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nonparametric statistical inference

Nonparametric Statistical Inference 4th Edition Gibbons, Jean Dickinson, Chakraborti, Subhabrata - Solutions

=+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 are
=+ Subjects of both sexes were drawn from three different colleges and classified as smokers if they smoked any number of cigarettes on a regular basis. In one
=+8.10. Smokers are commonly thought of as nervous people whose emotionality is at least partly caused by smoking because of the stimulating effect tobacco has on the nervous system. Nesbitt (1972) conducted a study with 300 college students and concluded that smokers are less emotional than
=+Find a 95% confidence interval estimate of the difference between median costs at Best Eastern and Travelers' Inn motels. Best Eastern: $68,75,92,79,95 Travelers' Inn: $69, 76, 81, 72, 75, 80
=+ 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 five Best Eastern motels and an independent sample of six Travelers' Inn motels, all located at
=+Seven students were enrolled in each section, and the data are shown below. Is there a difference in median scores?Section 1: 20, 32, 22, 21, 27, 26, 38 Section 2: 34, 20, 30, 28, 25, 23, 29
=+ 8.8. Howard, Murphy, and Thomas (1986) (see Problem 5.12) also investigated whether pretest anxiety scores differed for students enrolled in two different sections of the in- troduction to computer courses.
=+each spread over a 5-week period. Workloads were tailored to abilities of the individual subjects but escalated in intensity over the period. The data in Table 1 are the changes (after minus before) in body mass, body fat, and composite body girth for the amino acid (Treatment) group and placebo
=+mass, body fat, and composite girth. Ten of the subjects received the diet supplement and eight received a placebo. All subjects participated in 15 resistance exercise workouts of one hour
=+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
=+ Determine whether median number of correct scores is larger for young men than for older men. Age 18-29: 11,13, 15, 15, 17, 19, 20, 21, 21, 22 Age 50-59: 8,9, 10, 11, 12, 13, 5, 17, 19,23
=+Scores were the number of correct recognitions of the double jumps. The scores below are for 10 men in age groups 18-29 and 10 men in age group 50-59.
=+ Subjects were told that double jumps would occur and asked to signal their recognition of occurrence by pressing a button.
=+white face and a black pointer that moves around the face in 100 discrete steps of 36 degrees each. During the test period the pointer made 23 double jumps, defined as moving twice the normal distance or 7.2 degrees in the same time period, at random and irregular intervals.
=+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
=+8.4. Verify the results given in (3.4) for the mean and variance of T, -B, when S = R and N is even and derive a similar result for SR when N is odd.
=+ 8.3. Generate by enumeration the exact null probability distribution of T, -B, as defined in (3.3) for m=n=3, all S=R
=+Explain fully how tables of the null distribution of Wy could be used to find the null distribution of T. Since for N large, m and n will both converge to the constant value N/2 in the null case, these two test statistics have equivalent properties asymp- totically.
=+ Therefore the null distribution of Wy is identical to the null distribution of T conditional upon the observed number of plus and minus signs.
=+ (b) If these two samples are from identical populations, the median of the symmetric D population must be zero.
=+ (a) Show that the two-sample Wilcoxon rank-sum statistic Wy of (2.1) for these two samples equals the Wilcoxon signed-rank statistic 7+ defined in (5.7.1).
=+ Assume there are m positive values, n negative values, and no zero values. Define the m+n = N random variables X = Di Y = |D\ if D > 0 if D; < 0 Then the X1, X2.....X and Y, Y2.... Y, constitute two independent random samples of sizes m and n.
=+8.2. A single random sample D1, D2,..., Dy of size N is drawn from a population which is continuous and symmetric.
=+8.1. Given independent samples of m X and n Yvariables, define the following random variables for i = 1,2,....m: K = rank of X; among X.X.....Xm R; = rank of X; among X1 X2.....X. Y. Y2.....Y Use Ki and Ri to prove the linear relationship between the Mann-Whitney and Wilcoxon rank-sum statistics
=+ 7.3. Prove the three properties stated in Theorem 3.7.
=+7.2. Express the two-sample median-test statistic U defined in Section 6.4 in the form of a linear rank statistic and use Theorem 3.2 to find its mean and variance. Hint: For the appropriate argument k, use the functions S(k) defined as for (2.1)
=+7.1. One of the simplest linear rank statistics is defined as N WN = izi i-1 This is the Wilcoxon rank-sum statistic to be discussed on the next chapter. Use Theorem 3.2 to evaluate the mean and variance of WN-
=+What is the highest possible level of confidence? What assump- tions are we making for this procedure?
=+(b) Use the median test to calculate a confidence interval for the difference between the medians.
=+approximate P value and a conclusion. What assumptions are we making?
