1. IE341 Midterm

2. 1. The effects of a 2 x 2 fixed effects factorial design are: A effect = 20 B effect = 10 AB effect = 16 = 35 (a) Write the fitted regression model for this design. Y = 35 + 10(X 1 ) + 5(X 2 ) + 8(X 1 X 2 ) Plot the interaction effect. You need to get all 4 corners of the design to do this. -- Y = 35 + 10(-1) + 5(-1) + 8(-1)(-1) = 28 +- Y = 35 + 10(+1) + 5(-1) + 8(+1)(-1) = 32 -+ Y = 35 + 10(-1) + 5(+1) + 8(-1)(+1) = 22 ++ Y = 35 + 10(+1) + 5(+1) + 8(+1)(+1) = 58

3. 1. (continued)

4. 2. The mean results for a completely balanced experiment are With these cells means, compute orthogonal contrasts and their SS for: (a) High Temperature vs the average of Low and Medium Temperature. (b) Low Temperature vs Medium Temperature (c) High Temperature vs Medium temperature

5. 2. Before you can compute the contrasts, you need the temp means. Low Medium High Mean 7 8 12 (a) contrast -1 -1 2 = 9n SS = n 2 9 2 / n(6) = 81n/6 (b) contrast -1 1 0 = 1n SS = n 2 1 2 / n(2) = n/2 (c) This contrast cannot be done because there are only 2 df in the temperature factor and the first two contrasts have used these 2 df. Note: Because I got the corrected version online so late, I didn’t deduct for errors in this problem having to do with multiplying by n.

6. 3. The E(MS) for the effects in an experiment are: (a) What type of design is this? This is a random effects 2-factor factorial. (b) Set up all the F-tests. A effect: F = MSA / MSAB B effect: F = MSB / MSAB AB effect: F = MSAB / MSE

7. 4. Because of limited resources, an experiment with 4 factors at 2 levels each is placed in an 8-run design. (a) What kind of design is it? 2 4-1 fractional factorial (b) What is its generator? D = ABC (c) What is the design resolution? Because I = ABCD, design resolution = IV (d) List all confounded effects. A + BCD AB + CD B + ACD AC + BD C + ABD AD + BC D + ABC ( e) What is the defining relation of the complementary fraction? I = -ABCD

8. 5. Fabric strength produced by 4 types of loom is being studied. The data are:

9. 5. (a) Do the ANOVA. Mean 1= 30 Mean 2= 30 Mean 3= 35 Mean 4= 25 Grand = 30 SS between = 4(0 + 0 + 25 + 25) = 200 MS between = 200 / 3 SS within = 50 + 50 + 50 + 50 = 200 MS within = 200 / 12 F = 200/3 * 12/200 = 4 p = 0.03459 Source SS df MS p Loom type 200 3 200/3 0.03459 Error 200 12 200/12 Total 400 15

10. 5. (b) Plot the effect. (c) For greatest fabric strength, which type of loom would you choose? Loom Type 3

11. 6. A textile mill has a problem. The strength of the cloth produced has too much variability. They wonder if it’s due mostly to the looms or to the operators, so they decide to study it. From the large number of looms they have, 3 looms are chosen randomly. Also 3 operators are chosen at random from all the operators in the plant. 3 replicates are run for each combination of loom and operator. The ANOVA table is Source SS df MS p Looms 94 2 47 0.089 Operators 220 2 110 0.007 AB 80 4 20 0.354 Error 306 18 17 Total 700 26

12. 6. (a) What type of ANOVA is this? A random effects two-way ANOVA (b) What proportion of variance is due to looms, to operators, to interaction, and to error? Looms: Operators: AB: Error: σ 2 = MSE = 17

13. 6. (continued) Total: A + B + AB + E 3 + 10 + 1 + 17 = 31 proportion due to Looms = 3/ 31 proportion due to Operators = 10/ 31 proportion due to interaction = 1 / 31 proportion due to error = 17 / 31

14. 7. A factory produces grain refiners in 3 different furnaces, each of which has its own unique operating characteristics. Each furnace can be run at 3 different stirring rates. The process engineer knows that stirring rate affects the grain size of the product, so he decides to run an experiment testing the three stirring rates on his 3 furnaces. (a) What type of design is this? A randomized blocks design (b) Why is it designed this way? So that differences between the three furnaces do not affect differences between stirring rates.

