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A computer scientist is studying four different algorithms used to perform numerical integration. The speed (in seconds) with which a problem can be solved is measured. Each algorithm is to be used to solve each of 10 different problems, and hence a problem serves as a block. Algorithm Problem II III IV 10 .11 .16 .10 11 .13 .17 .09 13 .14 .17 . 12 08 .11 .13 .09 15 .16 .19 .16 . 18 23 .16 05 .07 .11 07 .13 .15 .18 .14 .12 .14 .15 .12 .16 .15 .20 .15 If the Bonferroni technique is used to compare all pairs of means, how many comparisons are possible? If each is done at the o =.01 level, what is an upper bound for o', the overall probability of committing at least one Type I error?A computer scientist is studying four different algorithms used to perform numerical integration. The speed (in seconds) with which a problem can be solved is measured. Each algorithm is to be used to solve each of 10 different problems, and hence a problem serves as a block. Algorithm Problem II III IV 10 .11 .16 10 11 .13 .17 .09 13 .14 .17 . 12 .11 .13 09 .16 .19 .16 18 .16 .07 11 .07 13 . 15 . 18 .14 .12 .14 .15 .12 .16 .15 .20 .15 Use Bonferroni 7 tests with o = .01 to determine whether there appears to be a "fastest" algorithm.Give an example of a 4 x 4 Latin square that is different from that given in Table 13.7. Give the data layout for your design. TABLE 13.7 Factor (bus) Factor (tire position) II III IV AThe design given in Table 13.12 is called a knight's move Latin square because one can get from each treatment type to another of the same type by making a knight's move as in the game of chess. The observed responses are given in parentheses in the table. Use these data to test Ho: P1.. = M2.. = My.. = /4.. at the a = .05 level. TABLE 13.12 A knight's move Latin square H II III IV A (3) B (42) C (9) D (5.3) C(1.0) D (5.6) A (3.8) B (4.3) B(45) A (3.5) D (5.7) C(1.4) D (5.5) C(8) B (3.9) A (3.7)In a study to compare the average gasoline mileage of three different grades of a particular brand of gasoline, two extraneous sources of variation are recognized. These are the driver and the type of car used in the road trials. Three drivers and three new automobiles are obtained. Table 13.13 gives the mileage obtained over each test run. Test the null hypothesis of equal means among grades of gasoline at the .05 level. TABLE 13.13 Gasoline mileage obtained controlling for driver and automobile type (A = regular, B = extra, C = super) Factor (auto type) 4 cylinder 6 cylinder 8 cylinder A (36.0) B (33.0) C (26.5) WN- B (36.5) C (33.5) A (25.0) C (38.0) A (32.5) B (26.0)A company is considering three word processors for use by their secretarial staff. All of them are acceptable, so the choice depends on which one is the easiest to learn. Since speed is affected by the type of article being produced and the learner, a Latin square design is used. The data obtained are given in Table 13.14. Test TABLE 13.14 Time required to learn various subroutines in a word processor (in hours) Factor (type of article) I prose II technical III report B (5) A (12) C (LO) C (12) B(LO) A (95) A (.8) C (1.7) B (9)Consider the one-way classification model YEutatE i = 1, 2, 3; j = 1,2 For this model, find Y, a, X, and E. Find XX, and show that this matrix has no inverse.Consider the randomized complete block model i = 1, 2, 3: j = 1, 2, 3, 4 For this model, find Y, a, X, and E. Find XX, and show that this matrix has no inverse.Following a major accidental spill from a chemical manufacturing plant near a river, a study was conducted to determine whether certain species of fish caught from the river differ in terms of the amounts of the chemical absorbed. If differences are found, regulations on human consumption may be recommended. Samples from catches of three major species were measured in parts per million. The resulting data are given below: Species A B C 18.1 29.1 26.6 16.5 15.8 16.1 21.0 20.4 18.8 18.7 23.5 25.0 7.4 18.5 21.8 12.4 21.3 15.4 16.1 23.1 19.9 17.9 23.8 15.5 