New Semester Started Get 50% OFF Study Help! --h --m --s Claim Now

Question Answers

Textbooks

Find textbooks, questions and answers

Oops, something went wrong!

Change your search query and then try again

Result Not Found

Books FREE
Study Help
Tutors
AI Study Planner NEW
Sell Books

Search
Sign In
Register

study help
business
nonparametric statistical inference

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

3.13 A certain broker noted the following number of bonds sold each month for a 12 month period:Jan. 19 July 22 Feb. 23 Aug. 24 Mar. 20 Sept. 25 Apr. 17 Oct. 28 May 18 Nov. 30 June 20 Dec. 21(a) Use the runs up and down test to see if these data show a directional trend and make an appropriate
3.14 The following are 30 time lapses in minutes between eruptions of Old Faithful geyser in Yellowstone National Park, recorded between the hours of 8 a.m. and 10 p.m. on a certain day, and measured from the beginning of one eruption to the beginning of the next:68, 63, 66, 63, 61, 44, 60, 62, 71,
3.15 In a psychological experiment, the research question of interest is whether a rat ‘‘learned’’ its way through a maze during 64 trials.Suppose the time-ordered observations on number of correct choices by the rat on each trial are as follows:0, 1, 2, 1, 1, 2, 3, 2, 2, 2, 1, 1, 3, 2, 1,
3.16 The data below represent departure of actual daily temperature in degrees Fahrenheit from the normal daily temperature at noon at a certain airport on 7 consecutive days.Day 1 2 3 4 5 6 7 Departure 12 13 12 11 5 1 2(a) Give an appropriate P value that reflects whether the pattern of positive
3.17 The three graphs in Figure P.3.17 illustrate some kinds of nonrandom patterns. Time is on the horizontal axis. The data values are indicated by dots and the horizontal line denotes the median of the data. For each graph, compute the one-tailed P value for nonrandomness using two different
3.18 Bartels (1982) illustrated the RVN test for randomness using data on annual changes in stock levels of corporate trading enterprises in Australia for 1968–1969 to 1977–1978. The values (in $A million) deflated by the Australian GDP are 528, 348, 264, 20, 167, 575, 410, 4, 430, 122. He
4.1 Two types of corn (golden and green-striped) carry recessive genes.When these were crossed, a first generation was obtained, which was consistently normal (neither golden nor green-striped). When this generation was allowed to self-fertilize, four distinct types of plants were produced: normal,
4.2 A group of four coins is tossed 160 times, and the following data are obtained:Number of heads 0 1 2 3 4 Frequency 16 48 55 33 8 Do you think the four coins are balanced?
4.3 A certain genetic model suggest that the probabilities for a particular trinomial distribution are, respectively, u1 ¼ p2, u2 ¼ 2p(1 p), and u3 ¼ (1 p)2, 0 < p < 1. Assume that X1, X2, and X3 represent the respective frequencies in a sample of n independent trials and that these numbers
4.4 According to a genetic model, the proportions of individuals having the four blood types should be related by Type 0: q2 Type A: p2 þ 2pq Type B: r2 þ 2qr Type AB: 2pr where p þ q þ r ¼ 1. Given the blood types of 1000 individuals, how would you test the adequacy of the model?
4.5 If individuals are classified according to gender and color blindness, it is hypothesized that the distribution should be as follows:Male Female Normal p=2 p2=2 þ pq Color blind q=2 q2=2 for some p þ q ¼ 1, where p denotes the proportion of defective genes in the relevant population and
4.6 Show that in general, for Q defined as in (4.2.1), E(Q) ¼ E Xk i¼1(Fi ei)2 ei" #¼Xk i¼1 nui(1 ui)ei þ(nui ei From this we see that if the null hypothesis is true, nui ¼ ei and E(Q) ¼ k 1, the mean of the chi-square distribution.
4.7 Show algebraically that when ei ¼ nui and k ¼ 2, we have Q ¼X2 i¼1(Fi ei)2 ei ¼(F1 nu1)2 nu1(1 u1)so that when k ¼ 2, ffiffiffiffi pQ is the statistic commonly used for testing a hypothesis concerning the parameter of the binomial distribution for large samples. By the central-limit
4.8 Give a simple proof that Dn,Dþn, and Dn are completely distribution free for any continuous and completely specified FX by appealing to the transformation u ¼ FX(x) in the initial definitions of Dn,Dþn, and Dn .
4.9 Prove that Dn ¼ max max 1in F0(X(i)) i 1 n , 0
