Appendix 6¾Example Graded Homework Assignment
E538¾Homework on Bayesian Analysis and on Anova¾2001¾D. Parkhurst
©2003, D.F. Parkhurst
The tasks below are
homework due by 4 p.m. on Monday, April 23.
(Please give directly to me, or slip under my office door, SPEA
355. If you do the latter, please write
the time and date on the front page.)
No late papers will be accepted.
Please put only the last four digits of your student number (not your
name) on your paper. Since this is
graded homework, be sure to do the work yourself, as described on pp. 5–6 of
the syllabus. You are welcome to obtain
help from others in using SPSS to perform the types of analysis required here,
but not with the particular datasets you are to analyze in these
problems. Feel free to ask either Prof.
Parkhurst or Oksana for help if you need it (but don’t wait until the last day
to do this!)
A major purpose of these problems is to check your
ability to use SPSS, so you must use that software to receive credit for this
homework set. Please hand in a printout
of the SPSS results for each SPSS task described.
1. Bayesian analysis: Suppose you work in the group responsible for potentially leaky
underground storage tanks (the LUST program) of your state’s environmental
agency. Your group has identified over
a thousand abandoned UST’s, and these need to be checked for leakage. Eighty of these tanks are chosen at random
for testing, and of those, thirty are found to be leaking (the criterion being
“at least one liter per week,” in the tests).
A. Using a beta
distribution with a=b=1 as a (nearly) uninformative prior, determine the
following four probabilities: (i)
, (ii)
, (iii) 
, (iv) 
. (Remember that the
second of those quantities is 
minus the first, and
so on.) Describe your results in one or
more sentences that might be used in a report to the division director, or in a
press release.
B. Repeat the above
analysis using a prior obtained from a couple of consultants familiar with
these types of tanks¾their
prior was a beta distribution with a=1 and b=9. Again, state your results in one or more
sentences. Explain (to me, not for the
press release) why the two sets of results either are, or are not, substantially
different.
2. Frequentist anova: This problem is taken from Dixon, W. J. and J. F. Massey 1969. Introduction
to Statistical Analysis. New York, McGraw-Hill Book Company (although I’ve
changed the numbers). Here is their
problem statement:
An agency wished to determine whether five makes of
automobiles would average the same number of miles per gallon. A random sample
of three cars of each make was taken from each of three cities, and each car
had a test run with 1 gallon of gasoline. The table records the number of miles
traveled. (a) Why were three cities used instead of just one city? (b) What
populations are sampled from? (c) How would you go about getting such a random
sample of three cars from a city? (d) What assumptions are made about the
populations, and what hypotheses can be tested? (e) Perform the analysis of
variance and state fully your conclusions.
and here are the modified
numbers:
|
city |
A |
B |
C |
D |
E |
|
Los Angeles |
16.0 |
15.4 |
18.0 |
13.8 |
19.8 |
|
|
15.5 |
14.6 |
19.0 |
14.5 |
20.4 |
|
|
17.5 |
15.1 |
18.5 |
13.9 |
21.3 |
|
San Francisco |
17.7 |
15.8 |
16.3 |
15.4 |
13.7 |
|
|
18.6 |
15.7 |
17.2 |
15.2 |
14.1 |
|
|
17.2 |
16.6 |
15.5 |
16.5 |
14.1 |
|
Portland |
15.6 |
15.4 |
18.5 |
15.3 |
18.1 |
|
|
14.7 |
14.3 |
18.1 |
14.2 |
20.4 |
|
|
17.0 |
15.7 |
17.3 |
14.8 |
19.7 |
Your tasks here are to
answer Dixon and Massey’s questions d and e, (and to answer the others for
yourself.) Use SPSS to perform the
analysis of variance. (Be sure to
consider any possible interaction.
State the hypotheses you are testing, provide the SPSS output, and state
your conclusions in a paragraph or two.
These data will be available in text format at the class website, as a file named gasmiles.txt. You may have to make a few changes to the way the data are listed so as to meet the requirements of SPSS, but figuring that out is part of the exercise. You should be able to cut and paste the data from a browser window to the SPSS data window.