20.1 21.1 11.9 25.5 Test whether the median amounts of chemical absorbed by the three species of fish differ at the 05 level of significance.A laboratory manager plans to purchase machines used to analyze blood samples. Five types of machines are being considered for purchase. After trial use, each of the eight technicians is asked to rank the machines in order of preference, with a rank of 1 being assigned to the machine most preferred. The respective rankings were as follows: Machine Technician I II III IV WNAUNUNN NAUPONWU Use the Friedman test to determine whether the group of technicians rate the machines differently at the .10 level of significance.Carbon dioxide is known to have a critical effect on microbiological growth. Small amounts of CO, stimulate growth of some organisms, whereas high concentrations inhibit the growth of most. The latter effect is used commercially when perishable food products are stored. A study is conducted to investigate the effect of CO, on the growth rate of Pseudomonas fragi, a food- spoiling organism. Carbon dioxide is administered at five predetermined different atmospheric pressures. The response measured was the percentage change in cell mass after a 1-hour growing time. Ten cultures were used at each atmospheric pressure level, resulting in the following data: Factor level (CO, pressure) 0.0 -083 .29 .50 .86 62.6 50.9 45.5 29.5 24.9 59.6 44.3 41.1 22.8 17.2 64.5 47.5 29.8 19.2 7.8 59.3 49.5 38.3 20.6 10.5 58.6 48.5 40.2 29.2 17.8 64.6 50.4 38.5 24.1 22.1 50.9 35.2 30.2 22.6 22.6 56.2 49.9 27.0 32.7 16.8 32.3 42.6 40.0 24.4 15.9 62.8 41.6 33.9 29.6 8.8 State the assumptions required to test the null hypothesisStudies are conducted to investigate the use of slag in road pavement base, subbase, and surfacing. The composition of slag obtained from various sources is of interest. These data are obtained on the percentage of S, O, in samples from five different sources (based on information from "Steel Plant Slag in Road Pavements," B. S. Heaton, Australian Civil Engineering Transactions, March 1993, pp. 49-53): Source Blast furnace Steel Portland Fly Natural slag slag cement ash basalt 35.1 16.0 20.1 58.3 45.9 34.7 15.6 22.0 57.6 46.3 34.8 17.2 23.1 55.0 44.5 33.2 16.2 19.7 60,1 45.2 33.6 16.3 19.5 61,2 44.5 36.8 14.7 16.2 58.2 49.6 Analyze these data, and write a report summarizing your results. Be sure to defend your choice of test statistics.It is well known that power surges or line "spikes" can damage sensitive electronic equipment A study of the surges was conducted to ascertain whether or not there are differences in the average frequency of the surges among the seven days of the week. Over a randomly selected ten-week period, the number of spikes was observed for ten 24-hour periods for each of the seven days in the week. The resulting data are as follows: Mon. Tues. Wed. Thurs. Fri. Sat. Sun. 25 24 16 22 33 28 35 21 20 20 17 28 20 20 19 19 19 44 31 29 20 16 21 21 33 21 19 21 21 17 19 33 26 19 13 13 36 22 23 24 23 27 28 22 26 21 15 23 15 27 20 30 18 20 23 20 22 22 29 22 L 14 23 20 16 26 29 Using analysis of variance, state the correct hypothesis for testing equality of means and conduct the appropriate ANOVA test. State your conclusion.It is well known that power surges or line "spikes" can damage sensitive electronic equipment A study of the surges was conducted to ascertain whether or not there are differences in the average frequency of the surges among the seven days of the week. Over a randomly selected ten-week period, the number of spikes was observed for ten 24-hour periods for each of the seven days in the week. The resulting data are as follows: Mon. Tues. Wed. Thurs. Fri. Sat. Sun. 25 24 16 22 33 28 35 21 2:0 20 17 28 20 20 19 19 19 44 31 29 20 16 21 21 33 21 19 21 21 17 19 33 26 20 19 13 13 36 22 23 25 24 23 27 28 26 21 15 23 15 27 20 30 18 20 23 20 22 22 29 22 L 14 23 20 16 26 29 Repeat Exercise 65 using Tukey's test. Do you get different results in Exercises 65 and 66? If so, explain the reasons for these differences