4.10 Prove that the null distribution of Dn is identical to the null distribution of Dþn:(a) Using a derivation analogous to Theorem 4.3.4(b) Using a symmetry argument
4.11 Using Theorem 4.3.3, verify that lim n!1 P Dn >1:0ffiffi7ffi pn ¼ 0:20
4.12 Find the minimum sample size n required such that P(Dn < 0.05) 0.99.
4.13 Use Theorem 4.3.4 to verify directly that P Dþ5> 0:447¼ 0:10. Calculate this same probability using the expression given in (4.3.5).
4.14 Related goodness-of-fit test. The Cramér–von Mises type of statistic is defined for continuous F0(x) by v2 n ¼ð11[Sn(x) F0(x)]2fX(x)dx(a) Prove that v2 n is distribution free.(b) Explain how v2 n might be used for a goodness-of-fit test.(c) Show that nv2 n ¼1 12n þXn i¼1 F0 X(i) 2i
4.15 Suppose we want to estimate the cumulative distribution function of a continuous population using the empirical distribution function such that the probability is 0.90 that the error of the estimate does not exceed 0.25 anywhere. How large a sample size is needed?
4.16 If we wish to estimate a cumulative distribution within 0.20 units with probability 0.95, how large should n be?
4.17 A random sample of size 13 is drawn from an unknown continuous population FX(x), with the following results after array:3.5, 4.1, 4.8, 5.0, 6.3, 7.1, 7.2, 7.8, 8.1, 8.4, 8.6, 9.0 A 90% confidence band is desired for FX(x). Plot a graph of the empirical distribution function Sn(x) and resulting
4.18 In a vibration study, a random sample of 15 airplane components were subjected to server vibrations until they showed structural failures. The data given are failure times in minutes. Test the null hypothesis that these observations can be regarded as sample from the exponential population
4.19 For the data given in Example 4.5.1, use the most appropriate test to see if the distribution can be assumed to be normal with mean 10,000 and standard deviation 2,000.
4.20 The data below represent earnings (in dollars) for a random sample of five common stocks listed on the New York Stock Exchange.1.68, 3.35, 2.50, 6.23, 3.24(a) Use the most appropriate test to see if these data can be regarded as a random sample from a normal distribution.(b) Use the most
4.21 It is claimed that the number of errors made by a typesetter is Poisson distributed with an average rate of 4 per 1000 words. One hundred random samples of 1000 words from this typesetter are examined and the number of errors are counted as shown below. Are these data consistent with the
4.22 For the original data in Example 4.4.1 (not the square roots), test the null hypothesis that they come from the continuous uniform distribution, using level 0.01.
4.23 Use the Dn statistic to test the null hypothesis that the data in Example 4.2.1(a) Come from the Poisson distribution with m ¼ 1.5(b) Come from the binomial distribution with n ¼ 13, p ¼ 0.1 These tests will be conservative because both hypothesized distribution are discrete.
4.24 Each student in a class of 18 is asked to list three people he likes and three he dislikes and label the people 0, 1, 2, 3, 4, 5 according to how much he likes them, with 0 denoting least liked and 5 denoting most liked. From this list, each student selects the number assigned to the person he
4.25 During a 50 week period, demand for a certain kind of replacement part for TV sets is shown below. Find the theoretical distribution of weekly demands for a Poisson model with the same mean as the given data and perform an appropriate goodness-of-fit test.Weekly Demand Number of Weeks 0 28 1
4.26 Suppose that monthly collections for home delivery of the New York Times in a large suburb of New York City are approximately normally distributed with mean $150 and standard deviation $20. A random sample of 10 delivery persons in a nearby suburb is taken; the arrayed data for monthly
4.27 A bank frequently makes large installment loans to builders. At any point in time, outstanding loans are classified in the following four repayment categories:A: Current B: Up to 30 days delinquent C: 30–60 days delinquent D: Over 60 days delinquent The bank has established the internal
4.28 Durtco Incorporated designs and manufactures gears for heavy-duty construction equipment. One such gear, 9973, has the following specifications:(a) Mean diameter 3.0 in.(b) Standard deviation 0.001 in.(c) Output normally distributed The production control manager has selected a random sample
