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```IE 361 Homework Set #1 Fall 2002
1. One of the groups doing the in-class measurement study on foam peanuts obtained the following
results:
i) Single measurements on 8 œ ) different peanuts had sample mean C œ Þ&&* inch and
sample standard deviation =C œ Þ!## inch.
ii) 7 œ ' repeat measurements on an additional single peanut had sample mean C œ Þ&\$(
inch and sample standard deviation = œ Þ!!) inch.
In what follows, we'll analyze the results of this simple measurement study.
a) In Stat 231 you learned how to make (>) confidence intervals for means and (;# ) confidence
intervals for standard deviations. Use the results from ii) and make (5 degree 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 part i) of the study. If I applied the Stat 231 formula for a (>) confidence interval
for a mean, what would I be attempting to estimate?
c) Combine results from the two parts of the study and estimate 5B , the standard deviation of the
"real" sizes of the peanuts (not including measurement error). Find a "standard error" for your
estimate. What does this second quantity indicate about the precision with which you have pinned
down 5B ? If this value seems unacceptably large, what would you have to do in order to improve
the precision with which 5B is known?
2. Do problem 2.5 of the text using the range-based formulas. Then redo it using the ANOVAbased formulas on page 27 of the text. (When making an ANOVA-based estimate of 5R&R , use the
#
formula given in class rather than using simply computing É5
.)
s #repeatability € 5
s reproducibility
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. B is the "truth"/gold-standard-measurement.
C is the local laboratory measurement. All on 8 œ "% different specimens.
B
C
'%(
'!&
(#)
'(&
"!\$*
*'&
"!*&
**&
"""' "!")
""*% """(
"&&( "%##
"&*% "%(!
")*' "('#
"*)\$ "(\$*
#"\$' "*")
#"*# "*)\$
###% #!!)
##%% #!"!
1
a) Fit a simple linear regression model to these data. For a fixed B/specimen, what do you estimate
"5measurement " 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 C œ #!!! 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 C at various fixed
values of B. 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.public.iastate.edu/~stat/compguide/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 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 B and the second should give C. After entering the data, click on "Fit Y by X" under the
"Analyze" menu. Put the B variable in the "Factor" part of the dialogue box and the C variable in the
"Response" part of the dialogue box and click on "OK." This will bring up a scatterplot (that can be
resized 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 C at the various B'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 ÈQWI ) can be read from the JMP report. To get simple descriptive statistics for the
columns (like B and =B ) can be gotten by choosing "Distribution" from the "Analyze" menu.
2
IE 361 Homework Set #2 Fall 2002
Problems 3.18 and 3.22 of the text.
3
```
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