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STAT/ME 424 HW 1

1. Aluminum pins manufactured for the aviation industry have diameters
that are normally distributed with mean 10 mm and standard deviation
0.02 mm.

Holes are automatically drilled on aluminum plates, with
diameter having a normal distribution with mean η mm and standard
deviation 0.02. Find the value of η such that the probability is 0.01
that a randomly selected pin will not go through a randomly selected
hole (i.e., the hole is too narrow for the pin).

2. A ten-foot cable is made of 50 strands. Suppose that individually,
10-ft strands have breaking strengths with mean 45 lb and standard
deviation 4 lb. Suppose further that the breaking strength of a cable
is the sum of the strengths of the strands that make it up.
(a) Find, stating any assumptions, the mean and standard deviation
of the breaking strengths of such 10-ft cables.

(b) Find the probability that a 10-ft cable of this type will support a
load of 2230 lb. Again, state any assumptions that you make.

 

STAT/ME 424 HW 2

The mean drying time of paint in a certain application is 12 min. A new
additive is tested to see if it reduces the drying time. One hundred specimens
are painted, and the sample mean drying time ¯y recorded.

Assume that
the population standard deviation of drying time is 2 min. Let µ be the
mean drying time for the new paint. The null hypothesis H0 : µ ≥ 12 is
tested against the alternative H1 : µ < 12. Assume that, unknown to the
investigators, the true mean drying time of the new paint is 11.5 min.
1. It is decided to reject H0 if ¯y ≤ 11.7. Find the significance level and
the power of the test.

2. For what values of ¯y should H0 be rejected so that the power of the
test is 0.90? What then will the significance level be?
3. For what values of ¯y should H0 be rejected so that the significance level
of the test is 5%? What then will the power be?
4. What is the smallest sample size needed for a 5% level test to have

STAT/ME 424 HW 3

1. Three brands of batteries are under study. It is suspected that the
mean lives of the three brands are different. Five randomly selected
batteries of each brand are tested with the following results:

Weeks of life
Brand 1 Brand 2 Brand 3
100 76 108
96 80 100
92 75 96
96 84 98
92 82 100

(a) Is there evidence to suggest that the mean lives of these brands
of batteries are different? Test an appropriate hypothesis at level
α = 0.05.
(b) Analyze the residuals from the data and state your conclusions.
(c) Construct a 95 percent confidence interval for the mean life of
battery brand 2.

(d) Construct a 99 percent confidence interval for the difference between the mean lives of brands 2 and 3.
(e) Based on the data, which of the 3 brands has the longest mean
life?
(f) If the manufacturer will replace without charge any battery that
fails in less than 85 weeks, what percentage of batteries would the
company expect to replace for the brand you chose in (e)?

2. Consider testing the equality of the means of two normal populations,
where the variances are unknown but are assumed to be equal. The
appropriate test procedure is the pooled t-test. Show that the square
of the t-statistic is equal to the ANOVA F-statistic.
power at least 0.90?

STAT/ME 424 HW 4

1. Twelve orange pulp silage samples were divided at random into four
groups of three.

One group was left as an untreated control while
the other three groups were treated with formic acid, beet pulp, and
sodium chloride, respectively. The observed moisture content of the
silage samples are shown below.

Sodium chloride Formic acid Beet pulp Control
80.5 89.1 77.8 76.7
79.3 75.7 79.5 77.2
79.0 81.2 77.0 78.6
(a) Obtain an analysis of variance table for the data and test the null
hypothesis that all four treatments yield the same true average
moisture content. Use α = 0.05.

(b) Compute a 99% confidence interval for the difference in response
between the average of the three treatment groups (acid, pulp,
and salt) and the control group.
(c) Compute 95% simultaneous confidence intervals for the differences
in response between each of the three treatment groups versus the
control group. Use the method among Bonferroni, Tukey, and
Scheff´e, that give the shortest intervals.

2. In an experiment with five treatment groups and 25 residual degrees of
freedom, for what numbers of contrasts is the Bonferroni method more
powerful than the Scheff´e method?

STAT/ME 424 HW 5

An economist compiled data on productivity improvements last year for a sample
of firms producing electronic computing equipment.

