Question

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:	21	0	35	27	34	18	37	−17	−21	−20
y:	16	−7	21	20	16	15	17	−1	−8	−8

(a) Compute Σx, Σx², Σy, Σy².

Σx		Σx2
Σy		Σy2

(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 two decimal places.)

	x	y
x
s2
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.

Answer 1

(a-b)

For data set X:

Following table shows the calculations:

	X	x^2
	21	441
	0	0
	35	1225
	27	729
	34	1156
	18	324
	37	1369
	-17	289
	-21	441
	-20	400
Total	114	6374

---------------------------------------

For data set Y:

Following table shows the calculations:

	Y	Y^2
	16	256
	-7	49
	21	441
	20	400
	16	256
	15	225
	17	289
	-1	1
	-8	64
	-8	64
Total	81	2045

(c)

According to Chebyshev's inequality, at least 75% data values lie within 2 standard deviations of mean.

For X:

Lower limit:

Upper limit

For Y:

Lower limit:

Upper limit

Correct option:

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