Two independent sampling stations are chosen for a study, one located downstream from an acid mine discharge point and the other located upstream. For $12$ samples collected at the upstream station, the species diversity index has mean value $\bar x_1 = 3.11$ and st. dev. $s_1=0.771$, while $10$ samples collected at the downstream station had $\bar x_2=2.04$ and $s_2=0.448$. Determine a $90\%$ confidence interval for the difference of population means assuming that the populations are normally distributed. Does the mine acid discharge affect the bio-diversity?
In a study conducted at Virginia Tech, J.A. Wesson examined the influence of the drug succinylcholine on the circulation levels of androgens in the blood. Blood samples were taken from wild, free-ranging deer immediately after they have received an intramuscular injection of succinylcholine administered using darts and a capture gun. A second blood sample was obtained from each deer $30$ minutes after the first sample after which the deer was released. The levels of androgens at time of capture and $30$ minutes later, measured in nanograms per mililiter, for $8$ deer are given in the table below.
Construct a $90\%$ confidence interval for the difference of androgen concentrations at time of capture and $30$ minutes later. Draw a conclusion.$$ \begin{array}{c|cccccccc} \text{Deer} & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ \hline \text{At capture} & 2.76 & 5.18 & 2.68 & 3.05 & 4.10 & 7.05 & 6.60 & 4.79\\ 30\text{ min later} & 7.02 & 3.10 & 5.44 & 3.99 & 5.21 & 10.26 & 13.91 & 18.53 \end{array} $$