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nonparametric statistical inference

Questions and Answers of Nonparametric Statistical Inference

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
14.10 A random sample of 135 U.S. citizens were asked their opinion about the current U.S. foreign policy in Afghanistan. Forty-three reported a negative opinion and the others were positive. These
14.11 A small random sample was used in an experiment to see how effective an informative newsletter was in persuading people to favor a flat income tax bill. Thirty persons were asked their opinion
14.12 Twenty married couples were selected at random from a large population and each person was asked privately whether the family would prefer to spend a week’s summer vacation at the beach or in
14.13 A study was conducted to investigate whether high school experience with calculus has an effect on performance in first-year college calculus. A total of 686 students who had completed their
14.14 For the data in Problem 14.8, investigate whether firms with debt greater than equity tend to have more assets than other firms.
14.15 Derive the maximum likelihood estimators for the parameters in the likelihood function of (14.2.1).
14.16 Show that (14.2.2) is still the appropriate test statistic for independence in a two-way rk contingency table when both the row and column totals are fixed.
14.17 Verify the equivalence of the expressions in (14.3.1) through (14.3.4).
14.18 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
14.19 Prior to the Alabama-Auburn football game, 80 Alabama alumni, 75 Auburn alumni, and 45 residents of Tuscaloosa who are not alumni of either university are asked who they think will win the
14.20 Four different experimental methods of treating schizophrenia,(1) weekly shock treatments, (2) weekly treatments of carbon dioxide inhalations, (3) biweekly shock treatment alternated with
1. X(n), the maximum (largest) value in the sample, is of interest in the study of floods and other extreme meteorological phenomena.
2. X(1), the minimum (smallest) value, is useful for phenomena where, for example, the strength of a chain depends on the weakest link.
3. The sample median, defined as X[(nþ1)=2] for n odd and any number between X(n=2) and X(n=2þ1) for n even, is a measure of location and an estimate of the population central tendency.
4. The sample midrange, defined as (X(1)þX(n))=2, is also a measure of central tendency.
5. The sample range X(n)X(1) is a measure of dispersion.
6. In some experiments, the sampling process ceases after collecting r of the observations. For example, in life-testing electric light bulbs, one may start with a group of n bulbs but stop taking
2.1 Let X be a discrete random variable taking on only positive integer values. Show that E(X) ¼X1 i¼1 P(X i)
2.2 Let X be a nonnegative continuous random variable with cdf FX.Show that E(X) ¼ð1 0[1 FX(x)]dx(Hint: Use integration by parts on the definition of E(X)).
2.3 Show that Xn x¼a nx px(1 p)nx ¼1 B(a, n a þ 1)ðp 0ya1(1 y)na dy for any 0
2.4 Find the transformation to obtain, from an observation U following a continuous uniform (0, 1) distribution, an observation from each of the following continuous probability distributions:(a)
2.5 Prove the probability-integral transformation (Theorem 2.5.1) by finding the moment-generating function of the random variable Y¼FX(X), where X is absolutely continuous and has cdf FX.
2.6 If X is a continuous random variable with probability density function fX(x)¼2(1x) for 0
2.7 The order statistics for a random sample of size n from a discrete distribution are defined as in the continuous case except that now we have X(1) X(2) X(n). Suppose a random sample
2.8 A random sample of size 3 is drawn from the population fX(x)¼exp[(xu)] for x>u. We want to find a 95% confidence-interval estimate for the parameter u. Since the maximum-likelihood estimate
2.9 For the n-order statistics of a sample from the uniform distribution over (0, u), show that the interval (X(n), X(n)=a1=n) is a 100(1a)%confidence-interval estimate of the parameter u.
2.10 Ten points are chosen randomly and independently on the interval (0, 1).(a) Find the probability that the point nearest 1 exceeds 0.90.(b) Find the number c such that the probability is 0.5 that
2.11 Find the expected value of the largest order statistic in a random sample of size 3 from:(a) The exponential distribution fX(x)¼exp(x) for x 0(b) The standard normal distribution
2.12 Verify the result given in (2.7.1) for the distribution of the median of a sample of size 2m from the uniform (0, 1) distribution when m¼2.Show that this distribution is symmetric about 0.5 by
2.13 Find the mean and variance of the median of a random sample of n from the uniform (0.1) distribution:(a) When n is odd(b) When n is even and U is defined as in Section 2.7
2.14 Find the probability that the range of a random sample of size n from the population fX(x)¼2e2x for x 0 does not exceed 4.
2.15 Find the distribution of the range of a random sample of size n from the exponential distribution fX(x)¼4 exp(4x) for x 0.
2.16 Give an expression similar to (2.7.3) for the probability density function of the midrange for any continuous distribution and use it to find the density function in the case of a uniform (0, 1)
2.17 By making the transformation U¼nFX(X(1)), V¼n[1FX(X(n))] in(2.6.8) with r¼1, s¼n, for any continuous FX, show that U and V are independent random variables in the limiting case as n ! 1, so
2.18 Use (2.9.5) and (2.9.6) to approximate the mean and variance of:(a) The median of a sample of size 2mþ1 from a normal distribution with mean m and variance s2.(b) The fifth order statistic of a
2.19 Let X(n) be the largest value in a sample of size n from the pdf fX.(a) Show that limn!1 P(n1X(n) x) ¼ exp (a=px) if fX(x) ¼ a/[p(a2+x2)] (Cauchy).(b) Show that limn!1 P(n2X(n) x) ¼
2.20 Let X(r) be the rth-order statistic of a random sample of size n from a cdf FX.(a) Verify that P(X(r) t) ¼Pnk¼r nk [FX(t)]k[1 FX(t)]nk.(b) Verify the probability density function of
2.21 Let X(1)
2.22 Let X(1)
2.23 Find the probability that the range of a random sample of size 3 from the uniform distribution is less than 0.8.
2.24 Find the expected value of the range of a random sample of size 3 from the uniform distribution.
2.25 Find the variance of the range of a random sample of size 3 from the uniform distribution.
2.26 Let the random variable U denote the proportion of the population lying between the two extreme values of a sample of n from some unspecified continuous population. Find the mean and variance of
2.27 Suppose that a random sample of size m, X1, X2, . . . , Xm, is available from a continuous cdf FX and a second independent random sample of size n, Y1, Y2, . . . , Yn, is available from a
2.28 Exceedance statistics. Let X1, X2, . . . , Xm and Y1, Y2, . . . , Yn be two independent random samples from arbitrary continuous cdf’s FX and FY, respectively, and let Sm(x) and Sn(y) be the
2.32 Define the indicator variable e(x) ¼1 if x 0 0 if x < 0Show that the random function defined by Fn(x) ¼Xn i¼1 e(x Xi)n is the empirical distribution function of a sample X1, X2, . . . ,
2.33 Prove that cov[Sn(x), Sn(y)]¼c[FX(x), FX(y)]=n where c(s, t) ¼ min (s, t) st ¼s(1 t) if s t t(1 s) if s tand Sn(.) is the empirical distribution function of a random sample of size

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