Answer:
The range in which at least 88.9% of the data will reside, ($244,800, $295,200).
Step-by-step explanation:
The Chebyshev's theorem states that, if X is a random variable with mean µ and standard deviation σ then for any positive number k, we have
[tex]P \{|X -\mu| < k\sigma\} \geq (1-\frac{1}{k^{2}})[/tex]
Here [tex](1-\frac{1}{k^{2}}) = 0.889[/tex].
Then the value of k is:
[tex]k = [\frac{1}{ 1-0.889}]^{1/2} = [\frac{1}{0.111}]^{1/2} = 3.0015\approx 3[/tex]
Then we know that,
|X - µ| ≥ kσ
⇒ µ - kσ ≤ X ≤ µ + kσ.
Here it is given that mean (µ) = $270,000 and standard deviation (σ) = $8400.
Then, the price range is given by,
[tex]\mu - k\sigma \leq X \leq \mu + k\sigma[/tex]
[tex]270000-(3\times 8400)\leq X\leq 270000+(3\times 8400)[/tex]
[tex]244800\leq X\leq 295200[/tex]
Thus, the range in which at least 88.9% of the data will reside, ($244,800, $295,200).
