Questions and Answers of Nonparametric Statistical Inference

8.1 Given independent samples ofmX and n Y variables, define the following random variables for i¼1, 2, . . . , m:Ki¼rank of Xi among X1, X2, . . . , Xm Ri¼rank of Xi among X1, X2, . . . , Xm, Y1,
8.2 A single random sample D1, D2, . . . , DN of size N is drawn from a population which is continuous and symmetric. Assume there are m positive values, n negative values, and no zero values. Define
8.3 Generate by enumeration the exact null probability distribution of TsBr as defined in (8.3.5) for m¼n¼3, all S¼R
8.4 Verify the results given in (8.3.6) for the mean and variance of TsBr when S¼R and N is even and derive a similar result for S¼R when N is odd.
8.5 Show that the median test of Section 6.4 is a linear rank test.
8.6 Giambra and Quilter (1989) performed a study to investigate gender and age difference in ability to sustain attention when given Mackworth’s clock-test. This clock is metal with a plain white
8.7 Elam (1988) conducted a double-blind study of 18 adult males to investigate the effects of physical resistance exercises and amino acid dietary supplements on body mass, body fat, and composite
8.8 Howard et al. (1986) (see Problem 5.12) also investigated whether pretest anxiety scores differed for students enrolled in two different sections of the introduction to computer courses. Seven
8.9 A travel agency wanted to compare the noncorporate prices charged by two major motel chains for a standard-quality single room at major airport locations around the country. A random sample of
8.10 Smokers are commonly thought of as nervous people whose emotionality is at least partly caused by the stimulating effect tobacco has on the nervous system. Nesbitt (1972) conducted a study with
8.11 A group of 20 mice are allocated to individual cages randomly. The cages are assigned in equal numbers, randomly, to two treatments, a control A and a certain drug B. All animals, in a random
8.12 The following data represent two independent random samples drawn from continuous populations that are thought to have the same form but possibly different locations.X: 79, 13, 138, 129, 59, 76,
8.13 A problem of considerable import to the small-scale farmer who purchases young pigs to fatten and sell for slaughter is whether there is any difference in weight gain for male and female pigs
8.14 How would you find the confidence-interval endpoints for the parameter of interest when the interval has confidence level nearest 0.90 and corresponds to(a) The sign test with n¼11(b) The
8.15 A self-concept test was given to a random sample consisting of six normal subjects and three subjects under psychiatric care. Higher scores indicate more self-esteem. The data are as
8.16 Verify the confidence interval estimate of mX mY with exact confidence coefficient at least 0.95 given in the MINITAB solution of Example 8.2.1.
9.1 Develop by enumeration for m ¼ n ¼ 3 the null probability distribution of Mood’s statistic MN of (9.2.1).
9.2 Develop by enumeration for m ¼ n ¼ 3 the null probability distribution of the Freund–Ansari–Bradley statistic of (9.3.3).
9.3 Verify the expression given in (9.2.3) for var(MN).
9.4 Apply Theorem 7.3.2 to derive the mean and variance of the statistic AN defined in (9.3.1).
9.5 Apply Theorem 7.3.2 to derive the mean and variance of the statistic BN defined in (9.3.5).
9.6 Verify the relationship between AN, BN and FN given in (9.4.3) for N even.
9.7 Use the relationship in (9.4.3) and the moments derived for FN for N even in (9.3.4) to verify your answers to Problems 9.4 and 9.5 for N even.
9.8 Use Theorem 7.3.2 to derive the mean and variance of Ts þ Br for N even, S 6¼ R, where S þ R N.
9.9 Verify the result given in (9.9.2) for the null probability distribution of Rosenbaum’s R statistic.
9.10 Olejnik (1988) suggested that research studies in education and the social sciences should be concerned with differences in group variability as well as differences in group means. For example,
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 the
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 small to keep the
9.13 Data on weekly rate of item output from two different production lines for 7 weeks are as follows:Line I: 36, 36, 38, 40, 41, 41, 42 Line II: 29, 34, 37, 39, 40, 43, 44 We want to investigate
10.1 Generate by enumeration the exact null distribution of the k-sample median test statistic for k ¼ 3, n1 ¼ 2, n2 ¼ 1, n3 ¼ 1. If the rejection region consists of those arrangements which are
10.2 Generate the exact null distribution of the Kruskal–Wallis statistic H for the same k and ni as in Problem 10.1. Find the critical region which consists of those rank sums R1, R2,R3, which
10.3 By enumeration, place the median test criterion (U1,U2,U3) and the H test criterion (R1, R2, R3) in one-to-one correspondence for the same k and ni as in Problems 10.1 and 10.2. If the two tests
10.4 Verify that the form of H given in (10.4.4) is algebraically equivalent to(10.4.2).
10.5 Show that H with k ¼ 2 is exactly equivalent to the large-sample approximation to the two-sample Wilcoxon rank-sum test statistic of Section 8.2 with m ¼ n1, n ¼ n2.
