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is the number of differences X;— 25.0 that; 3 (gin): 351.0 357.1. 366.2 3673
have positive signs, hence the name sign test.] 382.0 392.4 409.9
36. Refer to Exercise 35, and consider a confidence j= 4(10in.): 346.7 362.6 384.2 410.6
interval associated with the sign test, the sign 433.1 452.9 461.4
interval. The relevant hypotheses are now j;_ 5 (2 in): 4074 410.7 41994412
Ho: ji = figversus Hy: 7 # fig. Let’s use the fol- “418 4658 4734
lowing rejection region: either Y > 15 or Y < 5.
a. What is the significance level for this test? Use the K-W test to decide at significance level
b. The confidence interval will consist of all .O1 whether the true average axial stiffness index
values fig for which Hg is not rejected. Deter- depends somehow on plate length.
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Supplementary Exercises 785
38. The article “Production of Gaseous Nitrogen in Hew a ae ee
Human Steady-State Conditions” (J. Appl. Physiol., Happiness 22753297196
1972: 155-159) reports the following observations S
; iat ae Depression 22.5 53.7. 10.8 21.1
on the amount of nitrogen expired (in liters) under Calninesy 26 53.1 83 216
four dietary regimens: (1) fasting, (2) 23% protein,
(3) 32% protein, and (4) 67% protein. Use the 5 é 7 8
Kruskal-Wallis test (Exercise 37) at level .05 to a
test equality of the corresponding 4,;’s. Fear 19 54.6 21.0 20.3
Happiness 13.8 47.1 13.6 23.6
1. 4.079 4.859 3.540 5.047 3.298 yecresnen ee 3 tee ae
4.679 2.870 4.648 3.847 ee ee
2. 4.368 5.668 3.752 5.848 3.802
4844 3.578 5.393 4.374 Use Friedman’s test to decide whether emotion
3. 4.169 5.709 4416 5.666 4.123 has an effect on skin potential.
5.059 4.403 4.496 4.688 40. In an experiment to study the way in which differ-
4. 4928 5.608 4.940 5.291 4.674 ent anesthetics affect plasma epinephrine concen-
5.038 4.905 5.208 4.806 tration, ten dogs were selected and concentration
was measured while they were under the influence
of the anesthetics isoflurane, halothane, and
39. The model for the data from a randomized block cyclopropane (“Sympathoadrenal and Hemody-
experiment for comparing / treatments was namic Effects of Isoflurane, Halothane, and
Xj = "+; +f; +e, where the o’s are treat- Cyclopropane in Dogs,” Anesthesiology, 1974:
ment effects, the f’s are block effects, and the 465-470). Test at level .05 to see whether there
é’s were assumed normal with mean 0 and vari- is an anesthetic effect on concentration. [Hint: See
ance o°. We now replace normality by the Exercise 39.]
assumption that the ¢’s have the same continuous
distribution. A distribution-free test of the null TT
hypothesis of no treatment effects, called Fried- Dog
‘man’ s test, involves first ranking the observations igs ¢@¢F
in each block separately from 1 to /. The rank
average Rj is then calculated for each of the / Isoflurane 28 51 1.00 39 29
treatments. If Ho is true, the expected value of Halothane 30 39 63 38 21
each rank average is (J + 1)/2. The test statistic is Cyclopropane 1.07 135 69 28 1.24
ise ene 6 7 8 9 10
Fr “Way (@ 4) Isoflurane 36.32 69 17-33
Halothane 88 39 513242
For even moderate values of J, the test statistic has Cyclopropane 1.53 49 56 1.02 30
approximately a chi-squared distribution with
f= laf benny etme. ___ 41. Suppose we wish to test
The: article “Physiological. Effects “During Hp: the X and ¥ distributions are identical
Hypnotically Requested Emotions” (Psychoso-
matic Med., 1963: 334-343) reports the follow- versus
ing data. (rj) on skin. potesitial int millivolts when, H,: the X distribution is less spread out than the Y
the emotions of fear, happiness, depression, and distribution
calmness were requested from each of eight
sabiects. The accompanying figure pictures X and Y dis-
tributions for which H, is true. The Wilcoxon
A rank-sum test is not appropriate in this situation
Blocks (Subjects) because when Hy is true as pictured, the Y’s will
1: 2 3 4 tend to be at the extreme ends of the combined
—— sample (resulting in small and large Y ranks), so
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786 ciwpteR 14 Altemative Approaches to Inference
the sum of X ranks will result in a W value that is
neither large nor small. =
@@aribulion sips 40 44 48 49
ee ; Control 93.7 4.10 430 S156
ae distribution NUE EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE
Consult the Lehmann book (in the chapter bibli-
as ography) for more information on this test, called
anita a 4 tae @ a6 the Siegel-Tukey test.
42. The ranking procedure described in Exercise 41
Consider modifying the procedure for assigning is somewhat asymmetric, because the smallest
ranks as follows: After the combined sample of observation receives rank 1 whereas the largest
m + nobservations is ordered, the smallest obser- receives rank 2, and so on. Suppose both the
vation is given rank 1, the largest observation is smallest and the largest receive rank 1, the second
given rank 2, the second smallest is given rank 3, smallest and second largest receive rank 2, and so
the second largest is given rank 4, and so on. Then on, and let W" be the sum of the X ranks. The null
if H, is true as pictured, the X values will tend to distribution of W" is not identical to the null dis-
be in the middle of the sample and thus receive tribution of W, so different tables are needed.
large ranks. Let W! denote the sum of the X ranks Consider the case m = 3, n = 4. List all 35 possi-
and consider rejecting Ho in favor of H, when ble orderings of the three X values among the
w' > c. When Hg is true, every possible set of X seven observations (e.g., 1, 3, 7 or 4, 5, 6), assign
ranks has the same probability, so W' has the same ranks in the manner described, compute the value
distribution as does W when Hg is true. Thus c can of W" for each possibility, and then tabulate the
be chosen from Appendix Table A.13 to yield a null distribution of W”. For the test that rejects if
level % test. The accompanying data refers to w"” > c, what value of ¢ prescribes approximately
medial muscle thickness for arterioles from the a level .10 test? This is the Ansari-Bradley test;
lungs of children who died from sudden infant for additional information, see the book by Hol-
death syndrome (x’s) and a control group of chil- lander and Wolfe in the chapter bibliography.
dren (y's). Carry out the test of Ho versus H, at
level .05.