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Midterm2_2011 - Stat/ME 424 Spring 2011 Solutions to...

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Stat/ME 424, Spring 2011 Solutions to Problems for Midterm II (Total : 100 points) This is a closed-book exam, except for several sides of notes. You are allowed to use calculators. Problem 1 (20 pts) Consider a two-way layout with the model y i j = η + α i + β j + ε i j , and i = 1 , 2 and j = 1 , 2 , 3, where var ( ε i j ) = σ 2 , α i and β j represent the main effects of two qualitative factors. Suppose it is determined that α 2 = 0 and β 2 = 0 be chosen as the baseline constraints. Find estimates of α 1 , β 1 , and β 3 in terms of the y i j values. You may be able to define these estimates in different ways. Only give the best solution you can come up with. The model we use is: y i j = η + α i + β j + ε i j . E ( y 11 ) = η + α 1 + β 1 , E ( y 12 ) = η + α 1 , E ( y 13 ) = η + α 1 + β 3 , E ( y 21 ) = η + β 1 , E ( y 22 ) = η , E ( y 23 ) = η + β 3 , Therefore: ˆ α 1 = y 11 y 21 + y 12 y 22 + y 13 y 23 3 ˆ β 1 = y 11 y 12 + y 21 y 22 2 ˆ β 3 = y 13 y 12 + y 23 y 22 2 1
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Problem 2 (30 pts) An experiment is being performed to determine the optimum temperature and conveyor speed that will yield a desired level of hardness in copper after annealing (a heat treatment operation). Three levels of temperature viz. 1000, 1050 and 1100 degree F and three levels of conveyor motor rpm viz. 1400, 1600 and 1800 are chosen for the experiment. The experimenter has only 9 test pieces or experimental units, three each from three suppliers A, B and C. It is suspected that there may be a supplier-to-supplier variation with respect to the hardness of raw material. Further, three furnaces F1, F2 and F3 are available for the experiment, and the department head wants all of them to be used for conducting the experiment so that the total time required is less. The experimenter feels
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