15. 7 (c) Set up the experiment for the process engineer. Furnace 1Furnace 2Furnace 3 Stir rate 1 Stir rate 2 Stir rate 3

16. 8. Four factors are to be used in a manufacturing process for integrated circuits to improve yield. A is aperture setting (small, large), B is exposure time (20 sec, 30 sec), C is development time (30 sec, 45 sec), D is mask dimension (small, large). You are the statistical consultant for the firm and you are asked to design the experiment. You’d better do it or the boss will be angry and you know what that means.

17. 8. RunAperture setting Exposure time Develop time Mask dimension 1Small2030Small 2Large2030Small 3 30 Small 4Large30 Small 5 2045Small 6Large2045Small 7 3045Small 8Large3045Small 9 2030Large 10Large2030Large 11Small30 Large 12Large30 Large 13Small2045Large 14Large2045Large 15Small3045Large 16Large3045Large

18. 8. What kind of design did you create? A 2 4 factorial design with all fixed effects Note: Many of you did a 2 4-1 fractional factorial. Since there is no restriction on the number of runs, it is not acceptable to design an experiment that is troublesome when you can design the full factorial and have no confounding at all. However, if you did a fractional correctly and identified it as such, I took off only 5 instead of 10 points.

19. 9. The effect of 5 different ingredients on reaction time is being studied. Each batch of material is large enough for only 5 runs. Moreover, only 5 runs can be made in a day. Design the experiment. This is a Latin Square design.

20. 9. (continued) A, B, C, D, E are the five ingredients under test.

21. 10. What is the difference between a fixed effects model, a random effects model, and a mixed model? A fixed effects model is one in which we are testing the effects of treatments in terms of their means, and the conclusions apply only to those treatments under test. A random effects model is one in which all factors are random factors, that is, there are many levels of each factor, but only J of them are randomly selected for use in the experiment. In a random effects model, we are interested in estimating the variance components, that is, the portion of the total variance due to each of the factors. A mixed model is one is which at least one of the factors is random.

22. 11. What is the logic behind decomposition of the total SS in ANOVA? The logic is that the total SS includes variability due to one or more factors and variability due to error. We can separate these and test the effects of the factors.

23. 12. What are two ways of dealing with nuisance variables in the designs we have studied? We can deal with nuisance variables by blocking, by a Latin Square, or by a Graeco-Latin Square. Why do they work? They all work because all levels of the factors of interest are tested at each combination of levels of the nuisance factors.

24. 13. Why is a single-replicate design undesirable? A single replicate design does not allow for a pure error term to test effects of factors. Instead we must use higher-order interactions as error under the assumption that they are not significant. What is Cuthbert Daniel’s idea and how does it help in this situation? Daniel’s approach allows us to look at all effects by plotting them on normal probability paper. If the effects fall on a line, they behave like error and are not significant. This gives us a way of looking at all effects and choosing those that fall off the line to include in the model. All effects along the line are used as residual.

25. 14. How do you check ANOVA model adequacy? Checking model adequacy involves residual plots. Normality is checked by plotting residuals on normal probability paper and making sure they all fall along a line. Constant error variance is checked by plotting residuals against fitted values and making sure they all have about the same spread with no outliers.

26. 15. What do you mean by a completely randomized design? A completely randomized design is one in which all sources of bias are removed by (1) randomly assigning the experimental units to the various treatment levels and (2) doing the experimental runs in random order.

27. Grade distribution 95-100%9A+ 90-94%14A0 85-89%14A- 80-84%9B+ 75-79%3B0 70-74%1B- 65-69%2C+ 60-64%1C0 55-59%1C- <40%1D-

28. Question summary