Consider again the data in Exercise 4. Test the hypothesis

. Suppose that (X1, Y1), . . . , (Xn, Yn) are i.i.d. pairs of random variables with a continuous joint distribution. Let p = Pr(Xi Yi), and suppose that we want to test the hypotheses H0: p 1/2, H1: p > 1/2. (10.8.7) Describe a version of the sign test to use for testing these hypotheses.

Consider again the data in Example 10.8.4. Test the hypotheses (10.8.2) by applying the Kolmogorov-Smirnov test for two samples.

Consider again the data in Example 10.8.4. Test the hypotheses (10.8.2) by assuming that the observations are taken from two normal distributions with the same variance, and apply a t test of the type described in Sec. 9.6.

In an experiment to compare the effectiveness of two drugs A and B in reducing blood glucose concentrations, drug A was administered to 25 patients, and drug B was administered to 15 patients. The reductions in blood glucose concentrations for the 25 patients who received drug A are given in Table 10.44. The reductions in concentrations for the 15 patients who received drug B are given in Table 10.45. Test the hypothesis that the two drugs are equally effective in reducing blood glucose concentrations by using the Wilcoxon-Mann-Whitney ranks test. Table 10.44 Data for patients who receive drug A in Exercise 4 0.35 1.12 1.54 0.13 0.77 0.16 1.20 0.40 1.38 0.39 0.58 0.04 0.44 0.75 0.71 1.64 0.49 0.90 0.83 0.28 1.50 1.73 1.15 0.72 0.91 Table 10.45 Data for patients who receive drug B in Exercise 4 1.78 1.25 1.01 1.82 1.95 1.81 0.68 1.48 1.59 0.89 0.86 1.63 1.26 1.07 1.3

Consider again the data in Exercise 4. Test the hypothesis that the two drugs are equally effective by applying the Kolmogorov-Smirnov test for two samples.

Consider again the data in Exercise 4. Test the hypothesis that the two drugs are equally effective by assuming that the observations are taken from two normal distributions with the same variance and applying a t test of the type described in Sec. 9.6.

Suppose that X1,...,Xm form a random sample of m observations from a continuous distribution for which the p.d.f. f (x) is unknown, and that Y1,...,Yn form an independent random sample of n observations from another continuous distribution for which the p.d.f. g(x) is also unknown. Suppose also that f (x) = g(x ) for < x < , where the value of the parameter is unknown ( << ). Let F 1 be the quantile function of the Xis, and let G1 be the quantile function of the Yj s. Show that F 1(p) = + G1(p) for all 0

Consider again the conditions of Exercise 7. Describe how to carry out a one-sided Wilcoxon-Mann-Whitney ranks test of the following hypotheses: H0: 0, H1: > 0.

Consider again the conditions of Exercise 7. Describe how to carry out a two-sided Wilcoxon-Mann-Whitney ranks test of the following hypotheses for a specified value of 0: H0: = 0, H1: =

Consider again the conditions of Exercise 9. Describe how to use the Wilcoxon-Mann-Whitney ranks test to determine a confidence interval for with confidence coeffi- cient 1 0. Hint: For which values of 0 would you accept the null hypothesis H0 : = 0 at level of significance 0?

Let X1,...,Xm and Y1,...,Yn be the observations in two samples, and suppose that no two of these observations are equal. Consider the mn pairs (X1, Y1) . . . (X1, Yn), (X2, Y1) . . . (X2, Yn), . . . (Xm, Y1) . . . (Xm, Yn). Let U denote the number of these pairs for which the value of the X component is greater than the value of the Y component. Show that U = S 1 2 m(m + 1), where S is the sum of the ranks assigned to X1,...,Xm, as defined in this sectio

. Let X1,...,Xm be i.i.d. with c.d.f. F independently of Y1,...,Yn, which are i.i.d. with c.d.f. G. Let S be as defined in this section. Prove that Eq. (10.8.5) gives the mean of S.

Under the conditions of Exercise 12, prove that Eq. (10.8.6) gives the variance of S.

Under the conditions of Exercises 12 and 13, suppose further that F = G. Prove that Eqs. (10.8.5) and (10.8.6) agree with Eqs. (10.8.3) and (10.8.4), respectively

Consider again the conditions of Exercise 1. This time, let Di = Xi Yi. Wilcoxon (1945) developed the following test of the hypotheses (10.8.7). Order the absolute values |D1|,..., |Dn| from smallest to largest, and assign ranks from 1 to n to the values. Then SW is set equal to the sum of all the ranks of those |Di| such that Di > 0. If p = Pr(Xi Yi) = 1/2, then the mean and variance of SW are E(SW ) = n(n + 1) 4 , (10.8.8) Var(SW ) = n(n + 1)(2n + 1) 24 . (10.8.9) The test rejects H0 if SW c, where c is chosen to make the test have level of significance 0. This test is called the Wilcoxon signed ranks test. If n is large, a normal distribution approximation allows us to use c = E(SW ) + 1(1 0) Var(SW )1/2. a. Let Wi = 1 if Xi Yi, and Wi = 0 if not. Show that SW = n i=1 iWi. b. Prove that E(SW ) is as stated in Eq. (10.8.8) under the assumption that p = 1/2. Hint: You may wish to use Eq. (4.7.13). c. Prove that Var(SW ) is as stated in Eq. (10.8.9) under the assumption that p

In an experiment to compare two different materials A and B that might be used for manufacturing the heels of mens dress shoes, 15 men were selected and fitted with a new pair of shoes on which one heel was made of material A and one heel was made of material B. At the beginning of the experiment, each heel was 10 millimeters thick. After the shoes had been worn for one month, the remaining thickness of each heel was measured. The results are given in Table 10.46. Test the null hypothesis that material A is not more durable than material B against the alternative that material A is more durable than material B, by using (a) the sign test of Exercise 1, (b) the Wilcoxon signedranks test of Exercise 15, and (c) the paired t test. Table 10.46 Data for Exercise 16 Pair Material A Material B 1 6.6 7.4 2 7.0 5.4 3 8.3 8.8 4 8.2 8.0 5 5.2 6.8 6 9.3 9.1 7 7.9 6.3 8 8.5 7.5 9 7.8 7.0 10 7.5 6.6 11 6.1 4.4 12 8.9 7.7 13 6.1 4.2 14 9.4 9.4 15 9.1