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