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nonparametric statistical inference

Questions and Answers of Nonparametric Statistical Inference

=+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
=+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
=+(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
=+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
=+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
=+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
=+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
=+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
=+ (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
=+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
=+ 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
=+ 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
=+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
=+ 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
=+ 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
=+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
=+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
=+ 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
=+(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
=+(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
=+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
=+ 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
=+ 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
=+ 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
=+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
=+ 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
=+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).
=+(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
=+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
=+ (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
=+(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
=+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
=+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
=+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
=+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,
=+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
=+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
=+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
=+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

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