10.6 Show that H is equivalent to the F test statistic in a one-way ANOVA problem if applied to the ranks of the observations rather than the actual numbers.Hint: Express the F ratio as a function of
10.7 Write the k-sample median test statistic given in (10.2.2) in the form of(10.5.1) (cf., Problem 7.2).
10.8 How could the subsampling procedure described in Section 9.9 be extended to test the equality of variances in k populations?
10.9 In the context of the k-sample control median test defined in Section 10.3, show that for any i( ¼ 2, 3, . . . , k) and j( ¼ 0, 1, . . . , q) the random variable Vij has a binomial
10.10 Show that under H0 : u1 ¼ u2 ¼ ¼ uk the distribution of V*, the sum test statistic for comparisons with a control when u1 is known in the case of unequal sample sizes defined in
10.11 In the context of the Jonckheere–Terpstra test of Section 10.6, show that under H0 for i < j < r, cov(Uij,UirjH0) ¼ cov(Uji,UrijH0) ¼ninjnr 12 cov(Uij,UrijH0) ¼ cov(Uji,UirjH0) ¼ ninjnr
10.12 Many psychologists have developed theories about how different kinds of brain dominance may affect recall ability of information presented in various formats. Brown and Evans (1986) compare
10.13 Andrews (1989) examines attitudes toward advertising by undergraduate marketing students at universities in six different geographic regions. Attitudes were measured by answers to a
10.14 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 using
10.15 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
10.16 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
10.17 Below are four sets of five measurements, each set an array of data on the smoothness of a certain type of paper, each set obtained from a different laboratory. Find an approximate P value to
10.18 Verify the value of the Kruskal–Wallis test statistic given in the SAS solution to Example 8.2.1.
10.19 A sample of 100 female students at a large university were questioned about four experimental types of service clubs that have essentially the same goals. The types differed only with respect
11.1 A beauty contest has eight contestants. Two judges are each asked to rank the contestants in a preferential order of pulchritude. Answer parts (a) and (b) using (1) the Kendall tau-coefficient
11.2 Verify the result given in (11.4.9).
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 in ascending order of
11.4 Show that for the standardized bivariate normal distribution F(0, 0) ¼1 4 þ1 2p arcsin r
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 whether there is a direct
11.6 Company-financed expenditures in manufacturing on research and development (R&D) are currently about 2.7% of sales in Japan and 2.8% of sales in the United States. However, when these figures
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
11.8 For the time series data in Example 3.4.1, use the Mann test based on Spearman’s rank correlation coefficient to see if the data show a positive trend.
11.9 Do Problem 11.8 using the Daniels’ test based on Kendall’s tau.
11.10 The rainfall measured by each of 12 gauges was recorded for 20 successive days. The average results for each day are as follows:Day Rainfall Day Rainfall April 1 0.00 April 11 2.10 April 2 0.03
11.11 A company has administered a screening aptitude test to 20 new employees over a 2 year period. The record of scores and data on which the person was hired are shown below.1=4=08 75 9=21=08 72
11.12 Ten randomly chosen male college students are used in an experiment to investigate the claim that physical strength decreases with fatigue.Describe the relationship for the data below, using
11.13 Given a single series of time-ordered ordinal observations over several years, name all the nonparametric procedures that could be used in order to detect a long-term positive trend and
11.14 Six randomly selected mice are studied over time and scored on an ordinal basis for intelligence and social dominance.Mouse Intelligence Social Dominance 1 45 63 2 26 0 3 20 16 4 40 91 5 36 25
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 producers to describe
11.16 Derive the null distribution of both Kendall’s tau statistic and Spearman’s rho for n ¼ 3 assuming no ties.
11.17 A scout for a professional baseball team ranks nine players separately in terms of speed and power hitting, as shown below.Player Speed Ranking Power-Hitting Ranking A 3 1 B 1 3 C 5 4 D 6 2 E 2
11.18 Twenty-three students are classified according to their attitude toward elementary school integration. Then each is asked the number of years of schooling completed at that time, with numbers
