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nonparametric statistical inference

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

=+ (c) Comment on your answers in (a) and (b).
=+ (b) Test the null hypothesis that the number of defectives follows a binomial distribution.
=+(a) Test the null hypothesis that the number of defectives follows a Poisson distribution.
=+ 4.33. A quality control engineer has taken 50 samples, each of size 13, from a pro- duction process. The numbers of defectives are recorded below. Number of defects 0 1 2 3 4 5 Sample frequency 9 26 9 4 1 1 0 6 or more
=+second design and that the second design will be three times as popular as the third design. In a market test with 213 persons, 111 preferred the first design, 62 preferred the second design, and the remainder preferred the third design. Are these results consistent with the marketing manager's
=+ 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
=+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?
=+ 4.31. 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.
=+4.30. For the data x: 1.0.2.3.4.2, 7.1, 10.4, use the most appropriate procedure to test the null hypothesis that the distribution is (a) Exponential Fo(x) = 1-e-x (estimate by 1/x) (b) Normal In each part, carry the parameter estimates to the nearest hundredth and the dis- tribution estimates to
=+ 4.29. Compare and contrast the chi-square and Kolmogorov-Smirnov goodness-of-fit procedures.
=+Be brief but specific about which statistical procedure to use and why it is preferred and outline the steps in the procedure.
=+How would you determine statistically whether gear 9973 meets the 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 of 500 gears from the inventory and measured the diameter of each. Nothing more has been done to the data.
=+4.28. Durtco Incorporated designs and manufactures gears for heavy-duty construc- tion equipment. One such gear, 9973, has the following specifications:
=+Does it appear that installment loan operations are under control at this time?
=+follows: A:80% B: 12% C:7% D:1% They make frequent spot checks by drawing a random sample of loan files, noting their repayment status at that time and comparing the observed distribution with the stan- dard for control. Suppose a sample of 500 files produces the following data on number of loans
=+ 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
=+ A random sample of 10 delivery persons in a nearby suburb is taken; the arrayed data for monthly collections in dollars are: 90, 106, 109, 117, 130, 145, 156, 170, 174, 190 Test the null hypothesis that the same normal distribution model applies to this suburb, using the most appropriate test.
=+4.26. Suppose that monthly collections for home delivery of the New York Times in a large suburb of New York are approximately normally distributed with mean $150 and standard deviation $20.
=+Weekly demand 0 1 2 3 More than 3 Number of weeks 28 25610 50
=+Find the theoretical distribution of weekly de- mands for a Poisson model with the same mean as the given data and perform an ap- propriate goodness-of-fit test.
=+ 4.25. During a 50-week period, demand for a certain kind of replacement part for TV sets was distributed as shown below.
=+Test the null hypothesis that the students are equally likely to select any of the numbers 0, 1, 2, 3, 4, 5, using the most appropriate test and the 0.05 level of significance.
=+ 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
=+4.23. Use the D statistic to test the null hypothesis that the data in Example 2.1: (a) Come from the Poisson distribution with = 1.5 (b) Come from the binomial distribution with n = 13, p = 0.1 These tests will be conservative because both hypothesized distributions are discrete.
=+ 4.22. For the original data in Example 3.1 (not the square roots), test the null hy- pothesis that they come from the continuous uniform distribution, using level 0.01.
=+ No. of errors No. of samples 10 16 2 3 4 5 20 28 14
=+4.21. It is claimed that the number of errors made by a typesetter is Poisson dis- tributed with an average rate of 4 per 1000 words set. One hundred random samples of sets of 1000 words from this typesetter output are examined and the numbers of errors are counted as shown below. Are these data
=+(c) Determine the sample size required to use the empirical distribution function to estimate the unknown cumulative distribution function with 95% confidence such that the error in the estimate is (i) less than 0.25, (ii) less than 0.20.
=+(b) Use the most appropriate test to see if these data can be regarded as a random sample from a normal distribution with u=3, = 1.
=+(a) Use the most appropriate test to see if these data can be regarded as a random sample from a normal distribution.
=+4.20. The data below represent earnings per share (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
=+4.19. For the data given in Example 6.1 use the most appropriate test to see if the dis- tribution can be assumed to be normal with mean 10,000 and standard deviation 2,000.
