HOMEWORK #4 due Tuesday April 17, 2007
Text references are to Hollander and Wolfe, ``Nonparametric Statistical Methods'', 2nd edn.
IN THE FOLLOWING: Do Problems 1, 2, and 5 by hand. Problem 3 asks you to write a computer program. Problems 4 and 6 can be done either entirely by hand or else by using a computer program.
NOTES: Hand in your homework in the order
(a) Your written answers to all problems,
with references as needed to part (c) below,
(b) The computer source for any computer
programs that you used
(c) All output from the programs in
part (b)
This will put the emphasis on what you think the answers
should be and on your evidence for this. If a reader thinks that your
answers are reasonable, then he or she may or may not want to look at your
actual output and computer programs.
1. Two chemists studied the efficiency of conversion as a function of pressure of a compound methyl glucoside to monovinyl isomers. Specifically, they were interested in the percentage of methyl glucoside converted at 5 different pressures in the presence of acetylene. Due to limited laboratory space, they were only able to measure the conversion percentage at 3 different pressures in any one experimental run, but were able to carry out 10 different experiental runs under different choices of pressures. Their data is in Table 7.14 (page 316). See Problem 63 on page 316 for more information about the background of these data.
2. 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 1 below.
Table 1: Perceptions of colors by 12 subjects
Subject: 1 2 3 4 5 6 7 8 9 10 11 12
--------------------------------------------------------
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?
CochranTest.c 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.
3. (i) Write a computer program to estimate the exact P-value for the balanced-incomplete-block design (BIBD) test described in Section 7.6 in the text for the data in Problem 1. What is the estimated P-value? What is a 95% confidence interval for your estimated P-value? How does the P-value compare with the large-sample approximate P-value that you found in Problem 1?
TwoWayBibd.c on the Math408 Web site.
Add code to the program to find a P-value for L as well as for the BIBD
statistic ``Dscore'', for example by either defining two ``success''
counts nbig1,nbig2 instead of one count nbig
within a single loop, or else by using two permutation loops, one for
Dscore and one for L.)
4. In a study to determine the effect of light on the release of a hormone (luteinizing hormone, LH), rats were observed both under constant light and with 14hrs of light alternating with 10hrs of darkness. The rats were given one of five different levels of a luteinizing release factor (LRF) as a control for variable LRF. Six rats were studied for each of 5 levels of LRF and for each of two light regimes (constant or alternating), for a total of 60 rats, in a two-way experimental layout with six observations per cell.
5. Table 5.4 on page 171 in the text has blood platelet counts for 16 newborn infants, 10 of which were born to mothers who were given the steroid drug prednisone during their pregnancy. The mothers of the remaining 6 infants were not given prednisone. In addition to whether or not the mean platelet count in the two samples is different, whether or not the variance of the two samples is the same is also of interest. (See page 171 in the text for more background about these data.)
6. 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 2.
Table 2: Numbers of bad periods in Lake Michigan (1950-1964)
----------------------------------------------------------------
(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)