Block 0 2 0 Model 1519 4 379 A 0 1 0 B 770 1 770 C 280 1 280 AC 468 1 468 Residual 575 17 33 Cor Total 2095 23 Degrees of Freedom 1 1 1 1 1 1 1 2 14 23 F Value Mean Square 0 770 280 16 468 48 28 0 34 F0 1 Prob F 11 0 22 8 13 0 0 0 significant The Model of 11 implies the model is significant. There is only a chance that a this large could occur due to noise. Values of less than 0 indicate model terms are significant. In this case B, C, AC are significant model terms. These results agree with the results from Problem 6. Tool geometry, cutting angle and the interaction between cutting speed and cutting angle are significant at the level. The Design Expert program also includes factor A, cutting speed, in the model to preserve hierarchy. 7. Consider the experiment described in Problem 6. Analyze this experiment assuming that each one of the four replicates represents a block. Source of Variation Bit Size (A) Cutting Speed (B) AB Blocks Error Total Sum of Squares 1107 227 303 44 27 1709 Degrees of Freedom 1 1 1 3 9 15 Mean Square 1107 227 303 14 3 F0 Solutions from Montgomery, D. C. (2017) Design and Analysis of Experiments, Wiley, NY These results agree with those from Problem 6. Bit size, cutting speed and their interaction are significant at the level. Design Expert Output Response: Vibration ANOVA for Selected Factorial Model Analysis of variance table sum of Sum of Mean Source Squares DF Square Block 44 3 14 Model 1638 3 546 A 1107 1 1107 B 227 1 227 AB 303 1 303 Residual 27 9 3 Cor Total 1709 15 F Value Prob F 179 364 74 99 significant The Model of 179 implies the model is significant. There is only a chance that a this large could occur due to noise. Values of less than 0 indicate model terms are significant. In this case A, B, AB are significant model terms. 7. Consider the alloy cracking experiment described in Problem 6. Suppose that only 16 runs could be made on a single day, so each replicate was treated as a block. Analyze the experiment and draw conclusions. The analysis of variance for the full model is as follows: Design Expert Output Response: Crack Lengthin mm x ANOVA for Selected Factorial Model Analysis of variance table sum of Sum of Mean Source Squares DF Square Block 0 1 0 Model 570 15 38 A 72 1 72 B 126 1 126 C 103 1 103 D 30 1 30 AB 29 1 29 AC 128 1 128 AD 0 1 0 BC 0 1 0 BD 0 1 0 CD 0 1 0 ABC 78 1 78 ABD 0 1 0 ACD 1 BCD 0 1 0 ABCD 1 Residual 1 15 0 Cor Total 572 31 F Value Prob F 445 852 1478 1209 358 349 1502 0 0 0 0 920 0 0 0 0 0 0 0 0 0 0 0 0 significant The Model of 445 implies the model is significant. There is only a chance that a this large could occur due to noise. Values of less than 0 indicate model terms are significant. In this case A, B, C, D, AB, AC, ABC are significant model terms. The analysis of variance for the reduced model based on the significant factors is shown below. The BC interaction was included to preserve hierarchy. Solutions from Montgomery, D. C. (2017) Design and Analysis of Experiments, Wiley, NY Design Expert Output Response: Life in hours ANOVA for Selected Factorial Model Analysis of variance table sum of Sum of Mean Source Squares DF Square Block 91 1 91 Model 896 4 224 A 3 1 3 B 325 1 325 C 190 1 190 AC 378 1 378 Residual 61 2 30 Cor Total 1048 7 F Value Prob F 7 0 10 6 12 0 0 0 0 0 not significant The of 7 implies the model is not significant relative to the noise. There is a 12 chance that a this large could occur due to noise. Values of less than 0 indicate model terms are significant. In this case there are no significant model terms. This design identifies the same significant factors as Problem 6. 