A physical anthropologist performed a mineral analysis of nine ancient Peruvian hairs. The results for the chromium $\left(x_1\right)$ and strontium $\left(x_2\right)$ levels, in parts per million (ppm), were as follows:
$$
\begin{array}{r|rrrrrrrrr}
\hline x_1(\mathrm{Cr}) & .48 & 40.53 & 2.19 & .55 & .74 & .66 & .93 & .37 & .22 \\
\hline x_2(\mathrm{St}) & 12.57 & 73.68 & 11.13 & 20.03 & 20.29 & .78 & 4.64 & .43 & 1.08
\end{array}
$$
It is known that low levels (less than or equal to $.100 \mathrm{ppm}$ ) of chromium suggest the presence of diabetes, while strontium is an indication of animal protein intake.
(a) Construct and plot a $90 \%$ joint confidence ellipse for the population mean vector $\mu^{\prime}=\left[\mu_1, \mu_2\right]$, assuming that these nine Peruvian hairs represent a random sample from individuals belonging to a particular ancient Peruvian culture.
(b) Obtain the individual simultaneous $90 \%$ confidence intervals for $\mu_1$ and $\mu_2$ by "projecting" the ellipse constructed in Part a on each coordinate axis. (Alternatively, we could use Result 5.3.) Does it appear as if this Peruvian culture has a mean strontium level of 10 ? That is, are any of the points $\left(\mu_1\right.$ arbitrary, 10$)$ in the confidence regions? Is $[.30,10]^{\prime}$ a plausible value for $\mu$ ? Discuss.
(c) Do these data appear to be bivariate normal? Discuss their status with reference to $Q-Q$ plots and a scatter diagram. If the data are not bivariate normal, what implications does this have for the results in Parts a and b?
(d) Repeat the analysis with the obvious "outlying" observation removed. Do the infer. ences change? Comment.