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nonparametric statistical inference

Questions and Answers of Nonparametric Statistical Inference

=+12.13. Suppose in Problem 11.15 that an independent group of female consumers also ranked the products as follows:
=+12.12. A town has 10 different supermarkets. For each market, data are available on the following three variables: X1¼ food sales, X2 ¼ nonfood sales, and X3 ¼ size of store in thousands of
=+12.11. Show that if m ¼ n and l ¼ k in (5.3) so that the design is complete, then (5.3) is equivalent to Q ¼ 12S=kn(n þ 1), as it should be.Taster Varieties presented 1 ABD 2 BCE 3 CDF 4 DEG 5
=+(c) Will it be difficult to decide which students should be given a passing grade?
=+(b) Give an overall estimate of the relative performance of each student.
=+12.10. Ten graduate students take identical comprehensive examinations in their major field. The grading procedure is that each professor ranks each student’s paper in relation to all others
=+The ranks resulting from the three repetitions are shown below with each rank corresponding to the variety shown above. For example, the rank 3 by taster 12 is for variety E. Determine whether
=+presented together three times, to a group of seven of the tasters, with the design shown below used each of the three times.Student Score 1 2 3 4 5 6 7 8 Verbal 90 60 45 48 58 72 25 85
=+12.9. A manufacturer of ice cream carried out a taste preference test on seven varieties of ice cream, denoted by A, B, C, D, E, F, G. The subjects were a random sample of 21 tasters and each
=+comfort, and performance. The scores of the six best models are shown below. Determine whether the median scores are the same. Which model(s) would you buy, on the basis of this report?
=+12.8. Automobile Magazine (July 1989) published results of a comparison test of 15 brand models of $20,000 sedans. Each car was given a subjective score out of a possible 60 points (60 ¼ best) on
=+12.7. Eight students are given examinations on each of verbal reasoning, quantitative reasoning, and logic. The scores range from 0 to 100, with 100 a perfect score. Use the data below to find the
=+Table 2 Data for Problem 12.6 Group Mean age Optimism index Cynicism index Owners=managers 60 þ 55 1.1 Corporate executives 44 59 1.4 Middle managers 34 41 1.4 MBA students 25 30 1.8
=+optimism and cynicism but also noted a relation to organizational status of respondents as measured by age. Use the data in Table 2 to determine whether the relationship between optimism and
=+an optimism index, where larger numbers indicate an optimistic feeling about current and future ethical conditions, and a cynicism index that measures how subjects felt others would behave relative
=+Three more questions, also multiple choice, related to how subjects themselves would react to the same problematic situations. These answers were used to develop
=+school students to executives, were asked to respond to a questionnaire with general statements about ethics. Two questions related to subjects’ degree of agreement (5-point scale) with general
=+12.6. Webber (1990) reported results of a study to measure optimism and cynicism about the business environment and ethical trends. Subjects, ranging from high Contestant Judge A B C D E F G H 1
=+(b) Determine the relationship when the effect of technological alienation is removed.
=+present in persons who regard themselves as technologically alienated. The pretest scores are shown in Table 1 for 14 students, with larger scores indicating greater amounts of the trait.(a)
=+12.5. Howard, Murphy, and Thomas (1986) (see Problems 5.12 and 8.8) also wanted to determine whether there is a direct relationship between computer anxiety and math anxiety. Even though the two
=+(b) Calculate (6.2) for these same data to verify that it is an equivalent expression.
=+12.4. Given the following triplets of rankings of six objects:(a) Calculate the Kendall coefficient of partial correlation between X and Y from (6.1) and test for independence.
=+(b) Calculate the coefficient of rank correlation for each of the three pairs of rankings and verify the relation between rav and W given in (4.7).(c) Estimate the true preferential order of
=+12.2. A beauty contest has eight contestants. The three judges are each asked to rank the contestants in a preferential order of pulchritude. The results are:(a) Calculate Kendall’s coefficient
=+As a measure of the association between attitude and number of years of schooling completed:ðaÞ Compute Kendall’s tau with correction for ties.ðbÞ Compute Spearman’s R with correction for
=+11.18. Twenty-three subjects are asked to give their attitude toward elementary school integration and their number of years of schooling completed. The data are shown below.
=+ðaÞ Find the rank correlation coefficient and the appropriate one-tailed P value.ðbÞ Find the Kendall tau coefficient and the appropriate one-tailed P value.
=+11.17. A scout for a professional baseball team ranks nine players separately in terms of speed and power hitting, as shown below.
=+11.16. Derive the null distribution of both Kendall’s tau statistic and Spearman’s rho for n ¼ 3 assuming no ties.Minutes between rest periods Pounds lifted per minute 5.5 350 9.6 230 2.4 540
=+11.15. A board of marketing executives ranked 10 similar products, and an ‘‘independent’’ group of male consumers also ranked the products. Use two different nonparametric procedures to
=+ðdÞ Find the appropriate one-tailed P value for your result in ðcÞ.
=+ðcÞ Find the Kendall tau coefficient.
=+ðbÞ Find the appropriate one-tailed P value for your result in ðaÞ.
