4 The accompanying data resulted from an experiment comparing the degree of soiling
Example 10.4 The accompanying data resulted from an experiment comparing the degree of soiling
for fabric copolymerized with three different mixtures of methacrylic acid (similar data appeared in the article “Chemical Factors Affecting Soiling and Soil Release from Cotton DP Fabric,” American Dyestuff Reporter, 1983: 25–30).
Let m i denote the true average degree of soiling when the mixture i is used (i 5 1, 2, 3)
. The null hypothesis H 0 :m 1 5m 2 5m 3 states that the true average degree of soiling
is identical for the three mixtures. Let’s carry out a test at significance level .01 to see
whether H 0 should be rejected in favor of the assertion that true average degree of soil-
ing is not the same for all mixtures. Since I2152 and I(J 2 1) 5 12 ,H 0 will be
rejected if fF .01,2,12 5 6.93 . Squaring each of the 15 observations and summing
gives g gx 2 5 (.56) ij 2 1 (1.12) 2 1 c 1 (.93) 2 5 12.1351 . The values of the three
sums of squares are
The computations are summarized in the accompanying ANOVA table. Because
f 5 .99 , 6.93, H 0 is not rejected at significance level .01. The mixtures appear to be
indistinguishable with respect to degree of soiling (F .10,2,12 5 2.81 1 P-value . .10) .
Sum of
f
Source of Variation
df Squares
Mean Square
EXERCISES Section 10.1 (1–10)
1. In an experiment to compare the tensile strengths of I55
2. Suppose that the compression-strength observations on the
different types of copper wire, J54 samples of each type
fourth type of box in Example 10.1 had been 655.1, 748.7,
were used. The between-samples and within-samples esti-
662.4, 679.0, 706.9, and 640.0 (obtained by adding 120 to
mates of s 2 were computed as MSTr 5 2673.3 and
each previous x 4j ). Assuming no change in the remaining
MSE 5 1094.2 , respectively.
observations, carry out an F test with a 5 .05 .
a. Use the F test at level .05 to test H 0 :m 1 5m 2 5m 3 5 3. The lumen output was determined for each of I53 different
m 4 5m 5 versus H a : at least two m i ’s are unequal.
brands of 60-watt soft-white lightbulbs, with J58 bulbs of
b. What can be said about the P-value for the test?
10.1 Single-Factor ANOVA
each brand tested. The sums of squares were computed as
mixtures. There were 26 measurements on concrete cylin-
SSE 5 4773.3 and SSTr 5 591.2 . State the hypotheses of
ders for each mixture; these were obtained 28 days after
interest (including word definitions of parameters), and use
casting. The entries in the accompanying ANOVA table are
the F test of ANOVA (a 5 .05) to decide whether there are
based on information in the article “In-Place Resistivity of
any differences in true average lumen outputs among the
Bridge Deck Concrete Mixtures” (ACI Materials J., 2009:
three brands for this type of bulb by obtaining as much infor-
114–122). Fill in the remaining entries and test appropriate
mation as possible about the P-value.
hypotheses.
4. It is common practice in many countries to destroy (shred) refrigerators at the end of their useful lives. In this process
Sum of
material from insulating foam may be released into the
Source
df Squares
Mean Square f
atmosphere. The article “Release of Fluorocarbons from Insulation Foam in Home Appliances during Shredding”
Mixture
(J. of the Air and Waste Mgmt. Assoc., 2007: 1452–1460)
Error
gave the following data on foam density (gL) for each of two
Total
refrigerators produced by four different manufacturers:
8. A study of the properties of metal plate-connected trusses
used for roof support (“Modeling Joints Made with Light-
Does it appear that true average foam density is not the same
Gauge Metal Connector Plates,” Forest Products J., 1979:
for all these manufacturers? Carry out an appropriate test of
39–44) yielded the following observations on axial-stiffness
hypotheses by obtaining as much P-value information as pos-
index (kipsin.) for plate lengths 4, 6, 8, 10, and 12 in:
sible, and summarize your analysis in an ANOVA table.
5. Consider the following summary data on the modulus of elas-
ticity ( 10 6 psi) for lumber of three different grades [in
close agreement with values in the article “Bending Strength
and Stiffness of Second-Growth Douglas-Fir Dimension
Lumber” (Forest Products J., 1991: 35–43), except that the
Does variation in plate length have any effect on true aver-
sample sizes there were larger]:
age axial stiffness? State and test the relevant hypotheses
using analysis of variance with
a 5 .01 . Display your results in an ANOVA table. [Hint: g g x 2 ij 5 5,241,420.79.]
i .
9. Six samples of each of four types of cereal grain grown in a
certain region were analyzed to determine thiamin content,
resulting in the following data (mgg):
Use this data and a significance level of .01 to test the null
Wheat 5.2 4.5 6.0 6.1 6.7 5.8
hypothesis of no difference in mean modulus of elasticity for
Barley 6.5 8.0 6.1 7.5 5.9 5.6
the three grades.
Maize 5.8 4.7 6.4 4.9 6.0 5.2
6. The article “Origin of Precambrian Iron Formations” (Econ.
Oats
Geology, 1964: 1025–1057) reports the following data on
Does this data suggest that at least two of the grains differ
total Fe for four types of iron formation (1 5 carbonate,
with respect to true average thiamin content? Use a level
2 5 silicate, 3 5 magnetite, 4 5 hematite ).
a 5 .05 test based on the P-value method. 1: 20.5 28.1 27.8 27.0 28.0 10. In single-factor ANOVA with I treatments and J observa-
25.2 25.3 27.1 20.5 31.3 tions per treatment, let m 5 (1I)g m i . 2: 26.3 24.0 26.2 20.2 23.7 a. Express E(X ) in terms of m. [Hint: X 2 .. 5 (1I)g X i . ] 34.0 17.1 26.8 23.7 24.9 b. Determine . E(X i ) [Hint: For any rv Y, E(Y 2 )5
3: 29.5 34.0 27.5 V(Y) 1 [E(Y)] 29.4 2 27.9 2 .] 26.2 29.9 29.5 30.0 35.6 c. Determine . E(X )
4: 36.5 44.2 34.1 30.3 31.4 d. Determine E(SSTr) and then show that
2 E(MSTr) 5 s J 1 g (m
Carry out an analysis of variance F test at significance level .01, and summarize the results in an ANOVA table.
e. Using the result of part (d), what is E(MSTr) when H 0 is
7. An experiment was carried out to compare electrical resis-
true? When H 0 is false, how does E(MSTr) compare to
tivity for six different low-permeability concrete bridge deck
s 2 ?
CHAPTER 10 The Analysis of Variance