76. The one-sample t CI for m is also a con dence interval for the population median when the population distribution is normal.We now develop a CI for that is valid whatever the shape of the population distribution as long as it is continuous. Let X1, . . . , Xn be a random sample from the distribution and Y1, . . . , Yn denote the corresponding order statistics

(smallest observation, second smallest, and so on).

a. What is P(X1  )? What is P({X1  } 

{X2  })?

b. What is P(Yn  )? What is P(Y1

)? (Hint:

What condition involving all of the Xi s is equivalent to the largest being smaller than the population median?)

c. What is P(Y1 Yn)? What does this imply about the con dence level associated with the CI

(y1, yn) for ?

d. An experiment carried out to study the time (min)

necessary for an anesthetic to produce the desired result yielded the following data: 31.2, 36.0, 31.5, 28.7, 37.2, 35.4, 33.3, 39.3, 42.0, 29.9. Determine the con dence interval of

(c) and the associated con dence level. Also calculate the onesample t CI using the same level and compare the two intervals.