Math 8452, Homework #3. Due on June 20, 2013.
1. (To turn in.) Do problems 13, 14, and 15a from page 74 in the textbook. Problems 13 and 15a
may be done using SAS, R, or appropriate tables, but problem 14 should be done by listing.
2. (To turn in.) Using listing, find the exact distribution of the Ansari-Bradley statistic for the case
where n1 = 3, n2 = 4, and the statistic is obtained by summing the scores for values assigned to the
first sample. Is it possible to do a one-sided test at level 0.05 using this statistic? Please explain.
3. (To turn in.) By thinking about the sorts of alternatives that each test is sensitive to, create
small data sets (no more than 10 in each sample) with each of the following properties. In each
case, use exact tests rather than asymptotic approximations. (a) The rank sum test rejects equality
of distributions at level 0.05, but the Kolmogorov-Smirnov test does not. (b) The rank sum test
rejects equality of distributions at level 0.05, but the Ansari-Bradley test does not.
4. (To turn in.) (a) Using the one-sample Kolmogorov-Smirnov test, test the hypothesis that
the following data values come from the uniform distribution on (0, 1). (Hint: The CDF for this
distribution is F (x) = x for x ∈ (0, 1).) You may use the fact that the critical value for the twosided test when n = 10 and α = 0.05 is 0.409. (b) Sketch the 95% confidence band for the true CDF.
0.35 0.30 0.51 0.37 0.62 0.54 0.17 0.41 0.20 0.57
5. (Not to turn in yet.) In class, we used simulation to compare the rank sum test and the twosample t test in terms of power and the degree to which each test maintains its α level. Using R,
do a similar comparison for the F test for equal variances and the Ansari-Bradley test. To do the
two tests in R, use the functions var.test and ansari.test. To create appropriate alternatives to
equality of distributions, multiply one of the distributions by a constant greater than 1. Possible
base distributions to use: standard normal (rnorm(n,0,1)), standard Cauchy (rcauchy(n,0,1)),
Uniform on −1 to 1 (runif(n,-1,1)).
1