Chebyshev’s Inequality

CFA Level I Quant
#1

Im struggling alot with Chebyshevs Inequality, is the only topic I find really hard in reading 7, help me with this question:

Q. A sample of 438 observations is randomly selected from a population. The mean of the sample is 382, and the standard deviation is 14. Based on Chebyshev’s inequality, the endpoints of the interval that must contain at least 88.89% of the observations are closest to:

  1. 340 and 424.
  2. 396 and 480.
  3. 354 and 410.
#2

Chebychev’s theorem says that, for any random X, the probability that X is within k standard deviations of the mean is at least 1 - 1/k^2

(Probability for Risk Management 2nd ed by Matthew Hassett and Donald Stewart, pg 104)

that is, the probability that

mean - k sigma <= X <= mean + k sigma

is at least 1 - 1/k2

if k = 3, 1 - 1/32 = 88.89%

thus,

mean - k sigma = 382 - 3 x 14 = 340

also, mean + k sigma = 382 + 3 x 14 = 424

so the end points 340 and 424 contain at least 88.89% of the observations by the Chebychev’s theorem.

#3

Can you explain me that?

mean - k sigma <= X <= mean + k sigma

is at least 1 - 1/k2

#4

The thing IDK is how do you get the value of K…

#5

you need to find k such that 1 - 1/k2 = .8889

where k is an integer.

solving for k gives k = 3

(i made it sound more complicated than it needs to be. just plugging in 3 worked)

#6

Algebra:

1 – 1/k2 = 0.8889

1/k2 = 1 – 0.8889 = 0.1111

k2 = 1 / 0.1111 = 9.0009

k = √9.0009 = 3.0002

1 Like
#7

Thank you

:+1: :+1:

#8

You’re welcome.