HOMEWORK #4 due Tuesday April 7
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. A local agricultural company is interested in selling one or more of four new types of lamb chow (food) that were developed in the company's research division. Weight gains for yearling lambs on the four new lamb chows (labeled Ch01,Ch02,Ch03,Ch04) and on a standard lamb chow (Chstd) are given in Table 1.
Table 1: Lamb weight gains for five lamb chows
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Chstd: 58 68 28 14 150 98 138 78 124 84
Ch01: 148 176 90 52 132 32 128 32
Ch02: 168 218 158 238 72 100 192
Ch03: 44 206 132 12 108 148 156 182 68 70
Ch04: 92 150 124 136 180 128 132 216 168 220
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The sample medians of the weights of the lambs in the five groups in
Table 1 are significantly different (Kruskal-Wallis test, P=0.018,
large-sample approximation, 4 degrees of freedom).
OneWayMCCtrl.m on the Math408 Web
site.)
2. Table 8.6 in the text (page 380) has the following data:
Table 2: Spending per High-School Senior by State in 1987-1988
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4462 4164 3093 4789 3919 4457 5201 4369 2718 5329
3368 5051 3249 4149 3623 4989 3068 4246 7151 3786
6230 5207 4386 4747 3138 5017 3943 3434 2548 4092
2454 3744 3408 4692 3011 4246 3608 3998 3519 4076
2667 3858 6564 7971 5471 4124 3691 3794 2989 3840
Use the data from Table 2 to find a 95% confidence band for the true
distribution function F_Y(t)=P(Y<=t). That is, find increasing
functions F_1(t),F_2(t) such that
where Fhat_Y(t) is the empirical distribution function of Y determined by the values in Table 2 and
Draw a sketch of the three functions F_1(t),Fhat_Y(t),F_2(t) on the same graph.
3. A series of tests emphasizing dexterity in high places was carried out on cats, rats, and rabbits. The times taken in seconds to complete each of 14 different tasks are given in Table 3.
Table 3: Times taken to do 14 different tasks for animals from three species
Task: 1 2 3 4 5 6 7 8 9 10 11 12 13 14
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Cats 0.3 1.0 3.6 0.1 0.6 5.5 1.0 3.7 3.1 1.1 2.0 1.6 4.3 1.0
Rats 1.5 1.1 1.8 1.3 4.3 2.0 8.4 3.7 6.6 1.1 4.0 6.5 2.6 6.5
Rabbts 1.7 1.5 8.1 1.3 4.3 4.6 4.0 3.7 5.1 2.5 6.0 6.9 2.5 6.8
4. In a test of perceptions of color, a picture with ambiguous colors was shown to 12 subjects, who were asked if they saw various colors in the pictures. None of the 12 subjects showed evidence of red-green or blue color blindness. The results were scored as 1 for Yes and 0 for No. The picture was designed to have attributes of all six colors. The results are in Table 4 below.
Table 4: Perceptions of colors by 12 subjects
Subject: 1 2 3 4 5 6 7 8 9 10 11 12
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Red: 1 1 1 0 0 1 1 1 1 0 1 1
Green: 1 0 0 0 1 0 1 0 0 0 0 1
Blue: 1 1 1 1 0 1 1 0 1 0 1 1
Yellow: 0 0 0 1 0 0 1 0 0 0 0 1
Pink: 1 1 1 1 0 1 1 1 1 0 0 0
Orange: 1 0 1 0 0 1 1 0 1 0 1 1
Do some colors tend to stand out more to these subjects than other colors,
controlling for subject effects?
Cochran.m on the Math408 Web site for a discussion of
Cochran's test statistic Q. Use the large-sample approximation
for Q.) Recall that Cochran's test statistic is exactly the same as
Friedman's test statistic S' with tie correction for 0,1 data.
5. In a study of pollution in Lake Michigan, the number of ``odor periods'' was observed for each of the years 1950-1964. The numbers of days are in Table 5.
Table 5: Numbers of bad periods in Lake Michigan (1950-1964)
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(1950, 10) (1951, 20) (1952, 17) (1953, 16) (1954, 12)
(1955, 15) (1956, 13) (1957, 18) (1958, 17) (1959, 19)
(1960, 21) (1961, 23) (1962, 23) (1963, 28) (1964, 28)