There are four groups and a total of 31 observations.  As you will
see, the standard deviations range from 1.5 to 5.7.  One might choose
to ignore this variability, but read on! \\
1. Write done the usual assumptions and report the results of the
usual analysis of variance. \\
2. BY HAND, make a ``dot plot'' of the data, using letter symbols for
group.  Comment.\\
Now drop the assumption that variances are equal. \\
3. Note that lack of obvious relationship between mean and
variance by group (PLOT THEM). \\
4. Instead, use the inverse of group variance as
weight.  Briefly justify this choice (why might this be reasonable).
Now briefly critique it (why is it silly in this problem).  How does
the use of estimates of group variances affect the $p$-value?  [Hint:
examine the new SDs.] \\
5. Replace the observations by their ranks and rerun the usual
analysis of variance.  [This test is an approximation of the
Kruskal-Wallis test.]  Comment on whether this a reasonable approach
for this problem.  [Hint: what assumption(s) are helped by using
ranks?] \\
6. The exact test differs from all of these.  It is based on the
randomization principle.  That is, if there are no group differences,
then all assignments of group labels to the data are equally likely.
That is, one could (in theory) examine every permutation of the 31
responses (with 9 A's, 7 B's, 8 C's and 7 D's) and compute the $F$
statistic for each one.  The $p$-value is then the proportion of $F$
values that are as extreme or larger than the one observed.
According to my source, the ``right'' $p$-value for the raw data is
.030, or for the ranks (exact Kruskal-Wallis) is .06. \\
Comment on the disparity among $p$-values (you now have 5 different
ones!).  What is your conclusion about differences among groups?
[Again, be brief.  There is no ``right'' answer to my question,
anyway!]
\\[.125in]
Now think about inference on the variances themselves. \\
7. Calculate the ratio of the largest to smallest variances.  This is
Hartley's $F$-max test.  [``Liberal'' (using sample size 9, 4 groups)
critical values for 5\% and 1\% are, respectively, 7.18 and 11.7.  See
Milliken and Johnson Table A.1.]  This is a very easy test to perform.
Unfortunately, it relies heavily on the normal assumption, and is best
for balanced data.  Interpret results, with cautions for this data
set. \\
8. Conduct Levene's test for unequal variance (see SAS printout).
This test is not sensitive to departures from normality (based on
simulation studies -- see Milliken and Johnson), and can be used for
small samples.  Interpret results.\\
9. Comment briefly on the dilemna of testing for equal variance before
conducting analysis of variance.  How is this problem lessened (or
greatened) by increasing sample size?
