We are going to simulate an experiment where we are trying to see whether any of the four automated systems (labeled A, B, C, and D) that we use to produce our root beer result in a different specific gravity than any of the other systems. For this example, we would like the specific gravity of our root beer to be 1.025. We have found in taste tests that people will notice a difference if the specific gravity is different by more than 0.0015. From historical process control data, we believe that all of the systems have equal variances of 0.00062 for the specific gravity of the root beer they produce.
1. Identify the following:
a. The factor and its levels
b. The treatments
c. Any requirements on taking observations to ensure independence
2. Compute the number of observations per system you need to take for this experiment.
3. Randomly generate the number of observations you computed in #1 for each system in Minitab or whatever software package you are using. Store them in four columns labeled A - D. Use the following distributions for each system: A = N(1.025,0.0006^{2}), B = N(1.026,0.0006^{2}), C = N(1.0235,0.0006^{2}), and D = N(1.0240, 0.0006^{2}).
4. Conduct an ANOVA, generating a boxplot and a threeYinYone graph of the residuals. Is there any indication in the three in-one plot that the assumptions of the ANOVA have been violated? Are any differences suggested by the boxplot?
5. Given your simulated data, are there statistically significant differences between the four systems in terms of their ability to produce root beer that tastes the same to consumers?
6. Regardless of whether differences were found in #3, perform simultaneous comparisons using the Tukey procedure. If differences were found in #3, identify which systems are different than which other systems. If no differences were found in #3, in which case you would not normally conduct Tukey tests, do the Tukey tests support or not support the conclusion from #3? If it differs, which do you trust?
7. Now overwrite column D with a new set of random observations from N(1.024, 0.0018^{2}).
a. Repeat step 3 and indicate whether any assumptions of the ANOVA appear to have been violated. (Hint: There should be one!)
b. Even if assumptions have been violated, check the results of the ANOVA. Do they agree or disagree with your previous results? Given what was done to generate the new data, what does the similarity or dissimilarity of the results tell you about the effect of the violation?
8. Suppose that systems A and B are located in one factory, and systems C and D are located in another factory. If you do not care whether there are differences in specific gravity by factory, only by system, how might you separate the effect of factory from the effect due to system?