STA 404 Final Exam. Mathematics & Economics. Math Problem

STA 404 Name: 
Final Exam (120 pts) Class Number: 
When compiling your answers to the following questions, follow the usual guidelines for homework assignments listed in the syllabus.  A PDF copy of your work is to be submitted to UBLearns by 6:30 PM on Tuesday, 5/12/2020.  Some of the questions will place you in scenarios that we have not explicitly covered in class.  You are expected to use your accumulated knowledge of Minitab and experimental design to guide you.  This is an exam; you are not permitted to collaborate on these questions with anyone else – recall the statements you signed on day 1, committing not to cheat.
1. An experiment is designed to investigate four methods (T1, T2, T3, T4) of treating wood.  Large responses indicate a high degree of sheen, which is desirable for wooden furniture.  The four treatment compounds are applied to pieces of wood cut from the same length of plywood (batch).  After treatment, the lengths of wood must be sheltered from the elements; currently there is only space in the facility to store four treated lengths of wood.  The treatment compounds take a full day to dry completely, meaning the experiment is performed over four consecutive days.  The data appear below. (18 pts)
  Batch 1 Batch 2 Batch 3 Batch 4
Day 1 T4 = 192 T1 = 195 T3 = 292 T2 = 249
Day 2 T1 = 190 T4 = 203 T2 = 218 T3 = 210
Day 3 T3 = 214 T2 = 139 T1 = 245 T4 = 163
Day 4 T2 = 221 T3 = 152 T4 = 204 T1 = 134
Which design should be used to analyze these data, capable of treating both Day and Batch as homogeneous experimental conditions? (2)
Fit the appropriate model in Minitab, and provide the ANOVA table. (4)
Use this output to test whether the type of wood treatment has a significant impact on sheen.  Give the hypotheses, test statistic, p-value, and conclusion in context. (4)
Evaluate the model assumptions using a residual plot and a test for normality. (4)
Ignore Day and Batch, and fit a one-way ANOVA model addressing the same study question as the previous model.  Give the ANOVA table, and explain why the modeling technique from part (b) has an advantage over one-way ANOVA.  Be specific. (4) 2. Suppose you work in a chemistry lab that manufactures penicillin.  Five blends of penicillin were prepared using two mixing methods.  A single replicate of the experiment was performed, measuring penicillin yield as the response variable.  The data appear in the file penicillin.txt (UBLearns). (20 pts)
The data set contains abn=10 total responses.  State the values of a, b, and n. (3)
The lead investigator would like you to fit the full factorial model.  Before doing anything in software, state the statistical model along with any model assumptions. (4)
Attempt to fit the full model in Minitab, containing both main effects and their interaction, using Stat > ANOVA > GLM.  Do not do any model refinement at this point, and give the ANOVA table.  Your boss at the penicillin lab wants to know whether a single mixing method is preferred, and asks you about the main effect test for factor B.  Write a paragraph expressing your response to your boss. (6)
Fit a simpler model than the one stated in part (c) by eliminating one of the terms, and revisit your boss’s question from part (c).  Test (α=0.05) for a significant effect of mixing method, giving the hypotheses, test statistic, p-value, and conclusion in context. (4)
Using the model from part (d), provide a plot of residuals vs fitted values, test the residuals for normality, and comment on model adequacy. (3)
 3. The Umbrella Corporation is attempting to grow as much of a rare, but horrible virus as possible. They run an initial experiment in which there are two primary factors (factor A: incubation time; factor B: culture medium), each at two levels.  The factorial experiment was replicated six times, and the data appear in the table below: (20 pts)
Time Culture Medium
  1 2
6 21 25
23 24
20 29
22 26
28 25
26 27
12 37 31
38 29
35 30
39 34
38 33
36 35
Obtain the total effects by completing the following table. (4)
symbol I A B AB Total
(1) + - +
a + + - - b + - + - ab + + + +
Use Minitab to obtain the main effects of A, B, and the AB interaction, and present a table of output displaying these values.  Be sure that your model uses ±1 coding for the factor levels.  Demonstrate that you know how these values are calculated by manually obtaining the main effect of A. (4)
Provide the ANOVA table, and summarize whether we have significant main effects or interaction. (4)
Evaluate the model assumptions using residual diagnostics. (4)
The board of directors at the Umbrella Corporation is expecting you to show them a contour plot, as opposed to other commonly used plots – you do not want to disappoint them.  Present the contour plot, and suggest advantageous levels of Time and Medium. (4) 
4. A cardboard manufacturer was criticized about the physical strength of their product, and is investigating three of their production settings.  Factor A is the concentration of hardwood in the raw materials, with levels coded as 1, 2, or 3.  Factor B is pressure, with three levels coded 1, 2, and 3.  Factor C is cooking time, with levels coded as 1 or 2.  Three replicates of the full factorial experiment were carried out.  The data appear in the file cardboard_20.txt (UBLearns). (22 pts) 
