# Chebyshev's theorem

1. Feb 24, 2010

### cyod22

1. do bonds reduce the overall risk of an investment portfolio? let x be a random variable representing annual % return for Vangaurd Total Stock Index (all stocks). Let y be a random variable representing annual return for Vangaurd Balanced Index(60% stock and 40% bond). For the past several years we have the following data.

x: 11 0 36 21 31 23 24 -11 -11 -21
y: 10 -2 29 14 22 18 14 -2 -3 -10

a.) Compare Ex, Ex2, Ey and Ey2 (2 = squared)

b.) use results in part (a) to compute the sample mean, variance, and standard deviation for x and for y.

c.) Compute a 75 % Chebyshev interval about the mean for x values and also for y values. Use interval to compare funds.

I was able to do part a & b but have no idea what they want for c. i do have the answer but i am not sure what they used to get the answer.

2. Relevant equations

3. The attempt at a solution

2. Feb 25, 2010

### vela

Staff Emeritus
What is a Chebyshev interval?

3. Feb 26, 2010

First (and simplest): how many standard deviations around the mean does chebyshev's theorem say you must go to include 75% of the data values? (remember chebychev's theorem says the percentage of values between $$\bar x \pm ks$$ is at least
$$1 - {1}/{k^2}$$.