$4)$ Two machines are used for $$lling plastic bottles with a net volume of $16\mathrm{.}0$ ounces. The $$ll volume can be assumed to be normal with standard deviation $\backslash$sigma $1=0\mathrm{.}020$ and $\backslash$sigma $2=0\mathrm{.}025$ ounces. A member of the quality engineering sta$$ suspects that both machines $$ll to the same mean net volume, whether or not this volume is $16\mathrm{.}0$ ounces. A random sample of $10$ bottles is taken from the output of each machine.

Machine $1$ Machine $2$

$16\mathrm{.}0316\mathrm{.}0116\mathrm{.}0216\mathrm{.}03$

$16\mathrm{.}0415\mathrm{.}9615\mathrm{.}9716\mathrm{.}04$

$16\mathrm{.}0515\mathrm{.}9815\mathrm{.}9616\mathrm{.}02$

$16\mathrm{.}0516\mathrm{.}0216\mathrm{.}0116\mathrm{.}01$

$16\mathrm{.}0215\mathrm{.}9915\mathrm{.}9916\mathrm{.}00$

a$.$ Do you think the engineer is correct? Use $\backslash$alpha $=0\mathrm{.}05.$ What is the P$-$value for this test?

b$.$ Calculate a $95\%$ con$$dence interval on the di$$erence in means. Provide a practical interpretation of this interval.

c$.$ What is the power of the test in part $($a$)$ for a true di$$er$-$ ence in means of $0\mathrm{.}04?$

d$.$ Assume that sample sizes are equal. What sample size should be used to ensure that $\backslash$beta $=0\mathrm{.}05$ if the true di$$erence in means is $0\mathrm{.}04?$ Assume that $\backslash$alpha $=0\mathrm{.}05.$