=+1.6. 10.3, 3.5, 13.5, 18.4, 7.7, 24.3, 10.7, 8.4, 4.9, 7.9, 12.0. 16.2, 6.8, 14.7
=+Test the null hypothesis that these observations can be regarded as a sample from the exponential population with density function f(x) = e-x/10/10 for x > 0.
=+4.18. In a vibration study, a random sample of 15 airplane components were subjected to severe vibrations until they showed structural failures. The data given are failure times in minutes.
=+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 S, (x) and resulting confidence bands.
=+ 4.17. A random sample of size 13 is drawn from an unknown continuous population Fx(x), with the following results after array:
=+ might be used for a goodness-of-fit test. noxx(x-1)] 2 This statistic is discussed in Cramr (1928), von Mises (1931), Smirnov (1936), and Darling (1957).
=+ (a) Prove that (b) Explain how (c) Show that is distribution free.
=+ 4.14. Related goodness-of-fit test. The Cramr-von Mises type of statistic is defined for continuous Fx(x) by co=S,(x)-Fx(x)]-fx(x) dx
=+4.13. Use Theorem 3.4 to verify directly that P(D > 0.447) = 0.10. Calculate this same probability using the expression given in (3.5).
=+4.12. Find the minimum sample size n required such that P(D, < 0.05) > 0.99.
=+ 4.11. Using Theorem 3.3, verify that lim P(D, >1.07/n) = 0.20 18-10
=+4.10. Prove that the probability distribution of D is identical to the distribution of D (a) Using a derivation analogous to Theorem 3.4 (b) Using a symmetry argument
=+4.9. Prove that ax {, max [Fx (X) - - 1], 0} D =max
=+4.8. Give a simple proof that D.D, and D are completely distribution-free for any continuous Fx by appealing to the transformation u = Fx(x) in the initial definitions of DR. D, and D.
=+By the central-limit theo- rem, approaches the standard normal distribution as no and the square of any standard normal variable is chi-square-distributed with 1 degree of freedom. Thus we have an entirely different argument for the distribution of Q when k = 2.
=+4.7. Show algebraically that where e = no and k = 2, we have (F-e) (F-n0) Q= e n01(1-0) so that when k = 2. Qis the statistic commonly used for testing a hypothesis concerning the parameter of the binomial distribution for large samples.
=+ 4.6. Show that in general, for Q defined as in (2.1), E(Q) E (F-e) = [n0 (1-0) (ne) = 1-1 el 1-1 el From this we see that if the null hypothesis is true, ne =e, and E(Q) = k - 1, the mean of the chi-square distribution.
=+ How would the chi-square test be used to test the adequacy of the general model?
=+ 4.5. If individuals are classified according to gender and color blindness, it is hy- pothesized that the distribution should be as follows: Male Female Normal P/2 p/2+pq Color blind 9/2 92/2 for some p+q=1, where p denotes the proportion of defective genes in the relevant population and
=+ 4.4. According to a genetic model, the proportions of individuals having the four blood types should be related by Type 0: q Type A: p + 2pq Type B: +2qr Type AB: 2pr where p+q+r=1. Given the blood types of 1000 individuals, how would you test the adequacy of the model?
=+Assume that X1, X2, and X, represent the respective frequencies in a sample of n in- dependent trials and that these numbers are known. Derive a chi-square goodness-of-fit test for this trinomial distribution if p is unknown.
=+ 4.3. A certain genetic model suggests that the probabilities for a particular trinomial distribution are, respectively, 0 =p2,02 = 2p(1-p), and 03 (1-p)2,0
=+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?
=+62 A monk named Mendel wrote an article theorizing that in a second generation of such hybrids, the distribution of plant types should be in a 9:3:3:1 ratio. Are the above data consistent with the good monk's theory?
=+normal, golden, green-striped, and golden-green- striped. In 1200 plants this process produced the following distribution: Normal: 670 Golden: 230 Green-striped: 238 Golden-green-striped:
=+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 dis- tinct types of plants were produced:
=+(c) Fig. 1 Nonrandom patterns representing (a) cyclical movement, (b) trend movement, (c) clustering.
