Find the covariance s_(xy) using bivariate data.

The average credit card debt for a recent year was $9205. Five years earlier the average credit card
debt was $6618. Assume sample sizes of 35 were used and the population standard deviations of
both samples were $1928. Find the 95% confidence interval of the difference in means.

The following advanced exercise use a generalized ratio test to determine convergence of some series that arise in particular applications, including the ratio and root test, are not powerful enough to determine their convergence. The test states that if $ ${\displaystyle lim\{n\to \mathrm{\infty }\}\frac{a\left\{2n\right\}}{a_n}<\frac{1}{2}}$ then ${\displaystyle \sum a_n}$ converges, while if ${\displaystyle lim\{n\to \mathrm{\infty }\}\frac{a\{2n+1\}}{a_n}>\frac{1}{2}}$ then ${\displaystyle \sum a_n}$ $ diverges. Let ${\displaystyle a_n=\frac{n^{\ln n}}{{\left(\ln n\right)}^n}}$. Show that $\frac{a_{2n}}{a_n}\to 0$ as ${\displaystyle n\to \mathrm{\infty }}$

B. G. Cosmos, a scientist, believes that the probability is ${\displaystyle \frac{2}{5}}$ that aliens from an advanced civilization on Planet X are trying to communicate with us by sending high-frequency signals to Earth. By using sophisticated equipment, Cosmos hopes to pick up these signals. The manufacturer of the equipment, Trekee, Inc., claims that if aliens are indeed sending signals, the probability that the equipment will detect them is ${\displaystyle \frac{3}{5}}$. However, if aliens are not sending signals, the probability that the equipment will seem to detect such signals is $\frac{1}{10}$. If the equipment detects signals, what is the probability that aliens are actually sending them?