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HOMEWORK #2 due Tuesday Feb 17
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.
1. Grades on a college board test are collected for 10
students before and after a college board cram course. The grades are
TABLE 1: Scores before and after a college board prep course
-------------------------------------------------------------------
Student: #1 #2 #3 #4 #5 #6 #7 #8 #9 #10
Before: 15 19 35 43 47 50 57 37 56 41
After: 10 33 43 57 58 48 61 49 58 55
(i) Is there a significant improvement in the students' scores,
controlling for student-to-student variation? That is, use the Wilcoxon
signed rank test for the after-minus-before differences for each student
to test the hypothesis that there is no difference in the before and after
scores. Give two-sided P-values in two ways, using Table A4 ignoring
any ties (and interpolating in the table if need be) and using the normal
approximation, with tie correction if there are ties.
(ii) Find the Hodges-Lehmann estimator for the increase in score after
the college board cram course.
(iii) Find the exact nonparametric confidence interval with coverage
as close as possible to 95% based on the Wilcoxon signed-rank statistic.
2. Consider two samples of values from two different sources:
TABLE 2: Two independent samples, one of size 8 and one of size 10:
------------------------------------------------------------------------
X: 19.93 19.61 20.99 22.55 22.73 17.05 15.54 21.52
Y: 31.64 22.49 24.38 34.70 30.09 21.46 28.07 19.43 25.05 20.99
(i) Assuming the model Y dist X+theta, is there significant evidence
that theta ne 0; that is, that the Ys are shifted with respect to the Xs?
Use the Wilcoxon Rank Sum test in two ways, using Table A6 and the
large-sample normal approximation. as convenient. Give two-sided P-values.
(ii) Find the Hodges-Lehmann estimator for the shift of Y with respect
to X (that is, theta). How this this compare with the difference in means?
(iii) Find the exact nonparametric confidence interval with coverage
as close as possible to 95% based on the Wilcoxon signed-rank statistic.
Use Table A6 to find the limits as in Section 4.3 in the text.
3. Consider the ``Karate Kid'' data in Table 4.4 on
page 124 of the text. These data give the lengths of time that kids
who were supposedly baby-sitting two younger children spent before calling
an adult after their two younger charges supposedly became violent. A
control group of 21 kids (baby-sitters) had watched non-violent excerpts
from the 1984 Summer Olympics while a test group of 21 kids (baby-sitters)
had watched a violent TV program. The experimenters' hypotheses was that
the baby-sitters who had watched the violent TV program would take longer
to call an adult.
(i) Is there is significant difference in location (or time)
between the two samples? Use the Wilcoxon rank-sum test to find out. Use
the normal approximation with tie correction to find a two-sided P-value.
(ii) Find the Hodges-Lehmann Wilcoxon-rank-sum-like estimate of the
difference in medians. How does this compare with the difference in sample
means? Does the Hodges-Lehmann procedure appear to control better for
outliers?
(ii) Find the approximate exact nonparametric 95% confidence
interval based on the Wilcoxon Rank-Sum statistic.
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