Question

Suppose that prices of recently sold homes in one neighborhood have a mean of $200,000 with a standard deviation of $5200 . Using Chebyshev's Theorem, state the range in which at least 88.9% of the data will reside. Please do not round your answers.

Answer 1

Answer: Chebyshev's Theorem states that for any dataset, regardless of its shape, at least a certain percentage of the data will fall within a certain number of standard deviations from the mean.

In this case, we want to find the range in which at least 88.9% of the data will reside. Chebyshev's Theorem guarantees that at least (1 - 1/k^2) of the data will fall within k standard deviations from the mean, where k is any positive constant greater than 1.

Let's solve for k:

1 - 1/k^2 = 0.889

Now, isolate k:

1/k^2 = 0.111

Now, take the reciprocal of both sides:

k^2 = 1/0.111

k^2 ≈ 9.01

Now, take the square root of both sides to find k:

k ≈ √(9.01)

k ≈ 3

So, at least 88.9% of the data will fall within 3 standard deviations from the mean.

Now, let's find the range:

Lower bound = Mean - (3 * Standard Deviation) = $200,000 - (3 * $5200) = $200,000 - $15,600 = $184,400

Upper bound = Mean + (3 * Standard Deviation) = $200,000 + (3 * $5200) = $200,000 + $15,600 = $215,600

Therefore, at least 88.9% of the data will reside in the range of $184,400 to $215,600.