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.

x:

$18$

$0$

$37$

$14$

$23$

$24$

$29$

$10$

$23$

$18$

y:

$8$

$4$

$23$

$18$

$23$

$18$

$15$

$7$

$1$

$5$

A button hyperlink to the SALT program that reads: Use SALT.

$($a$)$ Compute $\backslash$Sigma x$,\backslash$Sigma x$2,\backslash$Sigma y$,\backslash$Sigma y$2.$

$\backslash$Sigma x

$\backslash$Sigma x$2$

$4607$

$\backslash$Sigma y

$\backslash$Sigma y$2$

$($b$)$ Use the results of part $($a$)$ to compute the sample mean, variance, and standard deviation for x and for y$.($Round your answers to four decimal places.$)$

x y

x

s$2$

s

$($c$)$ Compute a $75\%$ Chebyshev interval around the mean for x values and also for y values. $($Round your answers to two decimal places.$)$

x y

Lower Limit

Upper Limit

Use the intervals to compare the two funds.

$75\%$ of the returns for the balanced fund fall within a narrower range than those of the stock fund.

$75\%$ of the returns for the stock fund fall within a narrower range than those of the balanced fund.

$25\%$ of the returns for the balanced fund fall within a narrower range than those of the stock fund.

$25\%$ of the returns for the stock fund fall within a wider range than those of the balanced fund.

$($d$)$ Compute the coefficient of variation for each fund. $($Round your answers to the nearest whole number.$)$

x y

CV

$\%$

$\%$

Use the coefficients of variation to compare the two funds.

For each unit of return, the stock fund has lower risk.

For each unit of return, the balanced fund has lower risk.

For each unit of return, the funds have equal risk.

If s represents risks and x represents expected return, then s$/$x can be thought of as a measure of risk per unit of expected return. In this case, why is a smaller CV better? Explain.

A smaller CV is better because it indicates a higher risk per unit of expected return.

A smaller CV is better because it indicates a lower risk per unit of expected return.