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2. Use the Wilcoxon test to analyze the data given in 3 ;
Example 9.9. certified to work on a certain type of car was
selected, and the time (in minutes) necessary for
3. The accompanying data is a subset of the data re- each one to diagnose a particular problem was
ported in the article “Synovial Fluid pH, Lactate, determined, resulting in the following data:
Oxynen and Carbon. Dicwide Partial Pressure aoe s94 1536 267 201. 25K 350 308
Various Joint Diseases” (Arthritis Rheum., 1971: 319 532 125 232 88 249 302
476-477). The observations are pH values of syno-
vial fluid (which lubricates joints and tendons) taken Use the Wilcoxon test at significance level .10 to
from the knees of individuals suffering from arthri- decide whether the data suggests that true average
tis. Assuming that true average pH for non-arthritic diagnostic time is less than 30 minutes.
individuals is 7.39, test at level .05 to see whether the 5, Bort a gravimetric and a spectrophotometric method
data indicates a difference between average pH are under consideration for determining phosphate
values for arthritic and nonarthritic individuals. content of a particular material. Twelve samples of
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14.1 The Wilcoxon Signed-Rank Test 765
the material are obtained, each is split in half, and a correction for the variance which can be found
determination is made on each half using one of the in books on nonparametric statistics.]
two methods, resulting in the following data: ¢. A particular type of steel beam has been de-
signed to have a compressive strength (Ib/in?)
Sample 1 2 3 4 of at least 50,000. An experimenter obtained a
Gravimetric 547 585 668 461 random sample of 25 beams and determined the
strength of each one, resulting in the following
Spectrophotometric | 55.0 55.7 62.9 45.5 data (expressed as deviations from 50,000):
Sample 5.6 7 8 -10 -27 36 -S5 73-77 -81
Gravimetric 523 743 92.5 40.2 50, SS 99 NG 027 18 1RG
—150 —155 -159 165 —178 —183 —192
Spectrophotometric | 51.1 75.4 89.6 38.4 -199 -212 —217 -229
Sample 9 10 I 2 Carry out a test using a significance level of
Giaviinétiic 873 748 632 685 approximately .01 to see if there is strong evi-
dence that the design condition has been violated.
Spectrophotometric | 86.8 72.5 62.3 66.0 7 tye accompanying 25 observations on’ fracture
Use the Wileoxon test to decide whether one tech. ughness of base plate of 18% nickel maraging
nique gives on average a different value than the steel were reported in the article “Fracture Testing
3 . . of Weldments” (ASTM Special Publ. No. 381,
other technique for this type of material.
1965: 328-356). Suppose a company will agree to
6. The signed-rank statistic can be represented as purchase this steel for a particular application only
Si =Wi+W2+-+-+W,, where W; =i if the if it can be strongly demonstrated from experimen-
sign of the x; — fo with the ith largest absolute tal evidence that true average toughness exceeds
magnitude is positive (in which case / is included 75. Assuming that the fracture toughness distribu-
inS,) and W; = Oif this value is negative (= 1,2, tion is symmetric, state and test the appropriate
3,..., n). Furthermore, when Ho is true, the W;’s hypotheses at level .05, and compute a P-value.
are independent and P(W = i) = P(W = 0) =.5. [Hint: Use Exercise 6(b).]
a. Use these facts to obtain the mean and variance
of S,, when Ho is true. [Hint: The sum of the first 69.5 71.9 72.6 73.1 73.3 73.5 74.1 74.2 75.3
n positive integers is n(n + 1)/2, andthe sum of 75.5 75.7 75.8 76.1 76.2 76.2 76.9 77.0 77.9
the squares of the first n positive integers is 7g) 796 79.7 80.1 82.2 83.7 93.7
n(n + 1)(2n + 1)/6.]
b. The W;’s are not identically distributed (e.g., §, Suppose that observations X;, X>, ... , X, are made
possible values of W are 2 and 0 whereas pos- on a process at times 1, 2, . ..,2. On the basis of this
sible values of Ws are 5 and 0), so our Central data, we wish to test
Limit Theorem for identically distributed and
independent variables cannot be used here Ho: the X;’s constitute an independent and iden-
when n is large. However, a more general CLT tically distributed sequence
can be used to assert that when Hp is true and Weiss
n > 20, Sy has approximately a normal distri-
bution with mean and variance obtained in (a). Ha? Xie1 tends to be larger than X; fori = 1,....7
Use this to propose a large-sample standardized (an increasing trend)
signed-rank test statistic and then an appropriate Suppose the X/'s are ranked from 1 ton. Then when Hy
rejection region with level x for each of the three; sue, larger ranks tend to occur later in the sequence,
commonly encountered altetative hypotheses: whereas if Ho is true, large and small ranks tend
[Note: When there are ties in the absolute mag- to besmixed together. Tet R; be the rank of X;
nitudes, it is still correct to standardize S, by and consider the test statistic D = 7", (Ry — i)”.
subtracting the mean from (a), but there is a tek
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766 = cuarrer 14 Alternative Approaches to Inference
Then small values of D give support to H, (e.g., the to .10 as possible in the case n = 4. [Hint: List the 4!