7. Consider the data from the first replicate of Problem 6. Construct a design with two blocks of eight observations each with ABCD confounded. Analyze the data. Block 1 (1) ab ac bc ad bd cd abcd Block 2 a b c d abc abd acd bcd The significant effects are identified in the normal probability plot of effects below: Solutions from Montgomery, D. C. (2017) Design and Analysis of Experiments, Wiley, NY AC, BC, and BD were included in the model to preserve hierarchy. Design Expert Output Response: yield ANOVA for Selected Factorial Model Analysis of variance table sum of Sum of Mean Source Squares DF Square Block 42 1 42 Model 892 11 81 A 400 1 400 B 2 1 2 C 2 1 2 D 100 1 100 AB 81 1 81 AC 1 1 1 AD 56 1 56 BC 6 1 6 BD 9 1 9 ABC 144 1 144 ABD 90 1 90 Residual 25 3 8 Cor Total 959 15 F Value Prob F 9 47 0 0 11 9 0 6 0 1 17 10 0 0 0 0 0 0 0 0 0 0 0 0 significant The Model of 9 implies the model is significant. There is only a chance that a this large could occur due to noise. Values of less than 0 indicate model terms are significant. In this case A, D, ABC, ABD are significant model terms. 7. Repeat Problem 7 assuming that four blocks are required. Confound ABD and ABC (and consequently CD) with blocks. The block assignments are shown in the table below. The normal probability plot of effects identifies factors A and D, and the interactions AB, AD, and the ABCD as strong candidates for the model. For hierarchal purposes, factor B was included in the however, hierarchy is not preserved for the ABCD interaction allowing an estimate for error. Block 1 (1) ab acd bcd Block 2 ac bc d abd Block 3 c abc ad bd Block 4 a b cd abcd Solutions from Montgomery, D. C. (2017) Design and Analysis of Experiments, Wiley, NY The normal probability plot of effects identifies factors A, B, C, and the AB interaction as being significant. This is confirmed with the analysis of variance. Design Expert Output Response: Yield ANOVA for Selected Factorial Model Analysis of variance table sum of Sum of Mean Source Squares DF Square Block 0 1 0 Model 11585 4 2896 A 1 B 9214 1 9214 C 750 1 750 AB 504 1 504 Residual 78 26 3 Cor Total 11663 31 F Value Prob F 958 369 3049 248 166 significant The Model of 958 implies the model is significant. There is only a chance that a this large could occur due to noise. Values of less than 0 indicate model terms are significant. In this case A, B, C, AB are significant model terms. 7. Repeat Problem 7 assuming that four blocks are necessary. confounding scheme. Use ABC and CDE, and consequently ABDE. The four blocks follow. Block 1 (1) ab acd bcd ace bce de abde Block 2 a b cd abcd ce abce ade bde Block 3 ac bc d abd e abe acde bcde Block 4 c abc ad bd ae be cde abcde Suggest a reasonable Solutions from Montgomery, D. C. (2017) Design and Analysis of Experiments, Wiley, NY The normal probability plot of effects identifies the same significant effects as in Problem 7. Design Expert Output Response: Yield ANOVA for Selected Factorial Model Analysis of variance table sum of Sum of Mean Source Squares DF Square Block 13 3 4 Model 11585 4 2896 A 1 B 9214 1 9214 C 750 1 750 AB 504 1 504 Residual 65 24 2 Cor Total 11663 31 F Value Prob F 1069 412 3402 277 186 significant The Model of 1069 implies the model is significant. There is only a chance that a this large could occur due to noise. Values of less than 0 indicate model terms are significant. In this case A, B, C, AB are significant model terms. 7. Consider the data from the 25 design in Problem 6. Suppose that it was necessary to run this design in four blocks with ACDE and BCD (and consequently ABE) confounded. Analyze the data from this design. Block 1 (1) ae cd abc acde bce abd bde Block 2 a e acd bc cde abce bd abde Block 3 b abe bcd ac abcde ce ad de Block 4 c ace d ab ade be abcd bcde Solutions from Montgomery, D. C. (2017) Design and Analysis of Experiments, Wiley, NY The Model of 28 implies the model is significant. There is only a chance that a this large could occur due to noise. Values of less than 0 indicate model terms are significant. In this case A, B, C are significant model terms. The analysis is very similar to the original analysis in chapter 6. The same effects are significant. 