Chapter 5.5, Problem 5E

Let X equal the weight (in grams) of a nail of the type that is used for making decks Assume that the distribution of X is $N\left(8.78,0.16\right)$.

Let $\bar{X}$ be the mean of a random sample of the weights of $n=9$ nails

(a) Sketch, on the same set of axes, the graphs of the pdfs of X and of $\bar{X}$?

(b) Let $S^2$ be the sample variance of the nine weights Find constants a and b so that $P\left(a\le S^2\le b\right)=0.90$.

HINT $P\left(a\le S^2\le b\right)$ is equivalent to $P\left(\frac{8a}{0.16}\le \frac{8S^2}{0.16}\le \frac{8b}{0.16}\right)$, and $\frac{8S^2}{0.16}$ is $x^2\left(8\right)$.

Find $\frac{8a}{0.16}$ and $\frac{8b}{0.16}$ from Table IV in Appendix B.