Statistics Assignment deadline soon

Final Exam: STAT4504 Statistical Design and Analysis of Experiments

Due: 4:00 pm Thursday, April 28, 2022 The deadline for the final exam is set as per Carleton University’s take home exam policy. Instructions for submission: Convert your document to a PDF file. Upload the generated pdf via Gradescope under Final Exam. All programming needs to be completed with SAS or R as stated in the course outline. The assignment needs to be uploaded in Gradescope. Make sure when you upload, you specify which page corresponds to which question. The datasets for the final exam are available in an excel file “Data_Final.xlsx”. The dataset for each question is given in a different sheet so make sure you use the appropriate sheet for appropriate analysis. For each question on data analysis, you must

a. Select an appropriate statistical model to analyze the dataset based on the information provided in the question and provide justification for your choice of model.

b. Provide the appropriate ANOVA table with two additional columns that shows i. The expected values of the appropriate mean square terms under the model of your choice.

ii. The expected values of your test statistics as the ratio of the appropriate mean square terms based on your model.

c. Clearly state the null and alternate hypothesis, distribution of the test statistics under the null hypothesis, appropriate F-value or p-value that you are using to make the decision and your conclusion. Note: your null hypothesis is different for fixed effects and random effects.

d. Perform appropriate residual analysis and comment on the model adequacy. e. If model assumptions are not valid, suggest an appropriate transformation and provide justification

for the transformation. Refit the model with the appropriate transformation and state your conclusion including comments on the model adequacy. Make sure to state the appropriate null and alternate hypotheses.

Note: If you choose to use Box-Cox transformation in SAS using the proc transreg function, use the class() to specify dependent categorical variables and for continuous variables, use the identity() to specify the dependent continuous variables.

1. (15 points) An article in the Fire Safety Journal (“The Effect of Nozzle Design on the Stability and

Performance of Turbulent Water Jets,” Vol. 4, August 1981) describes an experiment in which a shape factor was determined for several different nozzle designs at six levels of jet efflux velocity. Interest focused on potential differences between nozzle designs, with velocity considered as a nuisance variable. The data are shown below:

Does the nozzle design influence the shape or the transformed shape (if transformation was needed)?

2. (20 points) To simplify production scheduling, an industrial engineer is studying the possibility of assigning one time standard to a particular class of jobs, believing that differences between jobs is negligible. To see if this simplification is possible, six jobs are randomly selected. Each job is given to the same three randomly selected operators. Each operator completes the job twice at different times during the week, and the following results are obtained. Use an appropriate model to test whether the engineer’s belief is justifiable.

3. (20 points) The yield of a chemical process is being studied. The two factors of interest are temperature and pressure. Three levels of each factor are selected; however, only nine runs can be made in one day. The experimenter runs a complete replicate of the design on each day. The data are shown in the following table:

Identify which factors affect the yield or the transformed yield (if transformation was needed).

4. (20 points) A structural engineer is studying the strength of aluminum alloy purchased from three vendors. Each vendor submits the alloy in standard-sized bars of 1.0, 1.5, or 2.0 inches. The processing of different sizes of bar stock from a common ingot involves different forging techniques, and so this factor may be important. Furthermore, the bar stock is forged from ingots made in different heat and thus, three different heat categories were selected randomly. Each vendor submits two tests specimens for each treatment combination categories. The resulting strength data is shown in the table below. Analyze the data and what combination of various treatments would you recommend if the engineer is interested in having aluminum alloy with highest strength.

5. (30 points) An engineer has performed an experiment to study the effect of four factors on the surface

roughness of a machined part. The factors (and their levels) are A = tool angle (12 degrees, 15 degrees), B

= cutting fluid viscosity (300, 400), C = feed rate (10 in/min, 15 in/min), and D = cutting fluid cooler used (no, yes). The data from this experiment (with the factors coded to the usual -1, +1 levels) are shown below.

a) Using the data provided, create a one-half fractional design such that the resulting design is of the

highest resolution. Estimate the model effects and select a tentative model and check for model adequacy. Perform any transformation if needed.

b) Drop any factor that does not seem to be important and analyze the data as a full factorial model with only significant factors from (a). Compare your results with those obtained in part (a). Note: For this question, only provide the additional two columns in the ANOVA table for the final model selected in (b). Provide a fitted regression model when the factors are coded variables.

6. (20 points) A process engineer is testing the yield of a product manufactured on three machines. Each

machine can be operated at two power settings. Furthermore, a machine has three stations on which the product is formed. An experiment is conducted in which each machine is tested at both power settings, and three observations on yield are taken from each station. The runs are made in random order, and the results are shown in Table below.

Identify which factors affect the yield or the transformed yield (if transformation was needed).

7. (15 points) In a completely randomized design (i.e., with one factor), show that 𝐸(𝑀𝑆!) = 𝜎"