Answer:
More than 88.889% of the values will fall between 100 and 124 for the data set that has a mean of 112 and a standard deviation of 4.
Step-by-step explanation:
The Chebyshev's theorem states that the probability of any random variable ''X'' assuming a value between a range of ''k times'' the standard deviation is at least [tex]1-\frac{1}{k^{2}}[/tex]
We can write mathematically this as :
P( μ - kσ < X < μ + kσ) [tex]\geq 1-\frac{1}{k^{2}}[/tex] (I)
Where μ is the mean and σ is the standard deviation.
In this exercise :
μ = 112
σ = 4
If we replace this values in the equation (I) :
[tex]P(112-k(4)<X<112+k(4))\geq 1-\frac{1}{k^{2}}[/tex]
The percent of the values falling between 100 and 124 can be written as :
[tex]P(100<X<124)[/tex] (II)
This probability must be equal to [tex]P(112-k(4)<X<112+k(4))[/tex] (III)
Therefore if we work with (II) and (III) ⇒
(II) = (III) ⇒
[tex]P(100<X<124)=P(112-k(4)<X<112+k(4))[/tex] ⇒
[tex]100=112-4k[/tex]
[tex]124=112+4k[/tex]
⇒ In any of the equations we find that
[tex]12=4k[/tex]
[tex]k=3[/tex]
Finally, we can write that
[tex]P(112-3(4)<X<112+3(4))=P(100<X<124)\geq 1-\frac{1}{k^{2}}=1-\frac{1}{3^{2}}=1-\frac{1}{9}[/tex]
[tex]P(100<X<124)\geq \frac{8}{9}=0.88888[/tex] ≅ 88.889%
According to Chebyshev's theorem, more than 88.889% of the values will fall between 100 and 124 for the data set.
