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**Q1:**1. In a completely randomized design, there are two factors, A with two levels and B with three levels. Suppose the 6 treatment means are 116, 12 = 10,13 = 8 H21 = 5, 22 = 5,/23 = 5. Note these are the true treatment means and are supposed to be known. Answer the following questions. a. [3pts] Are there interaction effects? Why? [Hint: Use the definition of interations.] b. [3pts] Find μ, ai, Bj and (aß)ij, i = 1, 2, j = 1, 2, 3, such that Hij = μl + ai + Bj + (aß)ij.See Answer**Q2:**2. Consider the reaction time experiment described in Exercise 4 of Chapter 4. a. [3pts] Write down the two-way complete model for the experiment. Remember to explain each term in the model. b. [3pts] Find the sums of squares that are accounted for by the factors and their interactions, i.e., ssA, ssB and ssAB. c.[5pts] Generate an interaction plot. Do you see an obvious interaction between the two factors? Carry out a formal hypothesis testing for the interaction. d. [3pts] Test the hypothesis that different elapsed times have the same effects on the reaction time. e.[3pts] Find a 95% confidence interval for the difference between the average reaction time from the auditory cue and the average reaction time from the visual cue [Hint: this is to compare the two main effects of factor cue]. f.[3pts] Find an appropriate confidence interval for the difference between auditory cue and the visual cue, when the elapse time is 5 seconds.See Answer**Q3:**1. Considering the two-way main effects model with two factors: Yijt = μ+αi+Bj+€ijt, i = 1, 2, 3; j = 1,2, 3, t = 1,2,,r. Anwser True or False to the following statements. (a) [1pt] μ+0₁+ B₂ is estimable. (b) [1pt] μ+ a₁ + (31 +3₂) is estimable. (c)[1pt] 31 (3₂ +33) is estimable. (d) [1pt] B₁ (32 +33)/2 is estimable. () CCCC () () ()See Answer**Q4:**2. An experiment is to be run with two factors: Factor A with 2 levels and factor B with 4 levels. The experimenter would like to examine the pairwise differences between the four levels of factor B, with a simultaneous confidence level of 90%. The experimenter is confident that the two factors do not interact and will employ a two-way main-effects model. Furthermore, the experimenter believes that the mse will be unlikely to exceed 25. Find the required sample size for the 90% simultaneous confidence intervals for the pairwise comparison of main effects of B to have a width at most 10. (a) [2pts, no partial credit]The required sample size is at least (b) [2pts] Provide the SAS code.See Answer**Q5:**3. The coating experiment, described in Excercise 7 of Chapter 7, was to study the effect of different spray parameters on thermal spray coating properties. In the experiment, the authors attempted to produce high-quality alumina (Al2O3) coatings by controlling the fuel ratio (factor A at 1:2.8 and 1:2.0), carrier gas flow rate (factor B at 1.33 and 3.21 L s-¹), frequency of detonations (factor C at 2 and 4 Hz), and spray distance (factor D at 180 and 220 mm). To quantify the quality of the coating, the researchers measured multiple response variables. In this example we will examine the porosity (vol. %). The data are shown in the table below and can be downloaded from http: //deanvossdraguljic.ietsandbox.net/DeanVossDraguljic/SAS-data.html. A B с D Yijkl A B C D 2 2 2 2 5.95 2 2 2 1 2 2 1 2 1 4.57 4.03 2.17 1 2 2 2 1 2 2 2 2 1 1 1 2 1 2 1 1 1 1 1 2 1 1 1 1 2 2 1 2 1 2 1 2 Yijkl 12.28 9.57 6.73 6.07 8.49 4.92 6.95 5.31 1 2 2 3.43 2 1 2 1 1.02 2 1 1 2 4.25 2 1 1 1 2.13 1 1 1 1 (a). [3pts]. Run a model with ALL main-effects and two-way interaction effects. Write down the SAS code copy the ANOVA table from SAS. and (b).[2pts] The 95% confidence interval for the difference between the two main effects of B is ( (c).[2pts] Do you believe there are significant interaction effects between B and C? Answer yes or no. (d).[3pts] Generate an interaction plot for the B*C interaction. State how the plot supports your answer in c. [Hint: use 1smeans B*C to get the least squares estimates, then plot it using any software of your choice.] (e).[3pts] The 90% confidence upper limit for the error variance o² is Show your work.See Answer

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