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nonparametric statistical inference

Nonparametric Statistical Inference 5th Edition Jean Dickinson Gibbons, Subhabrata Chakraborti - Solutions

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 describe them.
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 6 23 2(a) Find the coefficient of rank correlation.(b) Find the appropriate one-tailed P value for
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 the association between rankings and find a one-tailed P value in each case. State the hypothesis
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 6 F 7 8 G 8 9 H 4 5 I 9 7(a) Find the rank correlation coefficient and the appropriate one-tailed P
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 greater than 12 denoting some college or university experience For example, the first subject whose
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 ranks to test the null hypothesis that the four varieties of soybean all have the same effect on yield.
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 H 1 2 1 3 5 4 8 7 6 2 1 2 4 5 7 6 8 3 3 3 2 1 4 5 8 7 6(a) Calculate Kendall’s coefficient of
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 test for independence.(b) Calculate (12.6.2) for these same data to verify that it is an equivalent
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 somewhat different skills (clear, logical, and serial thinking versus quantitative talent), both
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 asked to respond to a questionnaire with general statements about ethics. Two questions related to
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 Kendall partial tau coefficient between quantitative and logic when the effect of verbal is removed.
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 characteristics that include factors of appearance, comfort, and performance. The scores of the six
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 had to compare only three varieties. Each pair of varieties is presented together three times, to a
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 the examination.Suppose that four professors give the following ranks, where 1 indicates the best
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 feet.Calculate the partial tau coefficient for X1 and X2, when the effects of X3 are eliminated.Size of
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 agreement between the three sets of rankings? Give a descriptive measure of agreement and find a P
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 drug for a month at a time. At the end of each month, each patient gave an estimate of the relative
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 stimulus and then rewarding the subject if it responds to an identical(matching) sample stimulus. Then
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) ¼ F½(x mX)=s (b) FX is uniform, with FX(x) ¼0 x 1=2 x þ 1=2 1=2 < x 1=2 1 x > 1=2 8 0
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 exponential: 3=2
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 þ ex)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 T3, when the relevant ARE’s exist, ARE(T1, T3) ¼ ARE(T1, T2) ARE(T2, T3) ¼ ½ARE(T3, T1)1 where
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 between the stage of a student’s academic career and his attitude with respect to whether the
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 samples. Do the three shifts have the same proportions of defects?Day Evening Night Defects 70 60 80
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 below determine whether the training film has a positive effect on the ratings.Rating after Film
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 selected for the study. During the previous year under the current policy, 20 of them had been
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. Determine whether the vaccination had a significant effect in protecting the cows against the disease.
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 curriculum?’’The superintendent then held a weeklong workshop on curriculum improvement and sent the same
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 area over a period of time, some 1500 certificates listed the neoplastic disease as primary cause of
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 classified into three groups according to asset size and the data below are obtained on the numbers with debt
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 according to these characteristics as shown below.Age Job Satisfaction (1¼Least Satisfied)1 2 3 Total
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 135 persons were then put on a mailing list to receive an informative newsletter about U.S. foreign
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 before receiving the letter and these same persons were then asked again after receiving the letter.
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 the mountains. The subjects were told to ignore factors such as relative cost and distance so that
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 first year of college calculus were classified according to their high school calculus experience as
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 intercourse (date rape). A survey of students produced 623 responses.Of the 355 female respondents, 79
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 game. The responses are as follows:Alabama Auburn Tuscaloosa Alabama win 55 15 30 Auburn win 25 60 15
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 biweekly carbon dioxide inhalations, and (4) tranquilizer drug treatment, are compared by assigning a
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 observations after the rth bulb burns out. Then information is available only on the first
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) Exponential distribution with mean 1.(b) Beta distribution with a¼2 and b¼1. The probability density
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 of size 5 is taken with replacement from the discrete distribution fX(x)¼1=6 for x¼1, 2, . . . ,
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 for u is X(1), the smallest order statistic, a logical choice for the limits of the confidence
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 the point nearest zero exceeds c.
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 writing (2.7.1) in the form fU(u) ¼ 8(0:5 ju 0:5j)2(1 þ 4ju 0:5j) for 0 < u < 1
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) population.
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 that the two extreme values of a random sample are asymptotically independent.
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 random sample of size 19 from the exponential distribution fX(x)¼exp(x) for x 0.
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 exp (
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 X(r) given in (2.6.4) by differentiation of the result in (a).(c) By considering P(X(r)>t=n) in the
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 U.
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 continuous cdf FY. Let Sj be the random variable representing the number of Y blocks I1, I2, . . . , Inþ1
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 corresponding empirical cdf’s. Consider, for example, the quantity m[1Sm(Y1)], which is simply 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, . . . , Xn, by showing that Fn(x) ¼ Sn(x) for all x
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 n from the population FX.
2.34 Let Sn(x) be the empirical distribution function for a random sample of size n from the uniform (0, 1) distribution. Define Xn(t) ¼ffiffiffi np jSn(t) tj Zn(t) ¼ (t þ 1)Xn[t=(t þ 1)] for all 0 t 1 Find E[Xn(t)], E[Zn(t)], var[Xn(t)] and var[Zn(t)], and conclude that var[Xn(t)]
3.1 Prove Corollary 3.2.1 using a direct combinatorial argument based on Lemma 3.2.1.
3.2 Find the mean and variance of the number of runs R1 of type 1 elements, using the probability distribution given in (3.2.2). Since E(R) ¼ E(R1)þ E(R2), use your result to verify (3.2.6).
3.3 Use Lemmas 3.2.2 and 3.2.3 to evaluate the sums in (3.2.5), obtaining the result given in (3.2.6) for E(R).
3.4 Show that the asymptotic distribution of the standardized random variable [R1 E(R1)]=s(R1) is the standard normal distribution, using the distribution of R1 given in (3.2.2) and your answer to Problem 3.2.
3.5 Verify that the asymptotic distribution of the random variable given in(3.2.9) is the standard normal distribution.
3.6 By considering the ratios fR(r)=fR(r 2) and fR(r þ 2)=fR(r), where r is an even positive integer and fR(r) is given in (3.2.3), show that if the most probable number of runs is an even integer k, then k satisfies the inequality 2n1n2 n< k
3.7 Show that the probability that a sequence of n1 elements of type 1 and n2 elements of type 2 begins with a type 1 run of length exactly k is(n1)kn2(n1 þ n2)kþ1 where (n)r ¼n!(n r)!
3.8 Find the rejection region with significance level not exceeding 0.10 for a test of randomness based on the length of the longest run when n1¼n2¼6.
3.9 Find the complete probability distribution of the number of runs up and down of various lengths when n¼6 using (3.4.1) and the results given for u5(r4, r3, r2, r1).
3.10 Use your answers to Problem 3.9 to obtain the complete probability distribution of the total number of runs up and down when n¼6.
3.11 Verify the statement that the variance of the RVN test statistic is approximately equal to 20=(5nþ7).
3.12 Analyze the data in Example 3.4.1 for evidence of trend using the total number of runs above and below(a) The sample median(b) The sample mean

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