Let X1, X2,, Xn be a random sample from a

Chapter 7, Problem 58

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Let X1, X2,, Xn be a random sample from a continuousprobability distribution having median m, (so thatP(Xi # m,) 5 P(Xi $ ,m) 5 .5).a. Show thatP(min (Xi) , m, , max (Xi)) 5 1 2 1122n21so that (min(xi), max(xi)) is a 100(1 2 a)% confidenceinterval for ,m with a 5 ( 12 ) n21. [Hint: The complementof the event {min (Xi) , m, , max (Xi)} is {max (Xi) #m,} {min (Xi) $ m,}. But max (Xi) # m, iff Xi # ,mfor all i.]b. For each of six normal male infants, the amount ofthe amino acid alanine (mg/100 mL) was determinedwhile the infants were on an isoleucine-free diet,resulting in the following data:2.84 3.54 2.80 1.44 2.94 2.70Compute a 97% CI for the true median amount ofalanine for infants on such a diet (The EssentialAmino Acid Requirements of Infants, Amer. J. ofNutrition, 1964: 322330).c. Let x(2) denote the second smallest of the xis andx(n21) denote the second largest of the xis. What isthe confidence level of the interval (x(2), x(n21)) form,?