91Ó°ÊÓ

Do bonds reduce the overall risk of an investment portfolio? Let \(x\) be a random variable representing annual percent return for Vanguard Total Stock Index (all stocks). Let \(y\) be a random variable representing annual return for Vanguard Balanced Index \((60 \%\) stock and \(40 \%\) bond). For the past several years, we have the following data (Reference: Morningstar Research Group, Chicago). \(\begin{array}{llllllllrr}x: & 11 & 0 & 36 & 21 & 31 & 23 & 24 & -11 & -11 & -21 \\ y: & 10 & -2 & 29 & 14 & 22 & 18 & 14 & -2 & -3 & -10\end{array}\) (a) Compute \(\Sigma x, \Sigma x^{2}, \Sigma y\), and \(\Sigma y^{2}\). (b) Use the results of part (a) to compute the sample mean, variance, and standard deviation for \(x\) and for \(y\). (c) Compute a \(75 \%\) Chebyshev interval around the mean for \(x\) values and also for \(y\) values. Use the intervals to compare the two funds. (d) Compute the coefficient of variation for each fund. Use the coefficients of variation to compare the two funds. If \(s\) represents risks and \(\bar{x}\) represents expected return, then \(s / \bar{x}\) can be thought of as a measure of risk per unit of expected return. In this case, why is a smaller \(C V\) better? Explain.

For mallard ducks and Canada geese, what percentage of nests are successful (at least one offspring survives)? Studies in Montana, Illinois, Wyoming, Utah, and California gave the following percentages of successful nests (Reference: The Wildlife Society Press, Washington, D.C.). \(x:\) Percentage success for mallard duck nests 56 \(\begin{array}{llll}85 & 52 & 13 & 39\end{array}\) \(y:\) Percentage success for Canada goose nests \(\begin{array}{lllll}24 & 53 & 60 & 69 & 18\end{array}\) (a) Use a calculator to verify that \(\Sigma x=245 ; \Sigma x^{2}=14,755 ; \Sigma y=224 ;\) and \(\Sigma y^{2}=12,070\) (b) Use the results of part (a) to compute the sample mean, variance, and standard deviation for \(x\), the percent of successful mallard nests. (c) Use the results of part (a) to compute the sample mean, variance, and standard deviation for \(y\), the percent of successful Canada goose nests. (d) Use the results of parts (b) and (c) to compute the coefficient of variation for successful mallard nests and Canada goose nests. Write a brief explanation of the meaning of these numbers. What do these results say about the nesting success rates for mallards compared to those of Canada geese? Would you say one group of data is more or less consistent than the other? Explain.

What was the age distribution of prehistoric Native Americans? Extensive anthropologic studies in the southwestern United States gave the following information about a prehistoric extended family group of 80 members on what is now the Navajo Reservation in northwestern New Mexico (Source: Based on information taken from Prehistory in the Navajo Reservation District, by F. W. Eddy, Museum of New Mexico Press). \begin{tabular}{l|cccc} \hline Age range (years) & \(1-10^{*}\) & \(11-20\) & \(21-30\) & 31 and over \\ \hline Number of individuals & 34 & 18 & 17 & 11 \\ \hline \end{tabular} "Includes infants. For this community, estimate the mean age expressed in years, the sample variance, and the sample standard deviation. For the class 31 and over, use \(35.5\) as the class midpoint.

In your biology class, your final grade is based on several things: a lab score, scores on two major tests, and your score on the final exam. There are 100 points available for each score. However, the lab score is worth \(25 \%\) of your total grade, each major test is worth \(22.5 \%\), and the final exam is worth \(30 \%\). Compute the weighted average for the following scores: 92 on the lab, 81 on the first major test, 93 on the second major test, and 85 on the final exam.