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

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Jan 9, 2024

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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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