=+11.14. Six randomly selected mice are studied over time and scored on an ordinal basis for intelligence and social dominance. The data are as follows:ðaÞ Find the coefficient of rank correlation.
=+11.13. Given a single series of time-ordered ordinal observations over several years, name some nonparametric procedures that could be used and how in order to detect a long-term positive trend.
=+11.12. Ten randomly chosen male college students are used in an experiment to investigate the claim that physical strength is decreased by fatigue. Describe the relationship for the data below,
=+Assuming that these test scores are the primary criterion for hiring, do you think that over this time period the screening procedure has changed, or the personnel agent has changed, or supply has
=+11.11 A company has administered a screening aptitude test to 20 new employees over a two-year period. The record of scores and date on which the person was hired are shown below.
=+(c) Compare and interpret the results of ðaÞ and ðbÞ.
=+The average results for each day are as follows:Use an appropriate test to determine whether these data exhibit some sort of pattern.Find the P value:(a) Using tests based on runs with both the
=+11.10. The rainfall measured by each of 12 gauges was recorded for 20 successive days.
=+11.8. For the time series data in Example 4.1 of Chapter 3, use the Mann test based on Spearman’s rank correlation coefficient to see if the data show a positive trend.Industry Japan United
=+11.7. The World Almanac and Book of Facts published the following divorce rates per 1000 population in the United States. Determine whether these data show a positive trend using four different
=+ðaÞ Use the signed-rank test to determine whether Japan spends a larger percentage than the United States on R&D.ðbÞ Determine whether there is a significant positive relationship between
=+ when these figures are looked at separately according to industry, the following data from Mansfield (1989) show some large differences.
=+11.6. Company-financed expenditures in manufacturing on research and development (R&D) are currently about 2.7 percent of sales in Japan and 2.8 percent of sales in the United States. However,
=+whether there is a direct relationship between number of Hispanics and percent of state population for the nine states below.Contestant Judge A B C D E F G H 1 2135487 6 2 1245768 3 State Hispanics
=+11.5. The Census Bureau reported that Hispanics are expected to overtake blacks as the largest minority in the United States by the year 2030. Use two different tests to see
=+the sample indicators are (i) sample numbers 1 and 2, (ii) 1 for the first sample and 0 for the second sample as in the Z vector of Chapter 7.
=+in ascending order of magnitude and (a) assigning ranks, (b) assigning sample indicators. Show that Kendall’s tau, calculated for these pairs without a correction for ties, is linearly related
=+11.3. Two independent random samples of sizes m and n contain no ties. A set of m þ n paired observations can be derived from these data by arranging the combined samples
=+11.2. Verify the result given in (4.9).
=+Spearman rank-correlation-coefficient procedures:ðaÞ Calculate the measure of association.ðbÞ Test the null hypothesis that the judges ranked the contestants independently (use tables).ðcÞ
=+11.1. A beauty contest has eight contestants. The two judges are each asked to rank the contestants in a preferential order of pulchritude. The results are shown in the table.Answer parts (a) and
=+10.23. Verify the value of the Kruskal-Wallis test statistic given in the SAS solution to Example 2.1 in Chapter 8.
=+Find an approximate P value to test whether the median smooth- ness can be regarded as the same for all laboratories. Laboratory Data ABCD 38.7 41.5 43.8 44.5 45.5 39.2 39.3 39.7 41.4 41.8 34.0
=+10.22. Below are four sets of five measurements, each set an array of data (for your convenience) on the smoothness of a certain type of paper, each set obtained from a different laboratory.
=+Test the null hypothesis that taste preference is the same for each cereal. Number who would buy Number who would not buy Cereal A B C D 785 75 50 69 80 57 80 60 43 70
=+presented to all members of each panel. After testing, each person is asked if he would purchase the product. The results are shown below.
