Chebyshev’s Inequality

[archived_user]

How you`re supposed to know Chebyshev’s inequality table??
I mean see the exercise:
According to Exhibit 27, the arithmetic mean monthly return and standard
deviation of monthly returns on the S&P 500 were 0.95 percent and 5.39 percent,
respectively, during the 1926–2017 period, totaling 1,104 monthly observations.
Using this information, address the following:
1 Calculate the endpoints of the interval that must contain at least 75 percent
of monthly returns according to Chebyshev’s inequality.
How I`m supposed to know that 75 percent is 2 standard deviations??
There`s a table in page 421 that states the percentage or k values of 1,25, 1,5, 2, 2,5 3 4
But how I`m supposed to know how many standard deviation I should put?
WTF

 

[archived_user]

Chebyshev’s inequality says that:
P(|X – μ| / σ ≤ k) ≥ 1 – 1/k2
So, if they give you, say, k = 3, then,
P(|X – μ| / σ ≤ 3) ≥ 1 – 1/32 = 1 – 1/9 = 8/9 = 88.89%
If they give you, say, P(|X – μ| / σ ≤ k) ≥ 75%, then,
75% = 1 – 1/k2
1/k2 = 1 – 75% = 25% = ¼
k2 = 4
k = √4 = 2
Don’t memorize it; understand it.

 

[archived_user]

The proportion of the observations within k standard deviations of the arithmetic mean is at least 1 − 1/k2 for all k > 1.
The book didn`t showed the proportion in the formula ex= x% = 1 - 1/K2
I just saw the formula how was it, but the book actually says it but its was wrote before.
Thanks your explanation clarified it all!

 

[archived_user]

I looked at the 2019 curriculum and it didn’t have it written out as I did.
Stupid.

 

[archived_user]

Also when solving the book didn`t showed the use of this formula:
Solution to 1:
According to Chebyshev’s inequality, at least 75 percent of the observations
must lie within two standard deviations of the mean, X ± 2s. For the monthly
S&P 500 return series, we have 0.95% ± 2(5.39%) = 0.95% ± 10.78%. Thus the
lower endpoint of the interval that must contain at least 75 percent of the
observations is 0.95% − 10.78% = −9.83%, and the upper endpoint is 0.95% +
10.78% = 11.73%.
It just assumed you knew it by the table, they should have showed the use of the formula you stated above.

 

[archived_user]

ImBruces wrote: It just assumed you knew it by the table, they should have showed the use of the formula you stated above.

Yup.
Stupid.

 

[archived_user]

S2000magician wrote:
I looked at the 2019 curriculum and it didn’t have it written out as I did.
Stupid.

Exactly they messed up with that they should have showed that!