STAT 496, Spring 2004
Homework Assignment #3
1. A paper products manufacturer is interested in improving the tensile strength of its
paper used in grocery bags. The current process uses 10% hardwood in the pulp,
and at this level, the paper has an average tensile strength of approximately 15 psi.
JMP analysis of the current process, control charts and capability analysis are given
on separate sheets.
(a) Comment on the stability of the current process. Refer to the control charts on
the accompanying page.
(b) Based on the control charts, what would be the estimate of the process standard
deviation?
(c) Comment on the capability of the current process. Be sure to interpret any
appropriate capability indices. Indicate what the estimate of the process standard
deviation is. What proportion of paper would you expect to see below the lower
specification of strength?
2. The paper process engineers and operators suspect that tensile strength is a function
of hardwood concentration in the pulp. Specifically, they believe that higher hardwood
concentrations would produce paper with higher tensile strengths. Because hardwood
is more expensive than the usual pulp wood, hardwood concentration must be 20% or
lower. The experimental team decides to investigate four levels of hardwood concentration, 5%, 10%, 15%, and 20%. For each hardwood concentration, six test specimens
will be made using a pilot plant. All 24 specimens are tested for tensile strength by
the same technician using a single machine. The testing of the specimens is done in
random order. Below are the resulting tensile strength measurements.
Hardwood Concentration (%)
5
10
15
20
7
12
15
19
8
16
19
24
14
13
20
22
12
17
18
23
9
18
16
18
10
14
20
20
(a) Comment on the use of control, randomization and replication in this experiment.
(b) Construct a plot of the data. From the plot, does there appear to be a relationship
between hardwood concentration and tensile strength of the paper? If so, in
which direction? Comment on the variation in tensile strength for each hardwood
concentration.
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(c) Code the hardwood concentration using the formula below. Calculate the simple
linear regression line relating tensile strength to coded hardwood concentration.
(Concentrationi − 12.5)
C1i = 2
5
!
(d) Uncode your equation in (c) to obtain the simple linear regression relating tensile
strength to hardwood concentration. Graph this line on your plot in (b).
(e) Predict the tensile strength of paper made with 17% hardwood.
(f) What percentage of the variability in tensile strength is explained by the linear
regression with hardwood concentration?
(g) Is there a significant linear relationship between tensile strength and hardwood
concentration? How do you know? Caution: Be sure to use M SRepError
(h) Add a quadratic term to you prediction equation by using the coded variable:
C2i =
C1i
2
2
15
−
12
!
(i) What percentage of variability in tensile strength can be explained by the addition
of this quadratic term?
(j) Is the quadratic term statistically significant? How do you know? Caution: Be
sure to use M SRepError
(k) Given the analysis above write a one paragraph summary about what you have
learned about the tensile strength of paper and the amount of hardwood used in
its manufacture.
You may use JMP or another computer program to verify what you have done above.
• Use Graph - Overlay Plot to obtain a plot of the data.
• Use Basic Stats - Distribution with Y, Columns - Tensile and By - Concentration
to obtain means and standard deviations for each hardwood concentration level.
• Use Basic Stats - Fit Y by X to fit a simple linear regression of tensile strength
on coded hardwood concentration, C1i . You can also use this to fit a simple linear
relation of tensile strength on hardwood concentration directly. Note that information
on Lack of Fit is hidden so you need to click on the blue triangle to show this. What
JMP calls Pure Error is Replication Error.
• Use Modeling - Fit Model to fit a polynomial regression of tensile strength on C1i
and C2i . Be sure to uncheck the Center Polynomial option in the Model Specification
pull down and choose the Minimal Report under Emphasis.
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