Stat424-F23-Prac-Exam3-Solution_final (1)

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Stat/ME 424, Solution to Practice Exam 3, Fall 2023 • This is a closed-book exam, except for eight sides of notes. • You are allowed to use calculators. Problem 1 (20 pts) Find out how many degrees of freedom are associated with the residual in the ANOVA or ANCOVA (whichever is appropriate) table for the following scenarios. (a) A randomized block design with 5 blocks and 5 treatments. ( b − 1 )( k − 1 ) = 4 × 4 = 16 . (b) A three-way layout with three replications ( n = 3, i.e. every level combination conducted three times) and three factors A , B , C , where A , B are three-level factors and C is a four-level factor. IJK ( n − 1 ) = 3 × 3 × 4 × ( 3 − 1 ) = 72. (c) In addition to the same layout as in (b), there is a covariate X . 71 df due to the covariate. (d) A 4 × 4 Latin square design. ( k − 1 )( k − 2 ) = 3 × 2 = 6. 1

Problem 2 (15 pts) An experiment is being performed to determine the effects of three different assembly methods on the throughput. The three assembly methods are denoted by M1, M2 and M3. The experiment is completed on the same day using the same machine. Three different operators, O1, O2 and O3, are chosen for the experiment. The experimenter will only do 9 trials. The manager wants all of operators to be used for conducting the experiment. The experimenter feels the three operators have different levels of experience. Owing to time and resource restrictions, it is only possible to conduct 9 trials as mentioned above. (a) Is there any factor(s) you intend to study on throughput? Assembly method (b) Is there any factor whose effect you would like to block ? If so, what are they? Operator (c) What kind of design (just state the name) would you use for this experiment? RBD. (d) Design the experiment. Your final output should be of the following form: Trial Number Operator Assembly Method 1 − − ··· ··· ··· N − − To get full credit, explain clearly how you derived this design. Trial Number Operator Assembly Method 1 O1 M1 2 O1 M2 3 O1 M3 4 O2 M1 5 O2 M2 6 O2 M3 7 O3 M1 8 O3 M2 9 O3 M3 2

Problem 3 (25 pts) You need to construct a 2 4 full factorial experiment with four factors A , B , C , D . The two levels of each factor are denoted by − (lower level) and + (higher level). Four machines (machine is not an experimental factor) are available for the experiment, and owing to resource constraints, you cannot conduct more than four trials on each machine. There may some difference among the four machines in terms of performances. (a) What would be the block size in this experiment? Four. There are four blocks (machines) of size four (trials) each. (b) How many factorial effects will be confounded with the block effects? Three. Since there are four blocks, there are 3 df associated with the block factor machine. These three df will be confounded with three factorial effects (each of the 15 factorial effects in a two-level factorial experiment has one df). (c) Using the generators B 1 = ABC and B 2 = ABD , design the experiment by showing which of the 16 trials (i.e., combinations of A , B , C , D ) will be conducted on machine 1, which on machine 2, etc. You need not randomize the trials . Run A B C D ABC ABD Machine 1 − − − + − + II 2 − − − − − − I 3 − − + + + + IV 4 − − + − + − III 5 − + − + + − III 6 − + − − + + IV 7 − + + + − − I 8 − + + − − + II 9 + − − + + − III 10 + − − − + + IV 11 + − + + − − I 12 + − + − − + II 13 + + − + − + II 14 + + − − − − I 15 + + + + + + IV 16 + + + − + − III Here machines 1,2,3,4 are allocated to ( − , − ) , ( − , +) , (+ , − ) , (+ , +) respectively. (d) Will any two-factor interaction be confounded with blocking effects? If so, which one? Yes, ABC × ABD = CD . 3

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Problem 4 (40 pts) In order to maximize the yield of a certain process, an engineer studied the impact of five factors A , B , C , D and E on yield using the following experimental design (with four replicates per trial). A B C D E − − − − − + − − + + − + − + − + + − − + − − + + + + − + − − − + + − + + + + + − (a) Write down a set of generators for this design. D = ABC , E = − AC . (b) What is the resolution of this design? III (c) Which effects are clear? None (d) How many points shall figure in each half-normal plot (one for location and one for disper- sion) ? Which factorial effects would they represent ? Clearly explain your answer. Seven. Five of them would correspond to the main effects of A , B , C , D , E . Since studying the aliasing structure it is found that A = − CE , B = − DE , C = − AE , D = − BE , E = − AC = − BD , we only have AB , CD , AD , BC left. Further, AB = CD and AD = BC . Thus, one point would represent either AB or CD and the other either AD or BC . Read the following additional information carefully before answering questions (e) to (h) . The following table gives the t PSE values for location (¯ y ) and dispersion (ln s 2 ) corresponding to different factorial effects. The values are arranged in descending order. The PSE values for location and dispersion are 0.132 and 0.079, respectively. The experimenter decides that the effects that are declared significant at level 0.05 by Lenth’s test will be considered for optimization of the process. Because the experimenter considers missing an important factor much more serious than misidentifying an inert factor, he/she decides to use IER critical values for the Lenth’s test. 4

Location Dispersion Effect t PSE Effect t PSE A 7.82 B 5.95 C -4.91 C -1.87 E 2.15 A - 1.24 -1.12 0.78 -0.92 -0.57 0.35 -0.29 0.24 0.08 (e) Based on the above table and all the stated information, which effects would the experi- menter consider for fitting location and dispersion models? (Hint: With I = 7 ,IER 0 . 05 = 2 . 30 , IER 0 . 025 = 3 . 43 ) Since IER 0 . 05 with I = 7 is 2.30, A and C are significant for location, and B is significant for dispersion. (f) Based on the information above, is it possible to fit the complete regression models linking ¯ y and ln ( s 2 ) to the significant factor effects? If not, which term(s) would be missing? No. Intercept terms will be missing. (g) Write down the regression models (as explicitly as possible) for ¯ y and ln s 2 in terms of the significant factor effects. The regression coefficient for the i th effect is ˆ θ i / 2, where ˆ θ i is the estimated factorial effect and is obtained by multiplying PSE with t PSE . Thus, ˆ y = constant + 0 . 516 x A − 0 . 324 x C , ln ( s 2 ) = constant + 0 . 235 x B . (h) What settings of the significant factors would you recommend ? To maximize average yield, choose A = + , C = − . To minimize dispersion, choose B = − . 5