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nonparametric statistical inference

Questions and Answers of Nonparametric Statistical Inference

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
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
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
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,
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
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
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
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
5.8 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
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
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
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
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
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
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
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
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
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
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
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
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.
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
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
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) <
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
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) <
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
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
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
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
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
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
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
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
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.
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
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
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

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