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# Mathematical Statistics with Applications, Seventh Edition

www.downloadslide.com Exercises 775 a b Is there sufficient evidence to indicate a difference in cadmium accumulation in plants grown in plots subjected to different levels of sludge application? Bound or determine the approximate p-value. What would you conclude at the α = .01 significance level? 15.39 Corrosion of metals is a problem in many mechanical devices. Three sealants used to help retard the corrosion of metals were tested to see whether there were any differences among them. Samples from ten different ingots of the same metal composition were treated with each of the three sealants, and the amount of corrosion was measured after exposure to the same environmental conditions for 1 month. The data are given in the accompanying table. Is there any evidence of a difference in the abilities of the sealants to prevent corrosion? Test using α = .05. Sealant Ingot I II III 1 4.6 4.2 4.9 2 7.2 6.4 7.0 3 3.4 3.5 3.4 4 6.2 5.3 5.9 5 8.4 6.8 7.8 6 5.6 4.8 5.7 7 3.7 3.7 4.1 8 6.1 6.2 6.4 9 4.9 4.1 4.2 10 5.2 5.0 5.1 15.40 A serious drought-related problem for farmers is the spread of aflatoxin, a highly toxic substance caused by mold, which contaminates field corn. In higher levels of contamination, aflatoxin is hazardous to animal and possibly human health. (Officials of the FDA have set a maximum limit of 20 parts per billion aflatoxin as safe for interstate marketing.) Three sprays, A, B, and C, have been developed to control aflatoxin in field corn. To determine whether differences exist among the sprays, ten ears of corn are randomly chosen from a contaminated corn field, and each is divided into three pieces of equal size. The sprays are then randomly assigned to the pieces for each ear of corn, thus setting up a randomized block design. The accompanying table gives the amount (in parts per billion) of aflatoxin present in the corn samples after spraying. Use the Friedman test based on F r to determine whether there are differences among the sprays for control of aflatoxin. Give approximate bounds for the p-value. Spray Spray Ear A B C Ear A B C 1 21 23 15 6 5 12 6 2 29 30 21 7 18 18 12 3 16 19 18 8 26 32 21 4 20 19 18 9 17 20 9 5 13 10 14 10 4 10 2 15.41 A study was performed to compare the preferences of eight “expert listeners” regarding 15 models (with approximately equal list prices) of a particular component in a stereo system. Every effort was made to ensure that differences perceived by the listeners were due to the component of interest and no other cause (all other components in the system were identical,

776 Chapter 15 Nonparametric Statistics www.downloadslide.com the same type of music was used, the music was played in the same room, etc.). Thus, the results of the listening test reflect the audio preferences of the judges and not judgments regarding quality, reliability, or other variables. Further, the results pertain only to the models of the components used in the study and not to any other models that may be offered by the various manufacturers. The data in the accompanying table give the results of the listening tests. The models are depicted simply as models A, B, ..., O. Under each column heading are the numbers of judges who ranked each brand of component from 1 (lowest rank) to 15 (highest rank). Rank Model 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 A 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8 B 0 0 0 1 0 2 1 1 1 0 0 0 0 2 0 C 0 1 1 1 4 0 0 1 0 0 0 0 0 0 0 D 1 0 1 1 0 1 0 0 1 0 1 0 1 1 0 E 0 2 1 3 0 2 0 0 0 0 0 0 0 0 0 F 0 0 0 0 0 0 0 0 1 2 2 3 0 0 0 G 0 0 0 0 0 0 0 0 0 1 0 2 4 1 0 H 1 2 1 1 0 0 2 1 0 0 0 0 0 0 0 I 3 2 1 0 0 0 0 0 0 1 0 1 0 0 0 J 0 0 1 0 2 0 2 0 0 0 2 0 1 0 0 K 0 0 0 0 0 0 1 1 0 2 1 1 1 1 0 L 0 0 0 0 0 0 1 1 4 0 1 0 1 0 0 M 1 1 2 1 1 2 0 0 0 0 0 0 0 0 0 N 2 0 0 0 0 0 1 1 0 0 0 1 0 3 0 O 0 0 0 0 1 1 0 2 1 2 1 0 0 0 0 a Use the Friedman procedure to test whether the distributions of the preference scores differ in location for the 15 component models. Give bounds for the attained significance level. What would you conclude at the α = .01 level of significance? [Hint: The sum of the ranks associated with the component of model O is 5 + 6 + 8 + 8 + 9 + 10 + 10 + 11 = 67; other rank sums can be computed in an analogous manner.] b If, prior to running the experiment, we desired to compare components of models G and H, this comparison could be made by using the sign test presented in Section 15.3. Using the information just given, we can determine that model G was preferred to model H by all eight judges. Explain why. Give the attained significance level if the sign test is used to compare components of models G and H. c Explain why there is not enough information given to use the sign test in a comparison of only models H and M. 15.42 An experiment is conducted to investigate the toxic effect of three chemicals, A, B, and C, on the skin of rats. Three adjacent 1 -inch squares are marked on the backs of eight rats, and 2 each of the three chemicals is applied to each rat. The squares of skin on each rat are ranked according to severity of irritation (1 = least severe, 3 = most severe). The resulting data are given in the accompanying table. Is there sufficient evidence to support the research hypothesis that the probability distributions of skin irritation scores corresponding to the three chemicals differ in location? Use α = .01. (Note: Ranking the severity of reactions to the chemicals for each rat is probably much more meaningful than assigning an arbitrary “irritation score” to each portion of skin.)

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