7. Consider the fill height deviation experiment in Problem 6. Suppose that only four runs could be made on each shift. Set up a design with ABC confounded in replicate 1 and AC confounded in replicate 2. Analyze the data and comment on your findings. Design Expert Output Response: Fill Deviation ANOVA for Selected Factorial Model Analysis of variance table sum of Sum of Mean Source Squares DF Square Block 1 3 0 Model 70 4 17 A 36 1 36 B 20 1 20 C 12 1 12 AB 2 1 2 Residual 5 8 0 Cor Total 78 15 F Value Prob F 24 50 28 17 3 0 0 significant The Model of 24 implies the model is significant. There is only a chance that a this large could occur due to noise. Values of less than 0 indicate model terms are significant. In this case A, B, C are significant model terms. The analysis is very similar to the original analysis of Problem 6 and that of problem 7. The AB interaction is less significant in this scenario. 7. Consider the putting experiment in Problem 6. Analyze the data considering each replicate as a block. The analysis is similar to that of Problem 6. Blocking has not changed the significant factors, however, the residual plots show that the normality assumption has been violated. The transformed data also has similar analysis to the transformed data of Problem 6. The ANOVA shown is for the transformed data. Design Expert Output Response: Distance from cupTransform:Square root ANOVA for Selected Factorial Model Analysis of variance table sum of Sum of Mean Source Squares DF Square Block 13 6 2 Model 37 2 18 A 21 1 21 B 15 1 15 Residual 245 103 2 Cor Total 295 Constant: 0 F Value Prob F 7 9 6 0 0 The Model of 7 implies the model is significant. There is only a chance that a this large could occur due to noise. Values of less than 0 indicate model terms are significant. In this case A, B are significant model terms. significant Solutions from Montgomery, D. C. (2017) Design and Analysis of Experiments, Wiley, NY 7. Using the data from the 24 design in Problem 6, construct and analyze a design in two blocks with ABCD confounded with blocks. Design Expert Output Response: UEC ANOVA for Selected Factorial Model Analysis of variance table sum of Sum of Mean Source Squares DF Square Block 1 Model 0 4 0 A 0 1 0 C 0 1 0 D 0 1 0 AC 0 1 0 Residual 0 10 Cor Total 0 15 F Value Prob F 32 56 38 27 6 0 significant The Model of 32 implies the model is significant. There is only a chance that a this large could occur due to noise. Values of less than 0 indicate model terms are significant. In this case A, C, D, AC are significant model terms. The analysis is similar to that of Problem 6. The significant effects are A, C, D and AC. 7. Consider the direct mail experiment in Problem 6. Suppose that each group of customers is in different parts of the country. Support an appropriate analysis for the experiment. Set up each Group (replicate) as a geographic region. The analysis is similar to that of Problem 6. Factors A and B are included to achieve a hierarchical model. Design Expert Output Response: Yield ANOVA for Selected Factorial Model Analysis of variance table sum of Sum of Mean Source Squares DF Square Block 0 1 0 Model 241 6 40 A 12 1 12 B 2 1 2 C 36 1 36 AB 42 1 42 AC 100 1 100 BC 49 1 49 Residual 27 8 3 Cor Total 269 15 F Value Prob F 11 3 0 10 12 28 14 0 0 0 0 0 0 significant The Model of 11 implies the model is significant. There is only a chance that a this large could occur due to noise. Values of less than 0 indicate model terms are significant. In this case C, AB, AC, BC are significant model terms. 7. Consider the isatin yield experiment in Problem 6. Set up the 24 experiment in this problem in two blocks with ABCD confounded. Analyze the data from this design. Is the block effect large? The block effect is very small. Solutions from Montgomery, D. C. (2017) Design and Analysis of Experiments, Wiley, NY Block 1 Block 2 Block 1 Block 2 Block 4 Block 3 Block 2 Block 1 Block 3 Block 4 (b) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 6 11 12 17 8 25 13 18 11 26 Analyze the data from this blocked design. Is blocking important? Blocking does not appear to be however, if the ADE or ABE interaction had been chosen to define the blocks, then blocking would have appeared as important. The ADE and ABE are significant effects