The coefficient of variation (CV) for a sample of values

Chapter 7, Problem 95SE

(choose chapter or problem)

Let \(X_{1}, X_{2}, \ldots, X_{\mathrm{n}}\) be independent \(\mathrm{X}^{2}\)-distributed random variables, each with 1 df. Define 𝑌 as

\(Y=\sum_{i=1}^{n} X_{i}\)

It follows from Exercise 6.59 that 𝑌 has a \(\mathrm{X}^{2}\) distribution with n df.

 Use the preceding representation of 𝑌  as the sum of the \(X^{\prime} s\) to show that

\(Z=(Y-n) / \sqrt{2 n}\) has an asymptotic standard normal distribution.

 A machine in a heavy-equipment factory produces steel rods of length 𝑌 , where 𝑌 is a normally distributed random variable with mean 6 inches and variance .2. The cost C of repairing a rod that is not exactly 6 inches in length is proportional to the square of the error and is given, in dollars, by \(C=4(Y-\mu)^{2}\). If 50 rods with independent lengths are produced in a given day, approximate the probability that the total cost for repairs for that day exceeds $48.

Equation Transcription:

   

 

   

Text Transcription:

X_1, X_2 \ldots, X_n

X^2

Y=\sum_i=1^n X_i

X^2

X^\prime s

Z=(Y-n) / \sqrt 2 n

C=4(Y-\mu)^2