Solved: Let X1, X2, ……., Xnbe a random sample from a

Chapter 7, Problem 58E

(choose chapter or problem)

Let \(X_1, X_2,\ldots ,X_n\) be a random sample from a continuous probability distribution having median \(\tilde \mu\), (so that \(P(X_i \leq \tilde \mu)= P(X_i \geq \tilde \mu) = .5\)).

a. Show that

\(P\left(\min \left(X_{i}\right)<\tilde{\mu}<\max \left(X_{i}\right)\right)=1-\left(\frac{1}{2}\right)^{n-1}\)

so that \(\left(\min \left(x_{\mathrm{i}}\right), \max \left(x_{i}\right)\right)\) is a \(100(1-\alpha)\)% confidence interval for \(\tilde \mu\) with \(\alpha=\left(\frac{1}{2}\right)^{n-1}\). [Hint: The complement of the event {min \((X_i)\) < \(\tilde \mu\) < max \((X_i)\)} is {max \((X_i) \leq \tilde \mu\)} \(\cup\) {min \((X_i) \geq  \tilde \mu\)}. But max \((X_i ) \leq \mu\) iff \(X_i \leq \tilde \mu\) for all i.]

b. For each of six normal male infants, the amount of the amino acid alanine (mg/100 mL) was determined while the infants were on an isoleucine-free diet, resulting in the following data: 2.84 3.54 2.80 1.44 2.94 2.70 Compute a 97% CI for the true median amount of alanine for infants on such a diet (“The Essential Amino Acid Requirements of Infants,” Amer. J. of Nutrition, 1964: 322–330).

c. Let \(x_{(2)}\) denote the second smallest of the \(x_i\)’s and \(x_{(n-1)}\) denote the second largest of the \(x_i\)’s. What is the confidence level of the interval \((x_{(2)}, x_{(n-1)})\) for \(\tilde \mu\)?