Let X1,X2, . . . ,Xn be a random sample of size n from the

QUESTION:

Let  \(X_{1}, X_{2^{\prime} \cdots}, X_{n}\) be a random sample of size \(n\) from the normal distribution \(N\left(\mu, \sigma^{2}\right)\) . Calculate the expected length of a  confidence interval for \(\mu\), assuming that \(n=5\) and the variance is

(a) known.

(b) unknown.

HINT: To find \(E(S)\), first determine \(E\left[\sqrt{(n-1) S^{2} / \sigma^{2}}\right]\), recalling that \((n-1) S^{2} / \sigma^{2}\) is \(\chi^{2}(n-1)\). (See Exercise 6.4-14.)

Equation Transcription:

Text Transcription:

X_1,X_2,...,X_n

n

N(mu,sigma^2)

mu

n=5

E(S)

E[sqrt(n-1)S^2/sigma^2

(n-1)S2/2

chi^2(n-1)