The firms were classified according to the level of their average expenditures for research and development in the
past three years (low, moderate, high). The results of the study follow (productivity
improvement is measured on a scale from 0 to 100).
i
j 1 2 3 4 5 6 7 8 9 10 11 12

1 Low 7.6 8.2 6.8 5.8 6.9 6.6 6.3 7.7 6.0
2 Moderate 6.7 8.1 9.4 8.6 7.8 7.7 8.9 7.9 8.3 8.7 7.1 8.4
3 High 8.5 9.7 10.1 7.8 9.6 9.5
1. Write down a model appropriate for the data.
2. Let µj denote the true mean productivity improvement for the jth level of
expenditure. Obtain the values for the estimates ˆµj
, j = 1, 2, 3.

3. Obtain the residuals and plot them against the ˆµj
. What are your findings?
4. Make a normal quantile plot of the residuals. Does the normality assumption
appear to be reasonable here?

5. The economist wishes to investigate whether location of the firm’s home office
is related to productivity improvement. The home office locations are as follows
(U=U.S.; E=Europe):
i
j 1 2 3 4 5 6 7 8 9 10 11 12
1 Low U E E E E U U U U
2 Moderate E E E E U U U U U E E E
3 High E U E U U E

Make side-by-side boxplots of residuals by location of home office. Does it
appear that the ANOVA model could be improved by adding location of home
office as a second factor? Explain.
6. Obtain the ANOVA table without using location of home office as a second
factor.
1

7. Test whether or not the mean productivity improvement differs according to
the level of research and development expenditures. Use a significance level of
α = 0.05. State your conclusion.
8. What is the significance probability (p-value) of the preceding test?

9. What appears to be the nature of the relationship between research and development expenditures and productivity improvement?
10. Estimate the mean productivity improvement for firms with high research and
development expenditure levels with a 95% confidence interval.
11. Obtain a 95% confidence interval for µ2 − µ1. Interpret your interval estimate.
12. Obtain confidence intervals for all pairwise comparisons of the treatment means;
use the Tukey procedure and a 90% simultaneous confidence level. State your
findings.

13. Is the Tukey procedure employed in the preceding question the most efficient
one that could be used here? Explain.
14. Obtain a 95% confidence interval for (µ1 + µ2)/2 − µ3, the difference in mean
productivity improvement between firms with low or moderate research and
development expenditures and firms with high expenditures. Interpret your
interval estimate.

15. The sample sizes for the three treatment levels are proportional to the population sizes. The economist wishes to estimate the mean productivity gain last
year for all firms in the population. Find a 95% confidence interval for it.
16. Using the Scheff´e method, obtain 90% simultaneous confidence intervals for
these 4 contrasts:

µ3 − µ2, µ2 − µ1,
µ3 − µ1, (µ1 + µ2)/2 − µ3.
What do you conclude?
17. Would the Bonferroni method be more powerful than the Scheff´e method for
the previous question? Explain.
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STAT/ME 424 Homework 6

 

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.
1

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?
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STAT/ME 424 Homework 7

A 24
complete factorial experiment was performed to study four factors on the taste of brownies.
The factors and their levels are given in Table 1. Six persons were asked to provide scores for each
batch of brownies and the average score for each batch is given in Table 2 together with the order of
the runs.

1. Estimate the main and interaction effects.
2. Use Daniel’s, Length’s, and Dong’s methods to determine the significant effects.
3. Which, if any, effects are significant at level α = 0.10?
4. Fit a model containing only the significant effects to the data, plot the residuals, and state your
conclusions.

5. Show a plot of the yields (average scores) versus run order.
6. Can the plot provide a possible explanation for the significant effects?
Table 1: Factor levels for cookie experiment
Factor Low level (–) High level (+)
Oven temperature (A) 300 degrees 400 degrees
Cooking oil (B) vegetable oil coconut oil

Mixing method (C) hand mixing with spoon electric mixer
Amount of sugar (D) 1 cup 2 cups
Table 2: Results from brownie experiment
Run Factor A Factor B Factor C Factor D Average
order Temperature Oil Mixing Sugar score
3 − − − − 15.3
9 + − − − 7.6

12 − + − − 11.2
6 + + − − 13.0
14 − − + − 10.8
16 + − + − 14.0
7 − + + − 17.8
1 + + + − 13.2
11 − − − + 9.0

13 + − − + 17.3
5 − + − + 11.0
15 + + − + 8.8

2 − − + + 15.0
8 + − + + 12.8
10 − + + + 13.0
4 + + + + 18.6