=+He tested randomness against the alternative of autocorrelation. Random stock level changes occur when (a) (b)
=+3.18. Bartels (1982) illustrated the rank non Neumann 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 SA million) deflated by the Australian GDP are 528, 348, 264, -20, -167, 575, 410, -4,
=+3.17. The three graphs in Figure 1 (see below) 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 non randomness using two
=+ (b) Given an appropriate P value that reflects whether the pattern of suc- cessive departures (from one day to the next) can be considered a random process or exhibits a trend for these seven days.
=+(a) Give an appropriate P value that reflects whether the pattern of positive and negative departures can be considered a random process or exhibits a tendency to cluster.
=+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 seven consecutive days. Day 1 2 3 4 5 6 7 Departure 12 13 12 11 5 -1 -2
=+(b) Would the runs up and down test be appropriate for these data? Why or why not?
=+(a) Test these data for randomness against the alternative of a tendency to cluster, using the dichotomizing criterion that 0, 1, or 2 correct choices indicate no learning, while 3 or 4 correct choices indicate learning.
=+3.15. In a psychological experiment, the research question of interest is whether a rat "learned" its way through a maze during 65 trials. Suppose the time-ordered observa- tions on number of correct choices by the rat on each trail are as follows: 0, 1, 2, 1, 1, 2, 3, 2, 2, 2, 1, 1, 3, 2, 1, 2,
=+and measured from the beginning of one eruption to the beginning of the next: 68, 63, 66, 63, 61, 44, 60, 62, 71, 62, 62, 55, 62, 67, 73, 72, 55, 67, 68, 65, 60, 61, 71, 60, 68, 67, 72, 69, 65, 66 A researcher wants to use these data for inference purposes, but is concerned about whether it is
=+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,
=+Dec. 21 (a) Use the runs up and down test to see if these data show a directional trend and make an appropriate conclusion at the 0.05 level. (b) Use the runs above and below the sample median test to see if these data show a trend and make an appropriate conclusion at the 0.05 level. (c) Compare
=+3.13. A certain broker noted the following number of bonds sold each month for a 12- month period: Jan. 19 Feb. 23 Mar. 20 Apr. 17 May 18 June 20 July 22 Aug. 24 Sept. 25 Oct. 28 Nov. 30
=+3.12. Analyze the data in Example 4.1 for evidence of trend using total number of runs above and below (a) The sample median (b) The sample mean
=+ 3.9. Find the complete probability distribution of the number of runs up and down of various length when n = 6 using (4.1) and the results given for us (r.rs.12.1).
=+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 n = n = 6.
=+3.7. Show that the probability that a sequence of n, elements of type 1 and n, elements of type 2 begins with a type 1 run of length exactly k is (ni) na (n1+2)+1 n! where (n), (n-r)!
=+3.6. By considering the ratios fa(r)/fa(r-2) and fa(r+2)/fa(r), where r is an even positive integer and fa(r) is given in (2.3) show that if the most probable number of runs is an even integer k, then k satisfies the inequality n 20112+2 n
=+3.5. Verify that the asymptotic distribution of the random variable given in (2.9) is the standard normal distribution.
=+3.4. Show that the asymptotic distribution of the standardized random variable R-E(R)/(R) is the standard normal distribution, using the distribution of R gi- ven in (2.2) and your answer to Problem 3.2.
=+3.3. Use Lemmas 2 and 3 to evaluate the sums in (2.5), obtaining the result given in (2.6) for E(R).
=+3.2. Find the mean and variance of the number of runs R of type 1 elements, using the probability distribution given in (2.2). Since E(R) E(R1)+E(R2), use your result to verify (2.6).
=+3.1. Prove Corollary 2.1 using a direct combinatorial argument based on Lemma 1.
=+Find E½XnðtÞ and E½ZnðtÞ; var½XnðtÞ and var½ZnðtÞ, and conclude that var½XnðtÞ 4var½ZnðtÞ for all 04t41 and all n.
=+2.34. Let SnðxÞ be the empirical distribution function for a random sample of size n from the uniform distribution on (0,1). Define XnðtÞ ¼ ffiffiffi n p jSnðtÞ tj ZnðtÞ¼ðt þ 1ÞXn tt þ 1for all 0 4t41
=+2.33. Prove that cov½SnðxÞ; SnðyÞ ¼ c½FX ðxÞ; FX ðyÞ=n where cðs;tÞ ¼ minðs;tÞ st ¼ sð1 tÞtð1 sÞ if s4t if s5t and Snð:Þ is the empirical distribution function of a random sample of size n from the population Fx.
