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nonparametric statistical inference

Questions and Answers of Nonparametric Statistical Inference

=+9.1. Develop by enumeration for m ¼ n ¼ 3 the null probability distribution of Mood’s statistic MN.
=+8.16. Verify the confidence interval estimate of Hx-Hy with exact confidence coeffi- cient at least 0.95 given in the MINITAB solution to Example 2.1.
=+(a) Find a P value relevant to the alternative that psychiatric patients have lower self-esteem than normal patients. 0.90). (b) Find a confidence interval for the difference of the locations
=+8.15. A self-concept test was given to a random sample consisting of six normal sub- jects and three subjects under psychiatric care. Higher scores indicate more self-esteem. The data are as
=+8.14. How would you find the confidence-interval end points for the parameter of in- terest when the interval has confidence level nearest 0.90 and corresponds to: (a) The sign text with n = 11 (b)
=+data below using both a test and a confidence-interval approach with confidence coefficient near 0.90. Female pigs: 9.31, 9.57, 10.21, 8.86, 8.52, 10.53, 9.21 Male pigs: 9.14,9.98, 8.46, 8.93,
=+They placed 8 young male pigs in one pen 8 young females in another pen and gave each pen identical feeding treatments for a fixed period of time. The initial weights were all between 35 and 50 lb,
=+female pigs when the two genders are subjected to identical feeding treat- ments. If there is a difference, the farmer can optimize production by buying only one gender of pigs for fattening. As a
=+8.13. A problem of considerable import to the small-scale farmer who purchases young pigs to fatten and sell for slaughter is whether there is any difference in weight gain for male and
=+ (a) The null hypothesis that the two populations are identical and find the P value (Do not use an approximate test) (b) The null hypothesis that the locations are the same and find the
=+8.12. The following data represent two independent random samples drawn from continuous populations which are thought to have the same form but possibly different locations. X: 79, 13, 138, 129,
=+ we can assume that the drug group cannot be worse (die sooner) than the control group under any reasonable conditions. Test the null hypothesis that the drug is without effect at a significance
=+infected, in a random sequence, with tuberculosis. The number of days until the mice die after infection are given as follows (one mouse got lost): Control A: 5,6,7,7,8,8,9,12 Drug B: 7,8,8,8,9, 9,
=+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)
=+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
=+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
=+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
=+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
=+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,
=+ 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
=+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
=+ 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
=+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
=+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
=+ 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
=+ 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
=+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
=+ 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) =
=+(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)
=+ 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
=+ 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
=+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
=+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
=+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
=+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
=+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
=+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
=+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
=+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
=+(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
=+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
=+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
=+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
=+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
=+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
=+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,
=+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
=+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
=+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
=+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
=+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.

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