8 The article “Some Mechanical Properties of Impregnated Bark Board” (Forest

Example 15.8 The article “Some Mechanical Properties of Impregnated Bark Board” (Forest

  Products J., 1977: 31–38) reports the following data on maximum crushing strength (psi) for a sample of epoxy-impregnated bark board and for a sample of bark board impregnated with another polymer:

  Epoxy (x’s)

  Other ( y’s)

  CHAPTER 15 Distribution-Free Procedures

  Let’s obtain a 95 CI for the true average difference in crushing strength between the epoxy-impregnated board and the other type of board.

  From Appendix Table A.16, since the smaller sample size is 5 and the larger sample size is 6, c 5 26 for a confidence level of approximately 95. The d ij ’s

  appear in Table 15.5. The five smallest d ij ’s [d ij(1) , c, d ij(5) ] are 4350, 4470, 4610,

  4730, and 4830; and the five largest d ij ’s are (in descending order) 9790, 9530, 8740, 8480, and 8220. Thus the CI is (d ij(5) ,d ij(26) ) 5 (4830, 8220) .

  Table 15.5 Differences for the Rank-Sum Interval in Example 15.8 y j

  When m and n are both large, the Wilcoxon test statistic has approximately a normal distribution. This can be used to derive a large-sample approximation for the value c in interval (15.12). The result is

  mn

  mn(m 1 n 1 1)

  c<

  1z

  2 a2 B (15.13) 12

  As with the signed-rank interval, the rank-sum interval (15.12) is quite effi- cient with respect to the t interval; in large samples, (15.12) will tend to be only a bit longer than the t interval when the underlying populations are normal and may be considerably shorter than the t interval if the underlying populations have heavier tails than do normal populations.

  EXERCISES Section 15.3 (17–22)

  17. The article “The Lead Content and Acidity of

  and then one segment was randomly selected for applica-

  Christchurch Precipitation” (N. Zeal. J. of Science, 1980:

  tion of the first solvent, with the other segment receiving

  311–312) reports the accompanying data on lead concen-

  the second solvent.

  tration (mgL) in samples gathered during eight different summer rainfalls: 17.0, 21.4, 30.6, 5.0, 12.2, 11.8, 17.3, and 18.8. Assuming that the lead-content distribution is

  Log

  symmetric, use the Wilcoxon signed-rank interval to

  obtain a 95 CI for m.

  18. Compute the 99 signed-rank interval for true average pH m (assuming symmetry) using the data in Exercise

  15.3. [Hint: Try to compute only those pairwise averages

  Calculate a CI using a confidence level of roughly 95 for

  having relatively small or large values (rather than all 105

  the difference between the true average amount extracted

  averages).]

  using the first solvent and the true average amount extracted

  19. An experiment was carried out to compare the abilities of

  using the second solvent.

  two different solvents to extract creosote impregnated in

  20. The following observations are amounts of hydrocarbon

  test logs. Each of eight logs was divided into two segments,

  emissions resulting from road wear of bias-belted tires

  15.4 Distribution-Free ANOVA

  under a 522 kg load inflated at 228 kPa and driven at

  21. Compute the 90 rank-sum CI for m 1 2m 2 using the data

  64 kmhr for 6 hours (“Characterization of Tire Emissions

  in Exercise 10.

  Using an Indoor Test Facility,” Rubber Chemistry and

  22. Compute a 99 CI for m 1 2m 2 using the data in

  Technology, 1978: 7–25): .045, .117, .062, and .072. What

  Exercise 11.

  confidence levels are achievable for this sample size using the signed-rank interval? Select an appropriate confidence level and compute the interval.