=+a relationship that is useful in calculating PðrÞ.ðcÞ Show that the number of tiles n to be put on a future test such that all of the n measurements exceed Xð1Þ with probability p is given by n ¼ mð1 pÞp 2.32. Define the random variable eðxÞ ¼ 1 0 if x 50 if x < 0 Show that the
=+ The manufacturer is interested in finding the probability that in a future test (performed by, say, an independent agency) of a random sample of n of these tiles, at least k ðk ¼ 1; 2; ... ; nÞ will have a heat resistance capacity exceeding Xð1Þ units. Assume that the heat resistance
=+2.31. A manufacturer wants to market a new brand of heat-resistant tiles which may be used on the space shuttle. A random sample of m of these tiles is put on a test and the heat resistance capacities of the tiles are measured. Let Xð1Þ denote the smallest of these measurements.
=+2.30. Let X1; X2; ... ; Xn be a random sample from the exponential distribution fX ðxÞ¼ð2yÞ1 ex=2y; x50; y > 0, and let the ordered X ’s be denoted by Y1 4Y2 4 4Yn. Assume that the underlying experiment is such that Y1 becomes available first, then Y2, and so on (for example, in a
=+2.29. Let SmðxÞ be the empirical cdf of a random sample of size m from a continuous cdf FX . Show that for 1 < x < y < 1, cov½SmðxÞ; SmðyÞ ¼ FX ðxÞ½1 FX ðyÞm
=+n þ 1 and varðT4Þ ¼ 2mðn 1Þðm þ n þ 1Þðn þ 1Þ2ðn þ 2Þ(Hackl and Katzenbeisser, 1984)The statistics T3 and T4 have been proposed as tests for H0:FX ¼ FY against the alternative that the dispersion of FX exceeds the dispersion of FY.
=+ðgÞ Let T4 be the number of X ’s in the interval I ¼ ðYðrÞ;Yðnþ1rÞ, where YðrÞ is the pth sample quantile of the Y’s. The interval I is called the interquartile range of the Y’s. Note that T4 ¼ m½SmðYðnþ1rÞÞ SmðYðrÞÞ. Show that the distribution of T4 is given by
=+ðeÞ Let T1 be the number of X observations exceeding the largest Y observation, that is, T1 ¼ m½1 SmðYðnÞÞ ¼ m PðnÞ. Show that PðT1 ¼ tÞ ¼m þ n t 1 m t m þ n mðfÞ Let T2 be the number of X ’s preceding (not exceeding) the smallest Y observation; this is, T2 ¼
=+The quantity PðiÞ is the count of the number of X ’s that precede the ith-order statistic in the Y sample and is called the ‘‘placement’’ of YðiÞ among the observations in the X sample. Observe that PðiÞ ¼ r1 þ þ ri, where ri is the ith block frequency and thus ri ¼ PðiÞ
=+ðaÞ Show that SmðYiÞ;i ¼ 1; 2; ... ; n, is uniformly distributed over the set of points ð0; 1=m; 2=m; ... ; 1Þ.ðbÞ Show that the distribution of SmðYðjÞÞ SmðYðkÞÞ; k < j, is the same as the distribution of SmðYðjkÞÞ. (Fligner and Wolfe, 1976)ðcÞ Show that the
=+precede) Y1 and may be called an exceedance statistic. Several nonparametric tests proposed in the literature are based on exceedance (or precedence) statistics and these are called exceedance (or precedence) tests. We will study some of these tests later.72 CHAPTER 2 Let Yð1Þ < Yð2Þ < <
=+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 SmðxÞand SnðyÞ be the corresponding empirical cdf ’s. Consider, for example, the quantity m½1 SmðY1Þ, which is
=+A simple distribution-free test for the equality of FX and FY can be based on S0, the number of blocks that do not contain any X observation. This is the ‘‘empty block’’ test(Wilks, 1962, pp. 446–452).
=+ðcÞ In particular show that, if FX ¼ FY, the marginal distribution of S0 is given by n þ 1 s0 m þ 1 n s0 m þ n nfor s0 ¼ n m þ 1; n m þ 2; ... ; n. (Wilks, 1962)

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