Let X1, . . . , Xn be a random sample from a continuous population whose median

Question:

Let X1, . . . , Xn" be a random sample from a continuous population whose median is denoted by M. For testing H0: M = M0, we can use the sign test statistic S = No. of Xi > M0,i = 1, . . . , n. H0 is rejected at level a in favor of H1: M ‰ M0 if S ‰¤ r or S ‰¥ n - r + 1, where r is the largest integer satisfying

If we repeat this test procedure for all possible values of M0, a 100( 1 - or)% confidence interval for M is then the range of values M0 so that S is in the acceptance region. Ordering the observations from smallest to largest, verify that this confidence interval becomes
( r + 1 )st smallest to ( r + 1 )st largest observation
(a) Refer to Example 6. Using the sign test, construct a confidence interval for the population median of the differences A - B, with a level of confidence close to 95%.
(b) Repeat part (a) using Darwin's data given in Exercise 15.22.