Do bonds reduce the overall risk of an investment portfolio? Let x be a random variable
representing annual percent return for the Vanguard Total Stock Index (all Stocks). Let y be a
random variable representing annual return for the Vanguard Balanced Index (60% stock and
40% bond). For the past several years, assume the following data.
x: 13 0 39 22 32 24 25 –13 –13 -22
y: 7 –4 27 16 23 18 16 –4 –5 –7
The sample means for x and y are 10.70 and 8.70, respectively.
Compute a 75% Chebyshev interval around the mean for x-values and also for y-values.
Round your answers to the nearest hundredth.
To compute the 75% Chebyshev interval around the mean for x-values and y-values, we'll need to calculate the standard deviation for each dataset.
To find the standard deviation, follow these steps:
1. Calculate the variance by squaring the difference between each data point and the mean, summing up those squared differences, and dividing by the number of data points minus 1.
2. Take the square root of the variance to get the standard deviation.
Let's calculate the standard deviation for x-values:
1. Subtract the mean from each x-value and square the difference:
(13 - 10.70)^2 = 5.62
(0 - 10.70)^2 = 114.49
(39 - 10.70)^2 = 815.69
(22 - 10.70)^2 = 131.04
(32 - 10.70)^2 = 456.49
(24 - 10.70)^2 = 179.56
(25 - 10.70)^2 = 205.21
(-13 - 10.70)^2 = 592.90
(-13 - 10.70)^2 = 592.90
(-22 - 10.70)^2 = 933.90
2. Sum up those squared differences:
5.62 + 114.49 + 815.69 + 131.04 + 456.49 + 179.56 + 205.21 + 592.90 + 592.90 + 933.90 = 3931.70
3. Divide the sum by the number of data points minus 1:
3931.70 / (10 - 1) = 437.97
4. Take the square root of the variance:
√437.97 ≈ 20.93
The standard deviation for x-values is approximately 20.93.
Now let's calculate the standard deviation for y-values using the same process:
1. Subtract the mean from each y-value and square the difference:
(7 - 8.70)^2 = 2.89
(-4 - 8.70)^2 = 152.10
(27 - 8.70)^2 = 327.61
(16 - 8.70)^2 = 53.65
(23 - 8.70)^2 = 203.29
(18 - 8.70)^2 = 87.61
(16 - 8.70)^2 = 53.65
(-4 - 8.70)^2 = 152.10
(-5 - 8.70)^2 = 218.49
(-7 - 8.70)^2 = 310.41
2. Sum up those squared differences:
2.89 + 152.10 + 327.61 + 53.65 + 203.29 + 87.61 + 53.65 + 152.10 + 218.49 + 310.41 = 1559.20
3. Divide the sum by the number of data points minus 1:
1559.20 / (10 - 1) = 173.24
4. Take the square root of the variance:
√173.24 ≈ 13.15
The standard deviation for y-values is approximately 13.15.
Now that we have the means and standard deviations, we can compute the Chebyshev interval for each dataset.
The Chebyshev inequality states that for any dataset, at least (1 - 1/k^2) of the data must lie within k standard deviations from the mean, where k is a positive constant.
For a 75% Chebyshev interval, k = 1 / √(1 - 0.75) = 2.
Using the formula, we can calculate the Chebyshev intervals:
For x-values:
Lower bound: mean - (k * standard deviation)
Upper bound: mean + (k * standard deviation)
Lower bound: 10.70 - (2 * 20.93) = -30.16
Upper bound: 10.70 + (2 * 20.93) = 51.56
For y-values:
Lower bound: mean - (k * standard deviation)
Upper bound: mean + (k * standard deviation)
Lower bound: 8.70 - (2 * 13.15) = -17.60
Upper bound: 8.70 + (2 * 13.15) = 34.00
Rounded to the nearest hundredth, the 75% Chebyshev intervals for x-values are approximately (-30.16, 51.56), and for y-values are approximately (-17.60, 34.00).