# STAT/ME 424 Homework 6 solution

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1. An aluminum master alloy manufacturer produces grain refiners in ingot form.
The company produces the product in four furnaces. Each furnace is known
to have its own unique operating characteristics, so any experiment run in the
foundry that involves more than one furnace will consider furnaces as a block
variable.

The process engineers suspect that stirring rate impacts the grain
size of the product. Each furnace can be run at four different stirring rates. A
randomized block design is run for a particular refiner and the resulting grain
size data is shown in Table 1.

Table 1: Stirring rate data
Stirring Furnace
rate 1 2 3 4
5 8 4 5 6
10 14 5 6 9
15 14 6 9 2
20 17 9 3 6

(a) Is there any evidence that stirring rate impacts grain size? Test the
appropriate hypothesis at level α = 0.05.

(b) Make a normal quantile plot of the residuals from the experiment. Interpret the plot.

(c) Plot the residuals versus furnace number and versus stirring rate. Do the
plots convey any useful information?

(d) State the model you use to answer the above questions and carry out a
test to check its validity.

(e) What should the process engineers recommend concerning the choice of
stirring rate and furnace for this grain refiner if small grain size is desirable?

(f) Estimate the pairwise differences between mean values of the stirring
rates with 95% simultaneous Tukey confidence intervals.

2. Aluminum is produced by combining alumina with other ingredients in a reaction cell and applying heat by passing electric current through the cell. Alumina is added continuously to the cell to maintain the proper ratio of alumina
to other ingredients. Four different ratio control algorithms were investigated
in an experiment. The response variables studied were related to cell voltage.

Specifically, a sensor scans the cell voltage several times each second, producing thousands of voltage measurements during each run of the experiment.

The process engineers decided to use the average voltage and the standard
deviation of cell voltage over the run as the response variables. The average
voltage is important because it impacts cell temperature, and the standard
deviation of voltage (called “pot noise”) is important because it impacts the
overall cell efficiency.

The experiment was conducted as a randomized block design, where six time
periods were selected as the blocks, and all four ratio control algorithms were
tested in each time period. The average cell voltage and the standard deviation
of voltage (shown in parentheses) for each cell are given in Table 2.

Table 2: Voltage data
Control Time period
Algorithm 1 2 3 4 5 6

1 4.93 (0.05) 4.86 (0.04) 4.75 (0.05) 4.95 ( 0.06) 4.79 ( 0.03) 4.88 (0.05)
2 4.85 (0.04) 4.91 (0.02) 4.79 (0.03) 4.85 (0.05) 4.75 (0.03) 4.85 (0.02)
3 4.83 (0.09) 4.88 (0.13) 4.90 (0.11) 4.75 (0.15) 4.82 (0.08) 4.90 (0.12)
4 4.89 (0.03) 4.77 (0.04) 4.94 (0.05) 4.86 (0.05) 4.79 (0.03) 4.76 (0.02)

(a) Analyze the average cell voltage data. (Use α = 0.05.) Does the choice
of ratio control algorithm affect the average cell voltage?

(b) Perform an appropriate analysis on the standard deviation of voltage.
Does the choice of ratio control algorithm affect the pot noise?

(c) Conduct any residual analyses you deem appropriate.

(d) Which ratio control algorithm would you select if your objective is to
reduce both the average cell voltage and the pot noise?