4.29 Compare and contrast the chi-square and K–S goodness-of-fit procedures.
4.30 For the data 1.0, 2.3, 4.2, 7.1, 10.4, use the most approximate procedure to test the null hypothesis that the distribution is(a) Exponential FX(x) ¼ 1 ex=b (estimate b by x)(b) Normal In each part, carry the parameter estimates to the nearest hundredth and the distribution estimates to
4.31 A statistics professor claims that the distribution of final grades from A to F in a particular course invariably is in the ratio 1:3:4:1:1. The final grades this year are 26 A’s, 50 B’s, 80 C’s, 35 D’s, and 10 F’s. Do these results refute the professor’s claim?
4.32 The design department has proposed three different package designs for the company’s product; the marketing manager claims that the first design will be twice as popular as the second design and that the second design will be three times as popular as the third design. In a market test with
4.33 A quality control engineer has taken 50 samples, each of size 13, from a production process. The numbers of defectives are recorded below.Number of Defects Sample Frequency 0 9 1 26 2 9 3 4 4 1 5 1 6 or more 0(a) Test the null hypothesis that the number of defectives follows a Poisson
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 F0(x) ¼0 x < 0 x2(3 2x) 0x 1 1 x > 1
4.35 At the completion of a basketball training camp, participants are classified into categories, A (highest), B and C, according to their shooting ability. A sample of this year’s participants yielded the following data:
A: 136 B: 38 C: 26 Historically, the percentages of participants falling into each category has been A: 80% B: 12% C: 8%Does it appear that this year’s participants correspond to the historic percentages in their shooting ability?
4.36 Many genetic traits occur in large populations according to the Hardy–Weinberg Law, which is based on the binomial expansion(p þ q)2 ¼ p2 þ 2pq þ q2 The so-called M–N blood types of humans have three antigens, M only, N only, and both M and N, with respective probabilities p2, q2, and
4.37 According to test theory, scores on a certain IQ test are normally distributed. This test was given to 18 girls of similar age and their scores were 114, 81, 87, 114, 113, 87, 111, 89, 93, 108, 99, 93, 100, 95, 93, 95, 106, 108. Test the null hypothesis that these scores are normally
5.1 Give a functional definition similar to (5.5.1) for the rank r(Xi) of a random variable in any set of N independent observations where ties are dealt with by the midrank method. Hint: In place of S(u) in (5.5.2), consider the function c(u) ¼0 if u > 0 1=2 if u = 0 1 if u > 0(
5.2 Find the correlation coefficient between variate values and ranks in a random sample of size N from(a) The uniform distribution(b) The standard normal distribution(c) The exponential distribution
5.3 Verify the cdf of differences given in (5.4.14) and the resultM ¼ 2 þffiffiffi 3p .Find and graph the corresponding probability function of differences.
5.4 Answer parts (a) through (e) using (1) the sign-test procedure and (2)the Wilcoxon signed-rank test procedure.(a) Test at a significance level not exceeding 0.10 the null hypothesis H0:M¼2 against the alternative H1:M>2, where Mis the median of the continuous symmetric population from which
5.5 Generate the null sampling distributions of Tþ and T for a random sample of six unequal and nonzero observations.
5.6 Show by calculations from tables that the normal distribution provides reasonably accurate approximations to the critical values of one-sided tests for a¼0.01, 0.05, and 0.10 when(a) N¼12 for the sign test(b) N¼15 for the signed-rank test
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 8.(a) Plot the power curve using the exact distribution of K.(b) Plot
5.8 Prove that the Wilcoxon signed-rank statistic TþT based on a set of nonzero observations X1,X2, . . . ,XN can be written in symbols as XX 1ijN sgn(Xi þ Xj)where sgn(x) ¼1 if x > 01 if x < 0
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 ¼Xi if Di > 0 Yi if Di < 0Assume there are m X values and n Y values, where mþn¼N and the X and Y values are independent. Show that the
zX and Y populations are identical.
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 ratios of the number of single-vehicle nighttime fatalities to the number of licensed drivers in the affected age group before and
