Im struggling alot with Chebyshev
s 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:
- 340 and 424.
- 396 and 480.
- 354 and 410.
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.
Can you explain me that?
mean - k sigma <= X <= mean + k sigma
is at least 1 - 1/k2
The thing IDK is how do you get the value of K…
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)
Algebra:
1 – 1/k2 = 0.8889
1/k2 = 1 – 0.8889 = 0.1111
k2 = 1 / 0.1111 = 9.0009
k = √9.0009 = 3.0002
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