in the analysis below. Design Expert Output Response 1 y ANOVA for selected factorial model Analysis of variance table sum of squares Type Sum of Mean Source Squares df Square Block 2 3 0 Model 879 11 79 83 1 83 0 1 0 285 1 285 153 1 153 AB 48 1 48 AD 88 1 88 AE 33 1 33 BE 52 1 52 DE 61 1 61 ABE 44 1 44 ADE 26 1 26 Residual 29 17 1 Cor Total 912 31 F Value 45 47 0 162 86 27 50 19 29 35 25 14 The Model of 45 implies the model is significant. There is only a chance that a this large could occur due to noise. Values of less than 0 indicate model terms are significant. In this case A, D, E, AB, AD, AE, BE, DE, ABE, ADE are significant model terms. Prob F 0 0 significant Solutions from Montgomery, D. C. (2017) Design and Analysis of Experiments, Wiley, NY 7. Repeat Problem 7 using a design in two blocks. (a) Recommend a blocking scheme and set up the design. Interaction ABCDE is confounded with the blocks. The design is shown below. Block Block 1 Block 2 Block 2 Block 1 Block 2 Block 1 Block 1 Block 2 Block 2 Block 1 Block 1 Block 2 Block 1 Block 2 Block 2 Block 1 Block 2 Block 1 Block 1 Block 2 Block 1 Block 2 Block 2 Block 1 Block 1 Block 2 Block 2 Block 1 Block 2 Block 1 Block 1 Block 2 A 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 B 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 C 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 D 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 E 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 y 8 5 5 5 9 7 3 5 8 14 9 8 8 11 6 9 7 5 7 12 9 3 6 11 12 17 8 25 13 18 11 26 Solutions from Montgomery, D. C. (2017) Design and Analysis of Experiments, Wiley, NY The runs for the experiment are shown below with the corresponding blocks. Run 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Block Block 2 Block 1 Block 1 Block 2 Block 1 Block 2 Block 2 Block 1 Block 1 Block 2 Block 2 Block 1 Block 2 Block 1 Block 1 Block 2 Glucose (g 20 60 20 60 20 60 20 60 20 60 20 60 20 60 20 60 NH4NO3 (g 2 2 6 6 2 2 6 6 2 2 6 6 2 2 6 6 FeSO4 (g x 6 6 6 6 30 30 30 30 6 6 6 6 30 30 30 30 MnSO4 (g x 4 4 4 4 4 4 4 4 20 20 20 20 20 20 20 20 y 23 15 16 18 25 16 17 26 28 16 18 21 36 24 33 34 The analysis of the experiment shown below identifies the contribution of the blocks. reducing the SSE and MSE, the AD and CD interactions now appear to be significant. Design Expert Output Response 1 y ANOVA for selected factorial model Analysis of variance table sum of squares Type Sum of Mean Source Squares df Square Block 6 1 6 Model 713 8 89 42 1 42 1 196 1 196 182 1 182 AB 196 1 196 AD 12 1 12 BC 20 1 20 CD 64 1 64 F Value Prob F 50 24 112 104 112 7 11 36 0 0 0 significant Solutions from Montgomery, D. C. (2017) Design and Analysis of Experiments, Wiley, NY Residual Cor Total 10 729 6 15 1 The Model of 50 implies the model is significant. There is only a chance that a this large could occur due to noise. Values of less than 0 indicate model terms are significant. In this case A, C, D, AB, AD, BC, CD are significant model terms. 7. The design in Problem 6 is a 23 factorial replicated twice. Suppose that each replicate was a block. Analyze all of the responses from this blocked design. Are the results comparable to those from Problem 6? Is the block effect large? The block effect is not large and does not appear to be important for the analysis on any of the four the responses as shown below. The results are comparable to those from Problem 6. Design Expert Output Response 1 Fishbone Pb ANOVA for selected factorial model Analysis of variance table sum of squares Type Sum of Mean Source Squares df Square Block 1 Model 12 7 1 10 1 10 0 1 0 0 1 0 AB 0 1 0 AC 0 1 0 BC 0 1 0 ABC 0 1 0 Residual 7 Cor Total 12 15 Factor Intercept Block 1 Block 2 AB AC BC ABC Coefficient Estimate 0 0 0 0 df 1 1 1 1 1 1 1 1 1 F Value Prob F 2300 14514 459 350 475 249 29 25 0 0 significant Standard Error CI Low 0 CI High 0 VIF 0 0 0 0 0 0 1 1 1 1 1 1 1 Design Expert Output Response 1 Fishbone pH ANOVA for selected factorial model Analysis of variance table sum of squares Type