Math
408 Homework 3
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. Soybean plants were grown in 32
pots located on 4 different heavy laboratory tables. Each table (group) of
soybean plants was given a different amount of a particular nutrient. The
weights of the soybean plants in grams in the four groups after two weeks are
given in Table 2.
TABLE 1 -- Weights of Soybean Plants after Two Weeks
----------------------------------------------------------
LabTable #1 - 136 96 122 60 40 42 52 20
LabTable #2 - 74 52 152 76 12 170 128 82
LabTable #3 - 126 106 94 120 82 84 94 124
LabTable #4 - 102 168 220 126 196 84 166 140
(i)
Is there a significant variation in the sample medians of the soybean weights
in the table? Carry out the Kruskal-Wallis test to
find out. Use the large-sample approximation with tie correction.
(ii) The experimenter recalls
that the soybean plant treatment groups were, in fact, arranged by distance
from the window, so that treatment groups with higher LabTable
numbers might have received more light. Do the soybean weights in Table 2
vary by sample by either monotonically increasing or monotonically decreasing
with the lab-table number? (That is, with a different alternative hypothesis
than in part (i).) Carry out the Jonckheere-Terpstra test to find out. Use the large-sample
approximation, either with or without tie correction as you prefer. Is the test
significant?
(iii) How do the two P-values
in parts (i) and (ii) compare? If the P-value in
part (i) is significant,
should the experimenter rethink his conclusion that different amounts of the
nutrient has significantly different effects on the growth of the soybean
plants? Why?
2. The following table contains data
about the amount of drying during storage of 14 similar items that were
prepared for storage using 5 different methods:
TABLE 2 -- Percentage of Drying After Storage
-------------------------------------------------
Method #1 - 7.8 8.3 7.6 8.4 8.3
Method #2 - 5.4 7.4 7.1
Method #3 - 8.1 6.4
Method #4 - 7.9 9.5 10.0
Method #5 - 7.1
(i) Is there a significant difference among the
storage methods with regards to the percentage of drying? Carry out the Kruskal-Wallis test to find out. Use the large-sample
approximation with tie correction to find out.
(ii) A curmudgeon might argue
that a data set with 14 observations distributed over 5 treatment groups, with
only one observation in one of the treatment groups, might not be a good
candidate for a large-sample approximation that assumes that all samples sizes
are arbitrarily large.
Use n=10,000 random permutations of
the data among the 14 places in the 5 treatment groups to estimate the exact Kruskal-Wallis P-value, and give a 95% confidence interval
for your estimate of the exact P-value. Does this procedure find that the
differences are significant? Are your conclusions different than in part (i)? (Hint: See the program OneWayLayout.m with output OneWayLayout.txt on the Math408 Web
site.)
3. (See Table 6.10 p226 in the text,
and Problem 32 p225 for more biological detail.) Salmonella colonies were grown
under six different concentrations of AcidRed114. For each concentration, three
colonies and the number of mutant clones were counted (see Table 4). In
Table 4, where mug stands for micrograms per milliliter and Mg for
milligrams per milliliter, so that 1Mg=1000mug. The values at 0mug correspond
to the natural state of the organism. The low values at high concentrations of
the mutagen may be due to the toxic effects of AcidRed114, so that fewer
colonies survive to be mutant or not.
TABLE 3: Number of Mutant TA98 Salmonella Colonies
under Exposure to Various Levels of Acid Red 114