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. An experiment was performed where 7 treatments were randomly and equally assigned to 28 experimental units. The only information you obtain is that the sample variance among the responses in \(s_{y}^{2}=12.3\) and the \(\mathrm{MS}_{E}=7.1\). Construct the ANOVA table and test whether there is at least one treatment effect different from zero at the . 05 level.
2. In HW #2, you performed the overall \(F\) test for the data set described in Montgomery \(3-7\). It should be standard practice to first check the model conditions before drawing conclusions. Let's do that now.

a State each of the underlying model conditions that must be approximately met in order for the overall \(F\) test result to be trustworthy.

b For each condition listed in Part a), assess its validity using a diagnostic test or plot. Make sure you specify what test or plot you are using and why you think the condition is or is not reasonably met based on this plot.

Problem 3.7 listed on next page.

3.7. The tensile strength of Portland cement is being studied. Four different mixing techniques can be used economically. A completely randomized experiment was conducted and the following data were collected:

1. Test the hypothesis that mixing techniques affect the strength of the cement. Use \(\alpha=0.05\).
2. Construct a graphical display as described in Section 3.5.3 to compare the mean tensile strengths for the four mixing techniques. What are your conclusions?
3. Use the Fisher LSD method with \(\alpha=0.05\) to make comparisons between pairs of means.
4. Construct a normal probability plot of the residuals. What conclusion would you draw about the validity of the normality assumption?
5. Plot the residuals versus the predicted tensile strength. Comment on the plot.
6. Prepare a scatter plot of the results to aid the interpretation of the results of this experiment.

3. Explain why it is common practice to first fit the one-factor ANOVA model, obtain the residuals, and use them to check model conditions instead of using the response data \(y\) to check model conditions prior to fitting the model.

4. Montgomery Exercise 3.14. In addition, comment on whether you think this is an observational or experimental study. Explain your answer.

Problem 3.14 listed on next page.

3.14. I belong to a golf club in my neighborhood. I divide the year into three golf seasons: summer (June-September), winter (November-March), and shoulder (October, April, and May). I believe that I play my best golf during the summer (because I have more time and the course isn't crowded) and shoulder (because the course isn't crowded) seasons, and my worst golf is during the winter (because when all of the part-year residents show up, the course is crowded, play is slow, and I get frustrated). Data from the last year are shown in the following table.

1. Do the data indicate that my opinion is correct? Use \(\alpha\) \(=0.05\).
2. Analyze the residuals from this experiment and comment on model adequacy.

5. Montgomery Exercise 3.9

3.9. Reconsider the experiment in Problem 3.7. Find a 95 percent confidence interval on the mean tensile strength of the Portland cement produced by each of the four mixing techniques. Also find a 95 percent confidence interval on the difference in means for techniques 1 and 3 . Does this aid you in interpreting the results of the experiment?

Problem 3.7 listed on next page.

3.7. The tensile strength of Portland cement is being studied. Four different mixing techniques can be used economically. A completely randomized experiment was conducted and the following data were collected:

1. Test the hypothesis that mixing techniques affect the strength of the cement. Use \(\alpha=0.05\).
2. Construct a graphical display as described in Section 3.5.3 to compare the mean tensile strengths for the four mixing techniques. What are your conclusions?
3. Use the Fisher LSD method with $\alpha=0.05$ to make comparisons between pairs of means.
4. Construct a normal probability plot of the residuals. What conclusion would you draw about the validity of the normality assumption?
5. Plot the residuals versus the predicted tensile strength. Comment on the plot.
6. Prepare a scatter plot of the results to aid the interpretation of the results of this experiment.

6. Consider the data set boxcox1.dat located in the data directory.

a Construct a plot similar to the figure shown on Topic 5 Slide 21 (log \(S_{i}\) vs \(\log \bar{y}_{i}\) ) and estimate the appropriate transformation by linear regression.

b Perform the Box-Cox analysis using Proc transreg in SAS or similar procedure in other software. What is recommended as the best transformation? How does this compare you your answer in Part a)?

Topic 5, slide 21 is on next page. (boxcox1.dat) attached with email.

7. An experiment was conducted to study the effect of hormones injected into test rats. There are two distinct hormones \((A, B)\) each at a high and low level. For purposes here, we will consider this to be four different treatments labeled \(\{\mathrm{A}, \mathrm{a}, \mathrm{B}, \mathrm{b}\}\). Each treatment is applied to six rats with the response being the amount of glycogen (in \(\mathrm{mg}\) ) in the liver. Here is the summary table:

Three specific contrasts are of interest. They are:

\(\begin{array}{lcccc}\text { Comparison } & \text { A } & \text { a } & \text { B } & \text { b } \\ \text { Hormone A vs Hormone B } & 1 & 1 & -1 & -1 \\ \text { Low level vs High level } & 1 & -1 & 1 & -1 \\ \text { Equivalence of level effect } & 1 & -1 & -1 & 1\end{array}\)

a What is the \(MS_{E}\) for this study?

b Are the three contrasts orthogonal? Explain why or why not.

c Perform the test for the equivalence of level effect and state what you conclude using the \(\alpha=0.05\) significance level.

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