Your company operates a machine shop, and, having heard you had experience in statistics and design of experiments, consulted you for your opinion on an experiment they want to run. The machine shop has three (3) machines (A, B, or C) they use in precision grinding of cam rollers. Three (3) machinists (machinists X, Y, or Z) are employed to perform the grinding of the rollers on the machines. The company wanted to know whether there were differences in the machines or the operators with respect to the daily output of parts that met specifications. (Typically, it is expected that an operator could produce somewhere around 25 per day.) Any operator can work any of the machines on any weekday, but the operator must operate the machine the whole day.

1. Justify why an ANOVA is likely to be an appropriate analysis methodology for this experiment.

2. Identify an appropriate model for the experiment assuming you will be using ANOVA to analyze the data.

3. State, in words, the research hypotheses being tested. Also state, in symbols, the statistical hypotheses being tested.

4. Suppose we would replace a machine or re-train a worker if we identified a difference of 3 cam rollers per day. How many observations of each treatment would we have to take to have a power of at least 0.8 with α = 0.05?

5. We are going to use Minitab to design an experiment based on the above assumptions, including your answer to 1.d. We will do this in a way that ensures that any sources of variance other than your factors will show up as random error.

a. Design a "General Full Factorial" experiment.

b. The design should have the factors listed above with the appropriate number of levels. Name the factors and levels.

c. Choose the number of replications.

d. Do not randomize

e. Generate the unrandomized design.

f. To complete the design, randomly assign each person to a different machine each day. (This is not trivial - you'll have to figure out how to do it! There are many ways to accomplish this, but one way uses the"sample from columns" option under Calc→Random Data.) That is, if person 1 is using machine A on a given day, person 2 can only be using either machine B or C that day, with person 3 using whatever machine person 2 is not using; each day a random combination is chosen, but each combination should be used the same number of times.  Your Minitab spreadsheet can then be sorted by day to look something like this (only the first four days are shown):  

(You can delete the "StdOrder," "RunOrder," "PrType," and "Blocks" columns - they are not useful for this example.) The Minitab spreadsheet now should show a possible, random experiment that could be run to collect the data.  In the above example, on day one machinist X is assigned to Machine A, machinist Y is assigned to Machine B, and so on.  There should be 10 replications of each treatment combination.  

6.  Now we are going to simulate taking data for the experiment.  To do so, title several new columns as follows: "Productivity"; "RandomError"; "RandomErrorInt";

"MachineAEffect"; "MachineAEffectInt"; "MachinistZEffect"; "MachinistZEffectInt."  

a. Generate the appropriate number (one for each row of the experiment) of random numbers for the "Random Error" column from N(0,4).  Using the calculator, set a new column called "Random Error Int" equal to ROUND('RandomError',0).

b. Generate the same number of random numbers for the "MachineAEffect" column from N(-6,4).  Set "MachineAEffectInt" to ROUND('MachineAEffect',0).

c. Generate the same number of random numbers for the "MachinistZEffect" column from N(-4,4).  Set "MachinistZEffectInt" to ROUND('MachinistZEffect',0).  

d. Then, again using the calculator, set "Productivity" equal to  the following formula (you can cut and paste this formula):

25+IF('Machine'="A",'MachineAEffectInt',0)+IF('Machinist'="Z",'MachinistZEffectInt',0)+'RandomErrorInteger'. 

(This formula penalizes operators on machine A by an average of 5 units and machinist Z by an average of 3 units.)

e. Run a "Two-way ANOVA" and report on the results.  Your report should state whether the assumptions of ANOVA appear to have been supported by the residual plots, include a boxplot of the data, and state whether machine and/or machinist or their interaction is significant.  Include the output of the ANOVA and the plots.  If necessary, conduct tests on the normality of the residuals and for the constancy of variance.  

f.  Run a "Balanced Design" ANOVA, including Machinist as a random factor and without including any interaction effect in the model.  (You do not need to print out any plots for this step.)  Report on the results - are they different than 6.f?  (Hint: Check both the p-values and f-statistics.) What is different about the interpretation of these results?

g.  Run a "General Linear Model" ANOVA, using the same model as in 6.f.; also have Minitab do simultaneous comparisons using a Tukey procedure for the levels of Machine.  Have Minitab print a Main Effects Plot for "Machine." 

Which machines appear to be different from which other machines according to the Main Effects Plot?  Is your interpretation supported by the Tukey comparisons?

h.  Note that what you see is an example of a case where there are main effects but no interaction effect.  (It may be that your particular results do not agree with this - that's OK; what you'd be seeing in this case is an example of how, randomly, the sample may not give you good information about the population. )  Also, we know, whether or not your residual plots show this, that the assumptions of the ANOVA have been met.

7.  Now we will see an example of a pure interaction effect.

a. Populate a column titled "MachinistBMachineBEffect" with the appropriate number of observations from N(-7,4).

b. Set "Productivity" equal to

25+If(('Machine'="B" And 'Machinist'="Y"),'MachinistBMachineBEffectInt',0)+'RandomErrorInteger'.

c. Run a General Linear Model including both main factors and the interaction.  Have Minitab print the four-in-one graph, a main effects plot for both main effects, and an interaction plot.

d. Examine the residual plots and indicate whether any assumptions appear to have been violated.  Run tests if necessary to confirm any suspicions you have about the assumptions being met.

e. Interpret the output of the ANOVA using the ANOVA table and the main/interaction effects plot as appropriate.

f. The results should, but may not due to random chance, indicate an interaction effect without separate main effects. Also, the residual plots should, but may not, indicate that the assumptions have been met.

 8.  Next we will investigate the effect of a violation - this time that the residuals are normally distributed.

a. Overwrite the "RandomError" column to be equal to a uniform distribution between -7 and 7 with the appropriate number of rows.

b. Set productivity again equal to

25+IF('Machine'="A",'MachineAEffectInt',0)+IF('Machinist'="Z",'MachinistZ EffectInt',0)+'RandomErrorInteger'.

c.  Rerun the two-factor ANOVA, making sure Minitab prints out the three-in-one residual plot. Store the residuals.

d.  Examine the residual plots - does the violation appear on the plots?  Explain.  

e.  Have Minitab generate a normal probability plot of the residuals and comment on the results of the test.

f.  Examine the Minitab output.  Compare it to that found in part 6.e. and comment on the effect of the violation.