Sum of Mean Source Squares df Square Block 1 Model 21 7 3 9 1 9 8 1 8 1 1 1 AB 0 1 0 AC 1 1 1 BC 0 1 0 ABC 0 1 0 Residual 0 7 0 Cor Total 21 15 F Value Prob F 102 336 277 38 20 40 3 3 0 0 0 significant Solutions from Montgomery, D. C. (2017) Design and Analysis of Experiments, Wiley, NY Design Expert Output Response 1 Hydroxyapatite pH ANOVA for selected factorial model Analysis of variance table sum of squares Type Sum of Mean Source Squares df Square Block 1 Model 20 7 2 8 1 8 8 1 8 0 1 0 AB 3 1 3 AC 0 1 0 BC 0 1 0 ABC 1 Residual 0 7 Cor Total 20 15 Factor Intercept Block 1 Block 2 AB AC BC ABC Coefficient Estimate 3 0 0 0 df 1 1 Standard Error 0 1 1 1 1 1 1 1 CI Low 3 0 0 0 0 0 0 0 F Value Prob F 1494 4172 4515 43 1658 7 64 0 0 0 CI High 3 0 0 0 0 0 significant VIF 1 1 1 1 1 1 1 7. Design an experiment for confounding a 26 factorial in four blocks. Suggest an appropriate confounding scheme, different from the one shown in Table 7. We choose ABCE and ABDF, which also confounds CDEF. Block 1 a b cd abcd ace bce de abde cf abcf adf bdf ef abef acdef bcdef Block 2 c abc ad bd e abe acde bcde af bf cdf abcdf acef bcef def abdef Block 3 ac bc d abd ae be cde abcde f abf acdf bcdf cef abcef adef bdef Block 4 (1) ab acd bcd ce abce ade bde acf bcf df abdf aef bef cdef abcdef 7. Consider the 26 design in eight blocks of eight runs each with ABCD, ACE, and ABEF as the independent effects chosen to be confounded with blocks. Generate the design. Find the other effects confound with blocks. Solutions from Montgomery, D. C. (2017) Design and Analysis of Experiments, Wiley, NY Block 1 b acd ce abde abcf df aef bcdef Block 2 abc d ae bcde bf acdf cef abdef Block 3 a bcd abce de cf abdf bef acdef Block 4 c abd be acde af bcdf abcef def Block 5 ac bd abe cde f abcdf bcef adef Block 6 (1) abcd bce ade acf bdf abef cdef Block 7 bc ad e abcde abf cdf acef bdef Block 8 ab cd ace bde bcf adf ef abcdef The factors that are confounded with blocks are ABCD, ABEF, ACE, BDE, CDEF, BCF, and ADF. 7. Consider the 22 design in two blocks with AB confounded. Prove algebraically that SSAB SSBlocks. If AB is confounded, the two blocks are: Block 1 (1) ab (1) ab 2 SSBlocks ab a b ab a b 2 4 SSBlocks SS Blocks SS Blocks Block 2 a b (1) (1) 2 2 2 2 ( ab ) 2 (1) ab a 2 b 2 2ab 2 2 2 ( ab ) a 2 b 2 2 (1) ab 2 (1) a 2 (1) b 2a ( ab ) 2b ( ab ) 2ab 2 4 2 ( ab ) a 2 b 2 2 (1) ab 2ab 2 (1) a 2 (1) b 2a ( ab ) 2b ( ab ) (1) 4 2 1 ab a b SS AB 4 7. Consider the data in Example 7. Suppose that all the observations in block 2 are increased 20. Analyze the data that would result. Estimate the block effect. Can you explain its magnitude? Do blocks now appear to be an important factor? Are any other effect estimates impacted the change you made in the data? Block Effect yBlock1 yBlock 2 406 715 8 8 8 This is the block effect estimated in Example 7 plus the additional 20 units that were added to each observation in block 2. All other effects are the same.

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design and analysis of experiments 7th edition solutions

Design and Analysis of Experiments, Student Solutions Manual 7th edition

Lisa Custer , Douglas C Montgomery

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Design and Analysis of Experiments, Student Solutions Manual (7th edition)

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This bestselling professional reference has helped over 100,000 engineers and scientists with the success of their experiments. The new edition includes more software examples taken from the three most dominant programs in the field: Minitab, JMP, and SAS. Additional material has also been added in several chapters, including new developments in robust design and factorial designs. New examples and exercises are also presented to illustrate the use of designed experiments in service and transactional organizations. Engineers will be able to apply this information to improve the quality and efficiency of working systems.

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