IE 361 Homework Set #1 Fall 2004
1. An IE 361 in-class measurement demonstration (using a fairly nice micrometer) produced
size measurements for inexpensive binder clips. (The raw data are on the IE 361 Handout
Page for anyone who is interested.) The units of measure were " mm above 32.00 mm" (so
that 31.67 was read as −.33).
• Single measurements on n = 10 different binder clips had sample mean y = −.050 mm
and sample standard deviation sy = .0254 mm.
• m = 7 repeat measurements on an additional single binder clip had sample mean
y = −.051 mm and sample standard deviation s = .0035 mm.
In what follows, we’ll analyze the results of this simple measurement study using the methods
given on the "Elementary Statistical Considerations in Metrology" handout.
a) In Stat 231 you learned how to make (t) confidence intervals for means and (χ2 )
confidence intervals for standard deviations. Use the results from the 2nd part of the
study and make (6-degrees-of-freedom) 90% confidence intervals of these two types.
What two quantities do those intervals intend to bracket/estimate in this context?
b) Now consider the first part of the study. If I applied the Stat 231 formula for a (t)
confidence interval for a mean, what would I be attempting to estimate?
c) Combine results from the two parts of the study and estimate σ x , the standard deviation of the “real” sizes of the binder clips (not including measurement error). Then
make make approximate 90% confidence limits for σ x . What do these indicate about
the precision with which one has pinned down σ x ? If this precision seems unacceptably low, what would you have to do in order to improve the precision with which σ x
is known?
2. Consider problem 2.5 of the text.
a) Do parts b) through g) of this problem using the range-based formulas.
b) Redo a) using the ANOVA-based formulas on the "Some Additional Notes on ANOVAbased Gauge R&R Estimation" handout.
c) Use ANOVA-based formulas from the handout and find 90% confidence limits for
σ repeatability and approximate 90% confidence limits for σ reproducibility and for σ R&R . Then
use your confidence limits for σ R&R to make approximate 90% limits for the gage
capability ratio (the measurement "precision to tolerance" ratio). What is to be done
if a company finds this last interval to be uninformative?
3. Below is a data set from a real calibration study (taken from a paper by John Mandel
of NBS/NIST). Unfortunately, the units are not given in the paper. For sake of argument,
suppose that they are ppm (parts per million) of some contaminant. x is the “truth”/goldstandard-measurement. y is the local laboratory measurement. All on n = 14 different
specimens.
1
x
647
728
1039
1095
1116
1194
1557
1594
1896
1983
2136
2192
2224
2244
y
605
675
965
995
1018
1117
1422
1470
1762
1739
1918
1983
2008
2010
a) Fit a simple linear regression model to these data. For a fixed x/specimen, what do you
estimate “σ measurement ” to be at the local laboratory? What “conversion formula” do
you recommend for translating “local lab measurements” to estimated “gold standard
measurements”?
b) A new specimen is measured as y = 2000 at the local laboratory. Give an approximate
95% confidence interval for the “true”/gold standard value for this specimen. Do this
two way. First simply read one interval off a plot of 95% prediction intervals for an
additional y at various fixed values of x. Then use the approximate confidence interval
formula given in class and on the “Class Outline” handout.
You can get help with using the JMP statistical package by looking at
http://www.stat.iastate.edu/resources/jmp.html
or by using the statistical software primer written for Vardeman and Jobe’s Basic Engineering
Data Collection and Analysis available both at
http://www.duxbury.com/default.htm
(under the “Book Companions”) and in a local/development version at
http://www.public.iastate.edu/∼vardeman/book_site/index.html
To do the ANOVA part of Problem 2, you will need to enter 3 columns of length 60 into
the JMP worksheet. The first should give part numbers, the second should give operator
numbers and the third should give the measurements. After entering the data, click on the
“Part” column heading, go to the “Cols” menu and choose “Column Info.” There make sure
that the “Modeling Type” is “Nominal.” Do the same for the “Operator” column. Then
from the “Analyze” menu choose “Fit Model.” You’ll get a dialogue box. The “Measurement” variable gets entered into the “Y” part of the box. The “Part” and “Operator”
variables get entered into the “Construct Model Effects” part of the box. Then by highlighting the “Part” variable in the “Select Columns” list and the “Operator” variable already
in the “Construct Model Effects” part of the box and clicking the “Cross” button, you can
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add the interaction effects to the model. Then clicking “Run Model” will get you a JMP
analysis. The “Effect Tests” part of the report contains the necessary sums of squares and
degrees of freedom to make the necessary estimates.