=+a) Use the median test and the control median test to test the hypothesis. For each test give the null hypothesis, the alternative hypothesis, the value of the test statistic, the exact and the
=+6.15. A researcher is interested in learning if a new drug is better than a placebo in treating a certain disease. Because of the nature of the disease, only a limited number of patients can be found. Out of these, 5 are randomly assigned to the placebo and 5 to the new drug. Suppose that the
=+ Find the P value for the alternative that on the average the girls learn the task faster than the boys, and find a confidence interval estimate for the difference 0 My - Mx with a confidence coefficient at least equal to 0.85, using the median test.
=+ 6.14. A sample of three girls and five boys are given instructions on how to complete a certain task. Then they are asked to perform the task over and over until they complete it correctly. The number of repetitions necessary for correct completion are 1, 2, and 5 for the girls and 4, 8, 9, 10,
=+to Fx and Fy, respectively. (Gastwirth, 1968; Chakraborti and Mukerjee; 1990)
=+(b) Show that the random variable m/2m-Vmn -p] is asymptotically nor- mally distributed with mean zero and variance (1-p)+(1-p) (Kp) +(KP) where fx and fy are the density functions corresponding
=+where is a fixed quantity, and let {r,} be a sequence of positive integers such that lim of X observations that do not exceed Y()- x(r/n) = p. Finally let Vm be the number (a) Show that mVmn is a consistent estimator of Ep.
=+ 6.13. (+)/(1-2) = 0,1 (Sen, 1964) In some applications the quantity = Fx (Kp), where Kp is the pth quantile of Fy, is of interest. Let lim (m/n),
=+ 6.12. Show that when m,n such that m/(m +n), 0 <
=+6.11. For the control median test statistic V, use Problem 2.28, or otherwise, to show that when Fx Fy, m E(V)=2 and var(V) 2r+m+2 4m (2r+3) [Hint: Use the fact that E(X) = EYE(XY) and var(X) = vary E(XY)+ Ey var(XY)]
=+(b) Hence show that the null distribution of the control median test statistic V, with n = 2r+1, can be expressed as 2r+1 m ()( r m+2r+1(m+2r j=0,1,...,m
=+6.10. (a) Show that the distribution of the precedence statistic P() under the null hypothesis (Fx Fy), given in Problem 2.28(c), can be expressed as (m(n-1 P(P)=j|Ho) n m+n (m+n-1\ (+/-1) i (77) (7) 1m+n' j=0,1,...,m These relationships are useful in calculating the null distribution of
=+ These counties were divided into two mutually independent groups, rural and nonrural, according to population size of less than 25,000 in 2000 or not. Random samples of nine rural and seven nonrural counties gave the following data on percentage population change: Rural Nonrural 1.1,-21.7,-16.3,
=+ 6.9. The 2000 census statistics for Alabama give the percentage changes in population between 1990 and 2000 for each of the 67 counties.
=+6.8. Can you think of other functions of the difference S, (x)-S,(x) (besides the maximum) which could also be used for distribution-free tests of the equality of two population distributions?
=+6.6. Answer parts ðaÞ to ðcÞ using ðiÞ the median-test procedure and (ii) the MannWhitney test procedure (use tables) for the following two independent random samples drawn from continuous populations which have the same form but possibly a difference of y in their locations:
=+6.5. Verify the expressions given in (6.15) for the moments of U under H0.
=+6.4. Use the recursion relation for the Mann-Whitney test statistic given in (6.14) to generate the complete null probability distribution of U for all m þ n44.
=+6.1. Use the graphical method of Hodges to find PðDþm;n 5dÞ, where d is the observed value of Dþm;n ¼ maxx½SmðxÞ SnðxÞ in the arrangement xyyxyx.
=+5.31. If X(1) and X(n) are the smallest and largest values, respectively, in a sample of size n from any continuous population FX with median k0.50, find the smallest value of n such that:ðaÞ PðXð1Þ < k0:50 < XðnÞÞ50:99ðbÞ P½FX ðXðnÞÞ FX ðXð1ÞÞ50:550:95
=+5.30. For order statistics of a random sample of size n from any continuous population FX , show that the interval ðXðrÞ; Xðnrþ1Þ; r < n=2Þ, is a 100ð1 aÞ percent confidenceinterval estimate for the median of FX , where 1 a ¼ 1 2n n 1 r 1Z 0:5 0xnrð1 xÞr1 dx
=+whether American adults sleep less today than they did five years ago and justify your choice. You should at least test hypothesis concerning the quantiles of order 0.05, 0.50, and 0.95.
