To give some indication of how the values in Table 9.7-1

Chapter 9, Problem 5E

(choose chapter or problem)

To give some indication of how the values in Table 9.7-1 are calculated, values of \(A_{3}\) are found in this exercise. Let \(X_{1}, X_{2}, \ldots, X_{n}\) be a random sample of size \(n\) from the normal distribution \(N\left(\mu, \sigma^{2}\right)\). Let \(S^{2}\) equal the sample variance of this random sample.

(a) Use the fact that \(Y=(n-1) S^{2} / \sigma^{2}\) has a distribution that is \(\chi^{2}(n-1)\) to show that \(E\left[S^{2}\right]=\sigma^{2}\).

(b) Using the \(\chi^{2}(n-1)\) pdf, find the value of \(E(\sqrt{Y})\).

(c) Show that

\(E\left[\frac{\sqrt{n-1} \Gamma\left(\frac{n-1}{2}\right)}{\sqrt{2} \Gamma\left(\frac{n}{2}\right)} S\right]=\sigma\)

(d) Verify that

\(\frac{3}{\sqrt{n}}\left[\frac{\sqrt{n-1} \Gamma\left(\frac{n-1}{2}\right)}{\sqrt{2} \Gamma\left(\frac{n}{2}\right)}\right]=A_{3}\)

found in Table 9.7-1 for \(n=5\) and \(n=6\). Thus, \(A_{3} \bar{s}\) approximates \(3 \sigma / \sqrt{n}\).

Equation Transcription:

Text Transcription:

A_3

X_1,X_2,,X_n

n

N(mu,sigm^2)

S^2

Y=(n−1)S^2/sigma^2

chi^2(n−1)  

E[S^2]=sigma^2 .

E(sqrt Y)    

E[ sqrt n−1 gamma (n−1/2)/ sqrt 2 gamma (n/2)S]= sigma

3/ sqrt n [ sqrt n−1 gamma (n−1/2) sqrt 2  gamma (n/2)=A3

n=5  

n=6

A_3^ bar s

3 sigma / sqrt n