Write the statistical model for the full factorial experiment.  Provide the range of possible values for all subscripts, and state any model assumptions. (4)
Fit the full model, but do not give the ANOVA table yet.  Instead, present the Pareto plot to determine which effects are considered significant. (2)
Based on the Pareto plot, refine the model while remaining true to the hierarchy principle.  Provide the ANOVA table from the refined model. (4)
Analyze the residuals from the refined model. (4)
Present any appropriate main effect plots and/or interaction plots based on the refined model.  Based on your plot(s), identify which factor levels maximize cardboard strength. (4)
The interaction plot causes one of the plant managers to wonder whether mean cardboard strength at level 2 of pressure and level 2 of time is statistically different from mean cardboard strength at level 3 of pressure and level 1 of time.  Because this comparison was suggested by the interaction plot, perform all tests for pairwise differences among the means of groups determined by pressure and time, and apply a Tukey correction for multiple testing.  Highlight in the output the comparison suggested by the plant manager.  Note: carrying out this analysis may require you to refit your refined model using the earlier method of Stat > ANOVA > GLM. (4)
 5. Consider a factor screening experiment in which we investigate six primary factors, each having two levels, and a single response variable.  The data are located in the file screening_20.txt (UBLearns). (24 pts)
Give the name of the design that accommodates six primary factors, each at two levels.  How many observations would comprise a single full replicate of the experiment? (4)
Ideally we would fit the full factorial model in Minitab, but the firm running the experiment cannot afford the number of observations required for a full replicate.  Instead, we will analyze what is known as the 1⁄2 fraction.  Your first job is to present the Pareto plot of the factor effects, which requires building a Minitab worksheet from inside the DOE menu.  While in the “Create Factorial Design” menu, you may still choose 2-level factorial (default generators), but this time in the Designs menu, choose the 1⁄2 fraction instead of the full factorial design.  Use the resulting worksheet to fit an initial model that includes interactions up to order 3.  Produce the Pareto plot. (4)
Based on the Pareto plot, indicate which of the six main effect terms appear non-significant. (2)
Refine the model by eliminating non-significant terms that do not violate hierarchy.  Give the ANOVA table and Pareto plot for your final model. (6) 
Chapter 8 of the textbook discusses a principle known as “sparsity of effects.”  Write a paragraph describing whether the sparsity of effects principle can be seen in the current screening experiment.  Give details and be precise. (2)
Analyze the residuals from the final model. (4)
While we do not have a full replicate of the experiment, use Minitab’s response optimizer to suggest advantageous levels of all surviving main effects.  Assume we wish to minimize the response variable. (2)
 6. Isatin is a chemical compound used to create clothing dye.  A factorial experiment was performed to examine the effect of four factors (each at two levels) on isatin yield: acid strength, reaction time, acid volume, and temperature.  Due to budget constraints, only a single replicate of the 2^4 experiment could be obtained.  There are safety concerns when working with powerful acids, and workers’ rights law dictates that one lab worker cannot perform all 16 runs.  Two workers are required to complete this single replicate, and the statistical analysis plan states that an analysis should block on worker.  The data appear in the file isatin_yield.txt (UBLearns). (16 pts)
With only one replicate of the experiment, the full factorial model cannot be fit.  Instead, begin by fitting a model with all main effects and interactions up to order 3.  Present the initial Pareto plot. (4)
Refine the model, eliminating non-significant terms in accordance with the hierarchy principle.  Give the Pareto plot and ANOVA table for the refined model. (4)
A careless data entry specialist did not include a column for “worker” in the data set, and thus far we have not blocked on worker, as was planned in advance.  The senior statistician recalls that the blocking variable was generated to coincide with the ABC interaction.  Manually update the “Blocks” column generated by Minitab to confound worker with the ABC interaction.  Update the model from part (b), this time blocking on worker.  Give the updated Pareto plot and ANOVA table. (4)
Create a surface plot involving relevant effects.  Rotate the figure until you achieve a view that clearly shows which levels of the important factors maximize isatin yield.  State which factor levels are preferred. (4)