=+5.2, 9.1, and 5.8 hours. Use the most appropriate statistical procedures to determine Representative Sales before Sales after 1 90 97 2 83 80 3 105 110 4 97 93 5 110 123 6 78 84 ONE-SAMPLE AND PAIRED-SAMPLE PROCEDURES 229
=+5.27. A study 5 years ago reported that the median amount of sleep by American adults is 7.5 hours out of 24 with a standard deviation of 1.5 hours and that 5% of the population sleep 6 or less hours while another 5% sleep 9 or more hours. A current sample of eight adults reported their average
=+where W1; W2; ... ; WN are independent random variables with PðWi ¼ 0Þ ¼ PðWi ¼ iÞ¼ 0:5, i ¼ 1; 2; ... ; N.
=+5.26. Again consider the Wilcoxon signed-rank test discussed in Section 5.7. Show that under H0 the distribution of the test statistic Tþ is the same as that of W ¼ PN i¼1 Wi,
=+5.25. Using the result in Problem 5.24, show that for the Wilcoxon signed-rank test statistic Tþ discussed in Section 5.7, the 2N random variables Z1; rðjD1jÞ; Z2; rðjD2jÞ; ... ;ZN rðjDN jÞ are mutually independent under H0.
=+5.24. Let X be a continuous random variable symmetrically distributed about y. Show that the random variables jXj and Z are independent, where Z ¼ 1 if X > y 0 if X 4y
=+(d) Use the signed-rank test procedure to do (b).
=+(c) Use the signed-rank test to do (a). What assumptions must you make?
=+(b) Use the sign-test procedure at level nearest 0.90 to find a two-sided confidence-interval estimate of the median difference in sales (after – before). Give the exact level.
=+(a) State the null and alternative hypotheses and use the sign test to find a P value relevant to the question of whether the course is effective.
=+5.22. In order to test the effectiveness of a sales training program proposed by a firm of training specialists, a home furnishings company selects six sales representatives at random to take the course. The data below are gross sales by these representatives before and after the course.
=+even greater than last year. A random sample of 5625 calls is selected from recent records and 2890 of them are found to last more than 3 min 15 sec. Is the treasurer’s claim supported? Give the null and alternative hypotheses and the P value.
=+5.21. Last year the elapsed time of long-distance telephone calls for a national retailer was skewed to the right with a median of 3 min 15 sec. The recession has reduced sales, Storm Type A Type B Storm Type A Type B 1 1.38 1.42 8 2.63 2.69 2 9.69 10.37 9 2.44 2.68 3 0.39 0.39 10 0.56 0.53 4
=+(c) Do (a) and (b) without assuming symmetry.
=+(b) Find a confidence interval for the median difference, assuming symmetry and with confidence coefficient near 0.90.
=+Another user claims to have found that the type B gauge gives consistently higher average readings than type A. Do these results substantiate such a conclusion? Investigate using two different nonparametric test procedures, by finding the P value from(a) Tables of the exact distribution(b) Large
=+5.19. In a trial of two types of rain gauge, 69 of type A and 12 of type B were distributed at random over a small area. In a certain period 14 storms occurred, and the average amounts of rain recorded for each storm by the two types of gauge are as follows:Natural food Processed food Corn of the
=+5.18. For the data in Example 7.1, find a confidence interval estimate of the median difference Before minus After using the level nearest 0.90.
=+5.17. For the data in Example 4.3, test H0: M ¼ 0:50 against the alternative H1 : M > 0:50, using the(a) Sign test(b) Signed-rank test and assuming symmetry
=+The number of correctly written words was then counted and scaled such that a zero score represents the score a person not under the influence of alcohol would make, a positive score indicates increased writing speed and accuracy, and a negative score indicates decreased writing speed and
=+5.16. In an experiment to measure the effect of mild intoxication on coordination, nine subjects were each given ethyl alcohol in an amount equivalent to 15.7 ml=m2 of body surface and then asked to write a certain phrase as many times as they could in 1 min.
=+Listed below are the average sodium contents of five ordinary foods in processed form and natural form for equivalent quantities. Find a confidence interval estimate of the median difference (processed minus natural) with confidence coefficient at least 0.87 using two different procedures.
=+Student Before After Student Before After A 20 20 H 34 19 B 21 18 I 28 13 C 23 10 J 20 21 D 26 16 K 29 12 E 32 11 L 22 15 F 27 20 M 30 14 G 38 20 N 25 17 226 CHAPTER 5
=+5.14. Reducing high blood pressure by diet requires reduction of sodium intake, which usually requires switching from processed foods to their natural counterparts.Table 2 Data for Problem 5.11 State Affected age group 2529 age group Florida 0.060 0.025 Georgia 0.068 0.023 Illinois 0.025 0.004
=+Six showed improvement, 5 showed no change, and 13 had a reduced level of performance. Find the P value for an appropriate one-sided test.
