Suppose that T is defined as in Definition 7.2. Reference

Chapter 7, Problem 98SE

(choose chapter or problem)

The coefficient of variation (CV) for a sample of values 𝑌1, 𝑌2, . . . , 𝑌n is defined by

\(\mathrm{CV}=S / \bar{Y}\)

This quantity, which gives the standard deviation as a proportion of the mean, is sometimes informative. For example, the value S = 10 has little meaning unless we can compare it to something else. If S is observed to be 10 and \(\bar{Y}\) is observed to be 1000, the amount of variation is small relative to the size of the mean. However, if S is observed to be 10 and \(\bar{Y}\) is observed to be 5, the variation is quite large relative to the size of the mean. If we were studying the precision (variation in repeated measurements) of a measuring instrument, the first case

(CV = 10/1000) might provide acceptable precision, but the second case (CV = 2) would be unacceptable. Let \(Y_{1}, Y_{2}, \ldots, Y_{10}\) denote a random sample of size 10 from a normal distribution with mean 0 and variance \(\sigma^{2}\). Use the following steps to find the number c such that

\(P\left(-c \leq \frac{S}{\bar{Y}} \leq c\right)=.95\)

Use the result of Exercise 7.33 to find the distribution of

\(\text { (10) } \bar{Y}^{2} / \mathrm{S}^{2}\)

Use the result of Exercise 7.29 to find the distribution of

\(S^{2} /\left[(10) \bar{Y}^{2}\right]\)

      (c) Use the answer to (b) to find the constant c.

Equation Transcription:

Text Transcription:

Y1, Y2,...,Yn

CV=S / \bar Y

\bar Y

\barY

Y1, Y2,...,Y10

\sigma^2

P\(-c \leq S\bar Y \leq c)=.95

(10)  \bar Y^2 /S^2)

S^2[(10) \barY^2\right]