=+10.21. A company is testing four cereals to determine taste preferences of potential buyers. Four different panels of persons are selected independently; one cereal is
=+10.20. An office has three computers, A, B, and C. In a study of computer usage, the firm has kept records on weekly use rates for 7 weeks, except that computer A was out for repairs for part of 2
=+ Use the composite scores below to test the null hypothesis that mean scores for all three chefs are the same. Chef A Chef B Chef C 4.05 4.35 2.24 5.04 3.88 3.93 3.45 3.02 3.37 3.57 4.56 3.21 4.23
=+10.19. Eighteen fish of a comparable size in a particular variety are divided randomly into three groups and each group is prepared by a different chef using the same recipe. Each prepared fish is
=+ De- termine whether median education level is the same for the three groups at = 0.05. Education Insurance Transportation Media 10 19 31 33 20 20 37 34 30 36 20 21 40 25 12 12 Total 100 100 100
=+10.17. Random samples of 100 insurance company executives, 100 transportation company executives, and 100 media company executives were classified according to highest level of formal education
=+10.16. Prior to the Alabama-Auburn football game, 80 Alabama alumni, 75 Auburn alumni, and 45 residents of Tuscaloosa who are not alumni of either are asked who they think will win the game. The
=+Three statements on the questionnaire related to the social dimension were (1) most advertising insults the intelligence of the average consumer; (2) advertising often persuades people to buy
=+10.15. Andrews (1989) examines attitudes toward advertising by undergraduate marketing students at universities in six different geographic regions. Attitudes were measured by answers to a
=+Determine whether the data in Table 2 support this theory. Table 2 Chimp Training Object Fabric Food Whiskey Liza Opal Frieda 21 2222 20 23 18 2222 22 19 20 21 229 22 18 22 13 18 15 19 19
=+ Then the chimpanzees were tested on their ability to transfer the learning to three kinds of novel items, classified as Objects, Fabrics, and Food. The data were recorded as number of correct
=+10.14. A matching-to-sample (MTS) task is used by psychologists to understand how other species perceive and use identity relations. A standard MTS task consists of having subjects observe a sample
=+Information was presented to these subjects in tabular form about the number of physicians who practice in six different states. Recall was measured by how accurately the subjects were able to rank
=+ 10.13. Many psychologists have developed theories about how different kinds of brain dominance may affect recall ability of information presented in various for- mats. Brown and Evans (1986)
=+10.12. Struckman-Johnson (1988) surveyed 623 students in a study to compare the proportions of men and women at a small midwestern university who have been coerced by their date into having sexual
=+ 10.11. In the context of the Jonckheere-Terpstra test discussed in Section 10.6, show that under Ho for i Hint: 2 cov(Uij, Uir)=var(U, Uir)-var(U)+var(Uir) var(Uij+r)-var(U)+var(Uir), where Ujr is
=+ Enu- merate the probability distribution of V when (i) n = n2=n3 = 3 and (ii) n = 3,n2 = n3 = 2.
=+10.10. Show that under Ho: 01 02 == 0, the distribution of V, the sum test statistic for comparisons with a control when 01 is known in the case of unequal sample sizes defined in Section 10.7, is
=+10.9. In the context of the k-sample control median test defined in Section 10.3, show that for any i(= 2.3..... k) and j(= 0.1.....q) the random variable Vij has a binomial distribution. What are
=+10.7. Write the k-sample median test statistic given in (2.2) in the form of (5.1) (cf. Problem 7.2).
=+ratio as a function of H in the form given in (4.3) or (4.2) to show that F = k-1 (N This is an example of what is called a rank transform statistic. For related interesting results, see for
=+10.6. Show that H is equivalent to the F test statistic in one-way analysis-of-variance problem if applied to the ranks of the observations rather than the actual numbers. Hint: Express the F
=+10.5. Show that H with k = 2 is exactly equivalent to the large-sample approximation to the two-sample Wilcoxon rank-sum test statistic mentioned in Section 8.2 with m = n, n = n2.
=+10.4. Verify that the form of H given in (4.4) is algebraically equivalent to (4.2).
=+ If the two tests reject for the largest values of Q and H, respectively, which test seems to distinguish better between extreme arrangements?
=+ 10.3. By enumeration, place the median test criterion (U1, U2. Us) and the H test criterion (R1, R2, R3) in one-to-one correspondence for the same k and n; as in Problems 10.1 and 10.2.
=+Find the critical region which consists of those rank sums R1, R2, R3 which have the largest value of H and find exact a.
=+10.2. Generate the exact distribution of the Kruskal-Wallis statistic H for the same k and n, as in Problem 10.1.
=+Compute the values of the Q statistic for all arrangements and compare that critical region for the same value of a with the region R.
=+10.1. Generate by enumeration the exact null probability distribution of the k-sample median test statistic for k = 3,n = 2, ng =1,n3 = 1. If the rejection region consists of those arrangements
=+(a) Find a one-tailed P value using the Siegel-Tukey test and state all assumptions needed for an exact P.(b) Find the one-tailed P value using the Wilcoxon procedure assuming the population
=+We want to investigate the relative variability between the two lines.Class 1 Class 2 7 3 4 6 4 7 5 9 4 3 6 2 6 4 4 8 3 2 7 6 LINEAR RANK TESTS FOR THE SCALE PROBLEM 351
=+9.13. Data on weekly rate of item output from two different production lines for seven weeks are as follows:Line I: 36, 36, 38, 40, 41, 41, 42 Line II: 29, 34, 37, 39, 40, 43, 44
=+small to keep the characteristics of the end product within specifications. Samples of lead ingots to be used as raw material are taken from two different distributors; each distributor has a
=+9.12. In industrial production processes, each measurable characteristic of any raw material must have some specified average value, but the variability should also be
=+9.11. The psychology departments of public universities in each of two different states accepted seven and nine applicants, respectively, for graduate study next fall. Their respective scores on
=+teacher with a bachelor’s degree. The mean score is 5 for each class. is there a difference in variability of scores?
=+achievement scores on an examination (10 ¼ highest possible score) given to two classes of ten students. Class 1 was taught by a teacher with a master’s degree and class 2 by a
=+degree is not different from that of classes taught by teachers with a master’s degree. The present study was aimed at determining whether variability in student achievement is the same for these
=+9.3. Verify the expression given in (2.3) for varðMN Þ.

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