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HOMEWORK #1 due Tuesday Feb 6
Text references are to Hollander and Wolfe,
``Nonparametric Statistical Methods'', 2nd ed.
NOTE: In the following, ^ means superscript, _ (underscore) means
subscript, and Sum(i=1,9) means the sum for i=1 to 9.
IN THE FOLLOWING: Do Problems 1-4 by hand. Problem 5 asks you to
write a computer program.
1. Consider the data in Table 3.7, p71 of the text about
clotting times before and after the administration of 600mg of aspirin.
(i) Use a Student-t-test to test the hypothesis that there is no
difference in the before and after clotting times. Is the resulting
P-value significant (P<0.05)? highly significant (P<0.01)?
(ii) Use the sign test to test the same hypothesis. Obtain two-sided
P-values using (a) the exact distribution of the binomial in
Table A2 and (b) the normal approximation. Is the resulting P-value
significant (P<0.05)? highly significant (P<0.01)? Is it the same
as in part (i)?
(iii) Use the Wilcoxon signed rank test to test the same hypothesis.
Obtain two-sided P-values using (a) the exact distribution of the signed
rank statistic in Table A4 and (b) the normal approximation. Is the
resulting P-value significant (P<0.05)? highly significant
(P<0.01)? Is it the same as in parts (i) and (ii)?
(iv) Which of the three tests would you consider most reasonable?
Why?
2. Consider the data in Table 3.11, p83 of the text about
levels of 6-beta-hydrocortisol excreted by chemical company workers.
(i) Use the sign test to test the hypothesis that the median amounts
per dat can be distinguished from 175 micrograms. Obtain two-sided
P-values using (a) the exact distribution of the binomial in Table A2
and (b) the normal approximation. (Hint: Subtract 175 from each of the
observations and see if the differences can be distinguished from zero.)
(ii) Find the nonparametric estimate of the median amount per day
using the Hodges-Lehmann sign-test estimator described in
Section 3.5.
(iii) Find a (1-alpha)times 100% confidence interval for the median
amount per day, using the sign test-associated confidence interval
described in Section 3.6, where alpha is chosen so that 1-alpha is
as close as possible to 0.95. What is the size (as in 95%) of the
resulting confidence interval?
(iv) A company executive states that while about as many values in
the original data are larger than 175mug/day, the differences from
175mug/day seem to be larger for the positive values. As an alternative,
find the nonparametric Hodges-Lehmann estimator of the median based on
the Wilcoxon sign test described in Section 3.2. Is the resulting
estimator larger than in part (ii)?
(v) Also find the associated (1-alpha)times 100% confidence interval
for the median amount per day, using the Wilcoxon signed rank-associated
confidence interval described in Section 3.3, where alpha is chosen
so that 1-alpha is as close as possible to 0.95. What is the size (as in
95%) of the resulting confidence interval?
3. It is conjectured that tropical plants of a certain
genus tend to produce more flowers at higher altitudes than at lower
altitudes. Fifteen species in this genus are known to occur at both
altitudes in a particular country. To test the conjecture, one plant
from a lowland forest and one plant from higher altitudes were collected
from each of twelve species from this genus. The number of flowers on
each plant were counted, and the results were:
Species LowAlt HighAlt Species LowAlt HighAlt
1 4 10 7 4 14
2 11 3 8 7 4
3 7 10 9 15 3
4 17 17 10 7 7
5 5 19 11 3 17
6 4 12 12 7 10
(i) Use the Wilcoxon signed rank test to test whether or not plants
from higher altitude tend to have more (or fewer) flowers than plants
from lower altitudes. What is the value of the Wilcoxon statistic T^+?
What is the associated (two-sided) P-value? Use both (a) the tables and
(b) the normal approximation. (Be sure to handle ties correctly. Recall
that ties between nonzero absolute values are ignored when using the
table.)
(ii) Even though the data is from 24 different plants, why would it
be incorrect to assume that the plants from the lowlands and the plants
from higher altitude form two independent samples?
4. Change the value of X_3 in Table 3.1 on p39 of the text
(Hamilton Depression Scale Factor values) from 1.62 to 16.2.
What effect does this have on the value of Zbar=(1/9)Sum(i=1,9) Z_i
for Z_i=Y_i-X_i? (That is, compare the values of Zbar before and after
the change.) What effect does this have on the value of the
Hodges-Lehmann estimator thetahat based on the Wilcoxon signed-rank
statistic? (See Example 3.3 on page 52.) Which estimator seems
to be more strongly affected by outliers?
5. Write a short program in C (or a C-like computer
language) based on the data salary data in Table 3.2 (p41) of the text
that
(i) Includes the salaries in dollars for the 12 private-sector
individuals and for the 12 government-sector individuals as initial
values of two global arrays, so that (for example)
xval[0]=12500 and yval[0]=11750,
(ii) Computes the 12 salary differences and stores them in a third
global array zval[] whose values are initially zero.
(For example, as defined by double zval[20]; before
the main() function.)
(iii) Computes and displays (a) the sample mean Zbar ,
(b) the sample standard deviation, ss , (c) the sample
standard error of the mean, stderr , and (d) the one-sample
t-statistic T=Zbar/stderr, for the 12 salary differences in
zval[].
(iv) If the salary differences Z_i were normally distributed with
mean mu and variance sigma^2 , then, given H_0:mu=0 , the statistic T in
part (iii) would have a Student- t distribution with 11 degrees of
freedom. Assuming instead that T has a normal distribution with mean zero
and variance one given H_0 , find the two-sided P-value for H_0:mu=0
versus H_1:mu is not 0.
(v) The text on page 41 concludes that the two-sided P-value P=0.078
using the Wilcoxon signed-rank test. How does this compare with the value
that you computed in part (iv)? (The answer to this part need not be part
of your computer program.)
(Hint: See sample C programs on the Math408 Web site.)
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