11 The study of nondestructive forces and stresses in materials furnishes important

Example 10.11 The study of nondestructive forces and stresses in materials furnishes important

  information for efficient engineering design. The article “Zero-Force Travel-Time Parameters for Ultrasonic Head-Waves in Railroad Rail” (Materials Evaluation, 1985: 854–858) reports on a study of travel time for a certain type of wave that results from longitudinal stress of rails used for railroad track. Three measurements were made on each of six rails randomly selected from a population of rails. The investigators used random effects ANOVA to decide whether some variation in travel time could be attributed to “between-rail variability.” The data is given in the accom- panying table (each value, in nanoseconds, resulted from subtracting 36.1 m’s from the original observation) along with the derived ANOVA table. The value f is highly

  significant, so H 0 :s 2 A 50 is rejected in favor of the conclusion that differences

  between rails is a source of travel-time variability.

  x i.

  Source of

  df Sum of

  Mean f

  EXERCISES Section 10.3 (22–34)

  22. The following data refers to yield of tomatoes (kgplot) for

  24. The accompanying summary data on skeletal-muscle CS

  four different levels of salinity. Salinity level here refers to

  activity (nmolminmg) appeared in the article “Impact of

  electrical conductivity (EC), where the chosen levels were

  Lifelong Sedentary Behavior on Mitochondrial Function of

  EC 5 1.6, 3.8, 6.0 , and 10.2 nmhoscm.

  Mice Skeletal Muscle” (J. of Gerontology, 2009: 927–939):

  Sedentary Active

  6.0: 51.7 48.8 53.9 49.0 Sample size

  10.2: 44.6 48.5 41.0 47.3 46.1 Sample mean

  Sample sd

  Use the F test at level a 5 .05 to test for any differences in true average yield due to the different salinity levels.

  Carry out a test to decide whether true average activity differs

  23. Apply the modified Tukey’s method to the data in Exercise

  for the three groups. If appropriate, investigate differences

  22 to identify significant differences among the m i ’ s.

  amongst the means with a multiple comparisons method.

  CHAPTER 10 The Analysis of Variance

  25. Lipids provide much of the dietary energy in the bodies of

  accompanying data on folacin content for randomly selected

  infants and young children. There is a growing interest in the

  specimens of the four leading brands of green tea.

  quality of the dietary lipid supply during infancy as a major determinant of growth, visual and neural development, and

  long-term health. The article “Essential Fat Requirements of

  Preterm Infants” (Amer. J. of Clinical Nutrition, 2000:

  245S–250S) reported the following data on total polyunsat-

  urated fats () for infants who were randomized to four dif-

  (Data is based on “Folacin Content of Tea,” J. of the Amer.

  ferent feeding regimens: breast milk, corn-oil-based

  Dietetic Assoc., 1983: 627–632.) Does this data suggest that

  formula, soy-oil-based formula, or soy-and-marine-oil-based

  true average folacin content is the same for all brands?

  formula:

  a. Carry out a test using

  a 5 .05 via the P-value method. b. Assess the plausibility of any assumptions required for

  Sample

  your analysis in part (a).

  c. Perform a multiple comparisons analysis to identify sig- nificant differences among brands.

  Breast milk

  8 43.0 1.5 28. For a single-factor ANOVA with sample sizes J

  i (i 5 1, 2, c I) , show that SSTr 5 g J i (X i 2X ) 2 5gJ i X i 2 2 nX 2

  14 43.5 1.2 where . n5gJ i

  29. When sample sizes are equal (J i 5J) , the parameters a 1 ,a 2 , ca I of the alternative parameterization are restricted

  a. What assumptions must be made about the four total

  by a i 50 . For unequal sample sizes, the most natural

  polyunsaturated fat distributions before carrying out a

  restriction is gJ i a i 50 . Use this to show that

  single-factor ANOVA to decide whether there are any differences in true average fat content?

  E(MSTr) 5 s 2 1 g J 2

  i a i

  b. Carry out the test suggested in part (a). What can be said

  I21

  about the P-value?

  What is E(MSTr) when H 0 is true? [This expectation is cor-

  26. Samples of six different brands of dietimitation margarine

  rect if gJ i a i 50 is replaced by the restriction ga i 50 (or

  were analyzed to determine the level of physiologically

  any other single linear restriction on the a i ’s used to reduce

  active polyunsaturated fatty acids (PAPFUA, in percent-

  the model to I independent parameters), but gJ i a i 50 sim-

  ages), resulting in the following data:

  plifies the algebra and yields natural estimates for the model

  Imperial

  14.1 13.6 14.4 14.3 parameters (in particular, a ˆ i 5x i 2x ) .]

  Parkay

  12.8 12.5 13.4 13.0 12.3 30. Reconsider Example 10.8 involving an investigation of the

  Blue Bonnet

  13.5 13.4 14.1 14.3 effects of different heat treatments on the yield point of steel

  16.8 17.2 16.4 17.3 18.0 a. If J58 and s51 , what is b for a level .05 F test when

  Fleischmann’s

  18.1 17.2 18.7 18.4 m 1 5m 2 ,m 3 5m 1 21 , and m 4 5m 1 11 ?

  b. For the alternative of part (a), what value of J is neces-

  (The preceding numbers are fictitious, but the sample means

  sary to obtain b 5 .05 ?

  agree with data reported in the January 1975 issue of

  c. If there are I55 heat treatments, J 5 10 , and s51 ,

  Consumer Reports.)

  what is b for the level .05 F test when four of the m i ’s are

  a. Use ANOVA to test for differences among the true aver-

  equal and the fifth differs by 1 from the other four?

  age PAPFUA percentages for the different brands.

  b. Compute CIs for all (m i 2m j )’s .

  31. When sample sizes are not equal, the noncentrality parame-

  c. Mazola and Fleischmann’s are corn-based, whereas the

  ter is gJ i a 2 i s 2 and f 2 5 (1I )g J i a 2 i s 2 . Referring to

  others are soybean-based. Compute a CI for

  Exercise 22, what is the power of the test when

  32. In an experiment to compare the quality of four different

  4 2 brands of magnetic recording tape, five 2400-ft reels of each

  [Hint: Modify the expression for V( ˆ u) that led to (10.5) in

  brand (A–D) were selected and the number of flaws in each

  the previous section.]

  reel was determined.

  27. Although tea is the world’s most widely consumed beverage

  A: 10

  after water, little is known about its nutritional value. Folacin

  B: 14 12 17

  is the only B vitamin present in any significant amount in tea, and recent advances in assay methods have made accurate

  C: 13 18 10 15 18

  determination of folacin content feasible. Consider the

  D: 17 16 12 22 14

  Supplementary Exercises

  It is believed that the number of flaws has approximately a

  since , m i 5 np i V(X ij )5s i 2 5 np i (1 2 p i )5m i (1 2 m i n) .

  Poisson distribution for each brand. Analyze the data at

  How should the X ij ’s be transformed so as to stabilize the vari-

  level .01 to see whether the expected number of flaws per

  ance? [Hint: g(m i )5m i (1 2 m i n) .]

  reel is the same for each brand.

  34. Simplify E(MSTr) for the random effects model when

  33. Suppose that X ij is a binomial variable with parameters n and p i

  J 1 5J 2 5 c5 J I 5J .

  (so approximately normal when np i 10 and nq i 10 ). Then