=+5.13. Twenty-four students took both the midterm and the final exam in a writing course. Numerical grades were not given on the final, but each student was classified as either no change, improvement, or reduced level of performance compared with the midterm.
=+introduction to computers. The student subjects were given a test to measure computer anxiety at the beginning of the term and then again at the end of the 5-week summer course. High scores on this test indicate a high level of anxiety. For the data in Table 3 on 14 students, determine whether
=+5.12. Howard, Murphy, and Thomas (1986) reported a study designed to investigate whether computer anxiety changes between the beginning and end of a course on
=+corresponding difference ratios for the 25–29 age group, who were not affected by the law change, as shown in Table 2. Carry out an appropriate test and write a report of your conclusions.
=+counter measures, or advertising campaigns [like MADD (Mothers Against Drunk Drivers] may have affected the fatality ratios. In order to investigate further, these researchers compared the Before After ratios for the affected age group with the
=+5.11. The conclusion in Problem 5.10 was that the median difference (BeforeAfter)was positive for the affected age group, but this does not imply that the reduction was the result of laws that raised the minimum legal drinking age. Other factors, Table 1 Data for Problem 5.10 State Affected ages
=+The researchers hypothesized that raising the minimum drinking age resulted in a reduced median fatality ratio. Investigate this hypothesis.
=+ratios of the number of single-vehicle nighttime fatalities to the number of licensed drivers in the affected age group before and after the laws were changed to raise the drinking age, shown in Table 1.
=+5.10. Hoskin et al. (1986) investigated the change in fatal motor-vehicle accidents after the legal minimum drinking age was raised in 10 states. Their data were the
=+the two-sample problem to be discussed in Chapter 8. Show how Tþ might be used to test the hypothesis that the X and Y populations are identical.
=+Assume there are m X values and n Y values, where m þ n ¼ N and the X and Y value are independent. Show that the signed-rank test statistic Tþ calculated for these Di is equal to the sum of the ranks of the X observations in the combined ordered sample of mX’s and nY’s and also that Tþ
=+5.9. Let D1; D2; ... ; DN be a random sample of N nonzero observations from some continuous population which is symmetric with median zero. Define jDij ¼ Xi if Di > 0 Yi if Di < 0
=+5.8. Prove that the Wilcoxon signed-rank statistic Tþ T based on a set of nonzero observations X1; X2; ... ; XN can be written symbolically in the form XX 14 i4j4 N sgnðXi þ XjÞwhere sgnðxÞ ¼ 1 if x > 01 if x < 0
=+(c) Discuss how the power functions might help in the choice of an appropriate sample size for an experiment.
=+5.7. A random sample of 10 observations is drawn from a normal population with mean m and variance 1. Instead of a normal-theory test, the ordinary sign test is used for H0: m ¼ 0, H1: m > 0, with rejection region K 2 R for K 58.(a) Plot the power curve using the exact distribution of K.(b) Plot
=+5.6. Show by calculations from tables that the normal distribution provides reasonably accurate approximations to the critical values of one-sided tests for a ¼ 0:01; 0:05, and 0.10 when:N ¼ 12 for the sign test N ¼ 15 for the signed-rank test
=+5.5. Generate the sampling distributions of Tþ and T under the null hypothesis for a random sample of six unequal and nonzero observations.
=+ðbÞ Give the exact probability of a type I error in (a)ðcÞ On the basis of the following random sample of pairs:test at a significance level not exceeding 0.10 the null hypothesis H0: M ¼ 2 against the alternative H1: M 6¼ 2, where M is the median of the continuous and symmetric population
=+5.4. Answer parts (a) through (e) using (i) the sign-test procedure and (ii) the Wilcoxon signed-rank test procedure.ðaÞ Test at a significance level not exceeding 0.10 the null hypothesis H0: M ¼ 2 against the alternative H1: M > 2, where M is the median of the continuous symmetric population
=+p . Find and graph the corresponding probability function of differences.
=+5.3. Verify the cumulative distribution function of differences given in (4.14) and the result M ¼ 2 þ ffiffiffi 3
=+5.1. Give a functional definition similar to (5.1) for the rank rðXiÞ of a random variable in any set of N independent observations where ties are dealt with by the midrank method. Hint: In place of SðuÞ in (5.2), consider the function cðuÞ ¼0 if u < 0 1=2 if u ¼ 0 1 if u > 0(
=+4.34. Ten students take a test and their scores (out of 100) are as follows: 95, 80, 40, 52, 60, 80, 82,58,65,50 Test the null hypothesis that the cumulative distribution function of the proportion of right answers a student gets on the test is Fo(x)= 2(3-2x) x 1

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