Recall from class that we found that the fibonacci sequence, with $f_0 = 0$, $f_1 = 1$ and $f_n = f_{n - 2} + f_{n - 1}$, had a closed form $$f_n = \frac{1}{\sqrt{5}} \left( \phi^n - \widehat{\phi}^n \right),$$ where $$\phi = \frac{1 + \sqrt{5}}{2} \; \text{and} \; \widehat{\phi} = \frac{1 - \sqrt{5}}{2}.$$

the lucas numbers are defined in the same way, but with different starting values. let $l_0$ be the zeroth lucas number and $l_1$ be the first. if
\begin{align*}
l_0 & = 2 \\
l_1 & = 1 \\
l_n & = l_{n - 1} + l_{n - 2} \; \text{for}\; n \geq 2
\end{align*}
then what is the tenth lucas number? (note: we seek a numerical answer.)

Two researchers are attempting to create a new psychotherapy protocol that more reduces panic attacks. they believe that they have succeeded and are now in the clinical trial period of their protocol's development. they are working with several cohort groups, some of which have been through the new protocol, while others have done the standard protocol currently recommended. if these researchers are interested in evaluating the difference in the percentage of test subjects who successfully decreased their panic attacks in 2 months on the new protocol versus the percentage of test subjects who successfully decreased their panic attacks in 2 months with the traditional protocol, which of the following statistical tests would be most appropriate? a. analysis of variance (anova) b. correlation c. chi-square test d. paired t-test e. independent t-test