Math 408 Homework 2

Text references are to Hollander and Wolfe, ``Nonparametric Statistical Methods'', 2nd ed.

NOTES:
    (1)  Whenever you are asked to test a hypothesis, state the P-value, whether the P-value is for a one-sided or two-sided test if appropriate (that is, if the statistic has a large-sample normal approximation), and whether you accept or reject H_0.

    (2)  If you use MATLAB to do a problem, include (hard copy of) your MATLAB output AND your MATLAB program in an APPENDIX to your homework. That is, do not mix together the answers to the questions and your computer output. In that way, for problems in which you used MATLAB, your answers become an ``executive summary'' that gives your conclusions, and interested parties can then look or not look at your actual MATLAB code and output to get more information or to see what happened if you get a wrong answer.

    (3)  In the following, ^ means superscript, _ (underscore) means subscript, and Sum(i=1,9) means the sum for i=1 to 9.

1. 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' hypothesis was that the baby-sitters who had watched the violent TV program would take longer to call an adult.

(i)  Is there a 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?

(iii)  Find the approximate nonparametric 95% confidence interval based on the Wilcoxon Rank-Sum statistic.

 

2. Consider the chorioamnion permeability data in Table 4.1. on page 110 of the textbook. In section 4.1, we saw that the Wilcoxon rank sum test did not reject the null hypothesis that the human chorioamnion is as permeable to water transfer at 12 to 26 weeks gestational age as it is at term. With this in mind and using the same data, write a Matlab program to use the Ansari-Bradley to test the hypothesis of equal dispersions versus the alternative that the variation in tritiated water diffusion across human chorioamnion is different at term than at 12 to 26 weeks gestational age. (Hint: To obtain all possible ways of sampling k out of n without replacement, you can use the Matlab function nchoosek(1:15,5). You may consider to do the Monte Carlo test using the Matlab function randsample(15,5) to generate a random sample of size 5 without replacement.)

 

At Term     12-26 Weeks

0.80        1.15

0.83        0.88

1.89        0.90

1.04        0.74

1.45        1.21

1.38

1.91

1.64

0.73

1.46

 

3. Consider the chorioamnion permeability data given in Table 4.1 in the textbook. Write a Matlab program to find the approximate p-value for the jackknife test of the hypothesis of equal dispersions versus the alternative that the variation in tritiated water diffusion across human chorioamnion is different at term than at 12 to 26 weeks gestational age. Compare your findings with those obtained in Problem 2 using an Ansari-Bradley test.

 

4. Verify the expression for E_0(C) and Var_0(C) in equations (5.12) and (5.13) on page145 of the textbook, when N=(m+n) is an odd integer.

 

5. (Compare the significance level of two-sample t-tests with permutation tests.) Nieman, Groot, and Jansen (1952), ¡°The nutritive value of butter fat compared with that of vegetable fats, I,¡± Koninkl. Ned, Akad. Van. Wetenschap, Ser. C 55: 588¨C604.

 

In a comparison of the effect on growth of two diets A and B, a number of growing rats were placed on these two diets, and the following growth figures were observed after 7 weeks.

 

A 156 183 120 113 138 145 142

B 109 107 119 162 121 123 76 111 130 115

 

i)                    Use the two-sample t-test to test if the mean growth rate differs for the two diets, at level alpha = 0.05. (Do not assume that the two populations have equal variances. Use the Matlab function ttest2(x,y,alpha,¡¯both¡¯,¡¯unequal¡¯).)

ii)                   Let the test statistic be the difference of the two sample means. Use permutation tests to find the p-value.

iii)                 Let the test statistic be the difference of the two sample medians. Use permutation tests to find the p-value.