11.19 For the data in Problem 3.13, use the two methods of this chapter to see if there is a positive trend.
12.1 Four varieties of soybean are each planted in three blocks. The yields are:Variety of Soybean Block A B C D 1 45 48 43 41 2 49 45 42 39 3 38 39 35 36 Use Friedman’s analysis of variance by
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 as follows:Contestant Judge A B C D E F G
12.3 Derive by enumeration the exact null distribution of W for three sets of rankings of two objects.
12.4 Given the following triplets of rankings of six objects:X 1 3 5 6 4 2 Y 1 2 6 4 3 5 Z 2 1 5 4 6 3(a) Calculate the Kendall coefficient of partial correlation between X and Y from (12.6.1) and
12.5 Howard et al. (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 subjects involve
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 school students to executives, were
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 of find the
12.8 Automobile Magazine publishes results of a comparison test of 15 brand models of comparably priced sedans. Each car is given a subjective score out of possible 60 points (60¼best) on each of 10
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 taster
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 taking
12.11 Show that if m¼n and l¼k in (12.5.4) so that the design is complete, then (12.5.4) is equivalent to Q¼12S=kn(nþ1), as it should be from(12.2.8).
12.12 A town has 10 different supermarkets. For each market, data are available on the following three variables: X1¼food sales, X2¼nonfood sales, and X3¼size of store in thousands of square
12.13 Suppose in Problem 11.15 that an independent group of female consumers also ranked the products as follows:Product A B C D E F G H I J Independent female ranks 8 9 5 6 1 2 7 4 40 3(a) Is there
12.14 An experimenter is attempting to evaluate the relative effectiveness of four drugs in reducing the pain and trauma of persons suffering from migraine headaches. Seven patients are given each
12.15 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
13.1 Use the results of Theorem 7.3.8 to evaluate the efficacy of the twosample Wilcoxon rank-sum test statistic of (8.2.1) for the location model FY(x) ¼ FX(x u) where(a) FX is N(mX, s2) or FX(x)
13.2 Calculate the efficacy of the two-sample Student’s t test statistic in cases(b) and (c) of Problem 13.1.
13.3 Use your answers to Problems 13.1 and 13.2 to verify the following results for the ARE of the Wilcoxon rank-sum (or Mann–Whitney) test to Student’s t test:Normal: 3=p Uniform: 1 Double
13.4 Calculate the efficacy of the sign test and the Student’s t test for the location model FX(x) ¼ F(x u) where u is the median of FX and F is the cdf of the logistic distribution F(x) ¼ (1
13.5 Evaluate the efficiency of the Klotz normal-scores test of (9.5.1) relative to the F test statistic for the normal-theory scale model.
13.6 Evaluate the efficacies of the MN and AN test statistics of Chapter 9 and compare their efficiency for the scale model where, as in Problem 13.1:(a) FX is uniform.(b) FX is double exponential.
13.7 Use the result in Problem 13.4 to verify that the ARE of the sign test relative to the Student’s t test for the logistic distribution is p2=12.
13.8 Verify the following results for the ARE of the sign test relative to the Wilcoxon signed-rank test.Uniform: 1=3 Normal: 2=3 Logistic: 3=4 Double exponential: 4=3
13.9 Suppose there are three test statistic, T1, T2, and T3, each of which can be used to test a null hypothesis H0 against an alternative hypothesis H1.Show that for any pair of tests, say T1 and
14.1 An ongoing problem on college campuses is the instructor evaluation form. To aid in interpreting the results of such evaluations, a study was made to determine whether any relationship exists
14.2 A manufacturer produces units of a product in three 8 hour shifts: Day, Evening, and Night. Quality control teams check production lots for defects at the end of each shift by taking random
14.3 A group of 28 salespersons were rated on their sales presentations and then asked to view a training film on improving selling techniques.Each person was then rated a second time. For the data
14.4 An employer wanted to find out if changing from his current health benefit policy to a prepaid policy would change hospitalization rates for his employees. A random sample of 100 employees was
14.5 A sample of five vaccinated and five unvaccinated cows were all exposed to a disease. Four cows contracted the disease, one from the vaccinated group and three from the nonvaccinated group.
14.6 A superintendent of schools is interested in revising the curriculum.He sends out questionnaires to 200 teachers: 100 respond No to the question ‘‘do you think we should revise the
14.7 A retrospective study of death certificates was aimed at determining whether an association exists between a particular occupation and a certain neoplastic disease. In a certain geographical
14.8 A financial consultant is interested in testing whether the proportion of debt that exceeds equity is the same irrespective of the magnitude of the firm’s assets. Sixty-two firms are
14.9 In a study designed to investigate the relationship between age and degree of job satisfaction among clerical workers, a random sample of 100 clerical workers were interviewed and classified

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