5.11 The conclusion in Problem 5.10 was that the median difference (Before minus After) was positive for the affected age group, but this does not imply that the reduction was the result of raising the minimum legal drinking age. Other factors, countermeasures, or advertising campaigns[like MADD
5.12 Howard et al. (1986) reported a study designed to investigate whether computer anxiety changes between the beginning and end of a course on introduction to computers. The 14 student subjects were given a test to measure computer anxiety at the beginning of the term and then at the end of the 5
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, improved, or reduced level of performance compared with the midterm. Six showed improvement, 5 showed no change,
5.14 Reducing high blood pressure by diet requires reduction of sodium intake, which usually requires switching from processed foods to their natural counterparts. Listed below are the average sodium contents of five ordinary foods in processed form and natural form for equivalent quantities. Find
5.15 For the data in Problem 4.20, use both the sign test and the signedrank test to investigate the research hypothesis that median earnings exceed 2.0.
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 minute. The number of correctly written words
5.17 For the data in Example 5.4.3, test H0 :M¼0.50 against the alternative H1 :M>0.50, using the(a) Sign test(b) Signed-rank test and assuming symmetry
5.18 For the data in Example 5.7.1, find a confidence-interval estimate of the median difference Before minus After using the level nearest 0.90.
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 shown below. Another user claims to have found that
5.20 A manufacturer of suntan lotion is testing a new formula to see whether it provides more protection against sunburn than the old formula. The manufacturer chose 10 persons at random from among the company’s employees, applied the two types of lotion to their backs, one type on each side, and
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 minutes 15 seconds.The recession has reduced sales, but the company’s treasurer claims that the median length of long-distance calls now is even greater than last
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 are gross sales by these representatives before and after the course.Representative
5.23 In a marketing research test, 15 adult males were asked to shave one side of their face with a brand A razor blade and the other side with a brand B razor blade and state their preferred blade. Twelve men preferred brand A. Find the P value for the alternative that the probability of
5.24 Let X be a continuous random variable symmetrically distributed about u. Show that the random variables jXuj and Z are independent, where Z ¼1 if X > u 0 if X u
5.25 Using the result in Problem 5.24, show that for the Wilcoxon signedrank test statistic Tþ, the 2N random variables Z1, r(jD1j), Z2, r(jD2j), . . . ,ZN, r(jDNj) are mutually independent under H0.
5.26 Show that the null distribution of the Wilcoxon signed-rank test statistic Tþ is the same as that of W ¼PNi¼1Wi, where W1,W2, . . . ,WN are independent random variables with P(Wi ¼ 0) ¼ P(Wi ¼ i) ¼ 0:5, i ¼ 1, 2, . . . ,N.
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 h 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 amounts
5.28 Find a confidence-interval estimate of the median amount of sleep per 24 hours for the data in Problem 5.27 using confidence coefficient nearest 0.90.
5.29 Let X(r) denote the rth-order statistic of a random sample of size 5 from any continuous population and kp denote the pth quantile of this population.Find:(a) P(X(1) < k0:5 < X(5))(b) P(X(1) < k0:25 < X(3))(c) P(X(4) < k0:80 < X(5))
5.30 For order statistics of a random sample of size N from any continuous population FX, show that the interval (X(r),X(Nrþ1), r < N=2), is a 100(1a)% confidence-interval estimate for the median of FX, where 1 a ¼ 1 2N N 1 r 1 ð 0:5 0xNr(1 x)r1dx
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:(a) P(X(1) < k0:50 < X(N)) 0:99(b) P[FX(X(N)) FX(X(1)) 0:5] 0:95
5.32 Derive the sample size formula based on the normal approximation for the sign test against a two-sided alternative with approximate size a and power 1b.
