Assignment

Note: All answers to be completed in SAS. All SAS input and output must pasted in to the solutions.

Other Statistical software, such as Minitab, JMP, SPSS, R, and MATLAB, are also allowed to use.

1. Montgomery Problems 15-15. Make sure to check assumptions and perform pairwise comparisons of adjusted treatment means if appropriate.
   15.15. Four different formulations of an industrial glue are being tested. The tensile strength of the glue when it is applied to join parts is also related to the application thickness. Five observations on strength \((y)\) in pounds and thickness \((x)\) in $0.01$ inches are obtained for each formulation. The data are shown in the following table. Analyze these data and draw appropriate conclusions.
   
2. Suppose the study in the previous exercise was a pilot study and its results are to be used to determine the appropriate design (CRD or CRD with covariate) for a larger study involving six glue formulations.

1. Reanalyze the data but this time do not use the thickness covariate. State the MSE from this analysis.
2. Your answer to a) estimates the error variance under a CRD. Use this value to estimate the power for the overall treatment \(F\) test in the larger study when \(D=2\) hundredths of an inch and \(n=15\) samples per glue type.
3. Now use your MSE from your analysis in Exercise 1 for the same power calculation. This is an approximate power calculation for the ANCOVA model because it does not adjust the error degrees of freedom for the covariate and assumes \(\bar{x}_{i} .=\bar{x}_{. .}\) so the standard errors of the means are constant.
4. Which model do you recommend based on power? Does this decision change if we say that running an ANCOVA costs twice as much because there are two measurements per experimental unit? Explain your answer.

3. For the vascular graft experiment (Table 4-3, page 145 ), change the data in the following manner. To the data for batch 1 , subtract .5 (e.g. \(90.3 \rightarrow 89.8\) ), and to the data for batch 2 through 6 , add 0.2,-0.9,-1.3,0.0, and 0.5 respectively. What happens to the analysis of variance table (original ANOVA shown on page 145)? In terms of the treatment \(F\) test, does altering the data in this way affect the \(\mathrm{SS}_{\mathrm{Tr}}\), or \(\mathrm{SS}_{\mathrm{E}}\), or both? Explain why is does or does not affect each of these terms?

4. Montgomery Problem 4-13.

4.13. Consider the ratio control algorithm experiment described in Section 3.8. The experiment was actually conducted as a randomized block design, where six time periods were selected as the blocks, and all four ratio control algorithms were tested in each time period. The average cell voltage and the standard deviation of voltage (shown in parentheses) for each cell are as follows:

1. Analyze the average cell voltage data. (Use \(\alpha=0.05\).) Does the choice of ratio control algorithm affect the average cell voltage?
2. Perform an appropriate analysis on the standard deviation of voltage. (Recall that this is called "pot noise.") Does the choice of ratio control algorithm affect the pot noise?
3. Conduct any residual analyses that seem appropriate.
4. Which ratio control algorithm would you select if your objective is to reduce both the average cell voltage and the pot noise?

5. Montgomery Problem 15-21. Although this is a repeated measures design, we can analyze it as if it were a RCBD treating judge as a random effect. Make sure to check assumptions and perform pairwise comparisons of treatments if appropriate.

15.21. Three different Pinot Noir wines were evaluated by a panel of eight judges. The judges are considered a random panel of all possible judges. The wines are evaluated on a 100-point scale. The wines were presented in random order to each judge, and the following results obtained.

Analyze the data from this experiment. Is there a difference in wine quality? Analyze the residuals and comment on model adequacy.

6. A clay tile company is interested in studying the effects of cooling temperature on strength. Since the company has five ovens which produce the tiles, four tiles were baked in each oven and then randomly assigned to one of the four cooling temperatures \(\left({ }^{\circ} \mathrm{C}\right)\). The data are:

1. Compute the appropriate F-statistic to determine if there is a difference among the four cooling temperatures (use \(\alpha=.05\) ). Perform pairwise comparisons using Tukey's procedure if appropriate. What are your conclusions?
2. Suppose the company believes there is a jump in the strength at \(10.0^{\circ} \mathrm{C}\) but otherwise cooling temperature has no effect (i.e., step function ___- ). Find a set of orthogonal contrasts that would allow you to assess this (HINT: What is the relationship among the means before and after \(10.0^{\circ} \mathrm{C}\).
3. Test these contrasts using SAS (or by hand). State your conclusions.

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Solution: The downloadable solution consists of 18 pages, 1354 words and 27 charts.
Deliverable: Word Document