To do Problem 3, you will need to enter 2 columns of length 14 into a new data table. The
first should give x and the second should give y. After entering the data, click on “Fit Y by
X” under the “Analyze” menu. Put the x variable in the “Factor” part of the dialogue box
and the y variable in the “Response” part of the dialogue box and click on “OK.” This will
bring up a scatterplot (that can be re-sized to improve resolution if you wish). Click the red
triangle on the bar above the plot and bring up a menu. Select “Fit Line.” This will produce
a SLR analysis for these data (and plot the least squares line). If you then click on the red
triangle by the “Linear Fit” bar below the plot, you can bring up a menu that includes an
entry “Confid Curves Indiv.” Checking that option will put 95% prediction limits for an
additional y at the various x’s on the plot. You can then select the cross-hair tool from
the JMP toolbar
and read off values on these plots of prediction limits. Other values you’ll
√
need (like MSE) can be read from the JMP report. Simple descriptive statistics for the
columns (like x and sx ) can be gotten by choosing “Distribution” from the “Analyze” menu.
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IE 361 Homework Set #2 Fall 2004
Do problems 3.18 and 3.22 of the text. (You need not do parts 3.18f, 3.22b, or 3.22f.)
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IE 361 Homework Set #3 Fall 2004
1. Return to problems 3.18 and 3.22 from the last assignment and do 3.18f, 3.22b, and
3.22f. Also, do problems 3.27, 3.30 and 4.31.
2. Use JMP and generate 100 standard normal random numbers, placing them in column
1. (Click on the “Rows” menu and add 100 rows. Then after clicking on the Column 1
heading, use the “Cols” menu and select “formula.” From the “Random” functions group,
select “Random Normal.” This will put the random numbers in column 1.)
a) Plot column 1 vs row number as an example of “stable process” behavior. (Use the
“Overlay Plot” option in the “Graph” menu to do this.) Then use JMP to find the
sample standard deviation of the normal random values. (Use the “Distribution”
option on the “Analyze” menu.) This should be reasonably close to the “process short
term standard deviation” σ = 1.
b) Use the “moving range chart” function in JMP to calculate the average moving range
MR for the data in column 1. (You can find “Control Chart” under the “Graph”
menu. In the dialogue box choose “IR” for the “chart type.” The center line on that
plot is at the average moving range.) How does MR/1.128 serve as an estimate of
the “process short term standard deviation” σ = 1?
c) Make up a new column (column 2) by adding the row number to the random numbers
in column 1. (You get a new column using the “Cols” menu. The opportunity to
specify the entries using a formula is activated using the “New Property” pull down
menu. The row number function is in the “Rows” functions group.) Plot this new
column vs row number as an example of a situation where one has samples of size
n = 1 and the process mean is not constant (in fact µi = i). Use JMP to find the raw
standard deviation of the values now in column 2. How does this value serve as an
estimate of the “process short term standard deviation” σ = 1?
d) Repeat part b) using the new values in column 2. Which estimate of σ = 1 do you
like best? The one in c) or the one here?
e) What is the moral told in parts a)-d) of this problem?
3. For the data of part c) of Problem 4.2, give the best possible guess at σ (the process
short term standard deviation) in this n = 1 situation. If future monitoring of individual
flatness values is contemplated and a mean flatness of 2.8 is “standard,” exactly how do you
suggest monitoring these flatness values for an “all-OK” ARL of 370?
4. Do Problems 5.2; 5.3b,c; 5.17a,b; 5.18; 5.28, 5.31, 5.37. (Regarding problem 5.2, JMP
has a "Normal Quantile" function in its "probability" group of functions in its "formula"
capability. You can sort using the "sort" function in the "Table" menu.)
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IE 361 "Homework Set #4" Fall 2004
I strongly suggest that (as informal "HW," not to be turned in) you work the parts of posted
old exams that are relevant to the lecture material from Chapters 6 and 7 of Vardeman and
Jobe. (I suggest that you actually WORK the exams before downloading the solutions, that
you can use to check yourselves.) Much more is needed here than simply looking over old
solutions.
Relevant parts of the Old Exams are:
Spring 2004:
Fall 2002:
Fall 2001:
Fall 2000:
Spring 2000:
Fall 1998:
Fall 1997:
Fall 1996:
Exam 3
Exam 3
Exam 3
Exam 3
Exam 3
Problem 3 of Exam 2 and all of Exam 3
Problem 4 of Exam 2 and Problems 1 and 2 of Exam 3
Problem 3 of Exam 2 and Problems 1 and 2 of Exam 3
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