5.33 Derive the sample size formula based on the normal approximation for the signed rank test against a two-sided alternative with approximate size a and power 1b.
6.1 Use the graphical method of Hodges described in Section 6.3 to find P(Dþm,n d) under H0, where d is the observed value of Dþm,n ¼ maxx [Sm(x) Sn(x)] in the arrangement xyyxyx.
6.2 For the median-test statistic derive the complete null distribution of U for m ¼ 6, n ¼ 7, and set up one- and two-sided critical regions when a ¼ 0:01, 0:05, and 0.10.
6.3 Find the large-sample approximation to the power function of a twosided median test for m ¼ 6, n ¼ 7, a ¼ 0:10, when FX is the standard normal distribution.
6.4 Use the recursion relation for the Mann–Whitney test statistic given in(6.6.14) to generate the complete null probability distribution of U for all m þ n 4.
6.5 Verify the expressions given in (6.6.15) for the moments of U under H0.
6.6 Answer parts (a) to (c) using (i) the median-test procedure and (ii) the Mann–Whitney 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 u in their locations:X 79 13 138 129 59 76
6.7 Represent a sample of m X and n Y random variable by a path of m þ n steps, the ith step being one unit up or to the right according as the ith from the smallest observation in the combined sample is an X or a Y, respectively. What is the algebraic relation between the area under the path and
6.8 Give some other functions of the difference Sm(x) Sn(x) (besides the maximum) which could be used for distribution-free tests of the equality of two population distributions.
6.9 The 2000 census statistics for Alabama give the percentage changes in population between 1990 and 2000 for each of the 67 counties. These counties were divided into two mutually independent groups, rural and nonrural, according to population size of less than 25,000 in 2,000 or not.Random
6.10 (a) Show that the distribution of the precedence statistic P(i) under the null hypothesis FX ¼ FY, given in Problem 2.28(c), can be expressed as P(P(i) ¼ jjH0) ¼n m þ n mj n 1 i 1 m þ n 1 j þ i 1 ¼i j þ 1 mj n i m þ n j þ i j ¼ 0, 1, . . . ,m These relationships
6.11 For the control median test statistic V, use Problem 2.28, or otherwise, to show that when FX ¼ FY, E(V) ¼m 2and var(V) ¼2r þ m þ 2 4m(2r þ 3)[Hint: Use the fact that E(X) ¼ EYE(XjY) and var(X) ¼ varYE(XjY) þ EYvar(XjY)]
6.12 Show that when m, n!1such that m=(m þ n) ! l, 0 < l < 1, the null distribution of the precedence statistic P(i) given in Problem 6.10 tends to the negative binomial distribution with parameters i and l, or j þ i 1 i 1 lj(1 l)j j ¼ 0, 1, . . . ,m (Sen, 1964)
6.13 In some applications the quantity jp ¼ FX(kp), where kp is the pth quantile of FY, is of interest. Let limn!1 (m=n) ¼ l, where l is a fixed quantity, and let {rn} be a sequence of positive integers such that limn!1 (rn=n) ¼ p. Finally let Vm, n be the number of X observations that do not
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 numbers of repetitions necessary for correct completion are 1, 2, and 5 for the girls and 4, 8, 9, 10, and
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, five are randomly assigned to the placebo and five to the new drug.Suppose that the
7.1 One of the simplest linear rank statistics is defined as WN ¼XN i¼1 iZi This is the Wilcoxon rank-sum statistic to be discussed in the next chapter. Use Theorem 7.3.2 to evaluate the mean and variance of WN under H0.
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 7.3.2 to find its mean and variance. Hint: For the appropriate argument k, use the functions S(k)defined as for (7.2.1).
7.3 Prove the three properties stated in Theorem 7.3.7.
7.4 For m¼n¼2, derive the probability mass function of TN, the sum of the X ranks, under H0. Determine whether this distribution is symmetric and if so, identify the point of symmetry. Calculate the mean and variance of TN.

Showing 5300 - 5400 of 5397

First
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54

Step by Step Answers

Services

Sitemap
Fun
Definitions
Become Tutor
Used Textbooks
Study Help Categories
Recent Questions
Expert Questions
Campus Wear
Sell Your Books

Company Info

Security
Copyrights
Privacy Policy
Terms & Conditions
SolutionInn Fee
Scholarship
Online Quiz
Give Feedback, Get Rewards

Get In Touch

About Us
Contact Us
Career
Jobs
FAQ
Student Discount
Campus Ambassador

Secure Payment

Download Our App

© 2026 SolutionInn. All Rights Reserved