Question

sale price of homes the average sale price of new one - family houses in the united states for a recent year was $251,900. find the range of values in which at least 93.75% of the sale prices will lie if the standard deviation is $52,900. round your k to the nearest whole number. the range of values is between $\square$ and $\square$.

Explanation:

Step1: Use Chebyshev's Theorem formula

The formula for the proportion of data within $k$ standard deviations is $1-\frac{1}{k^2}$. Set this equal to 0.9375:
$$1-\frac{1}{k^2}=0.9375$$

Step2: Solve for $k$

Rearrange to isolate $\frac{1}{k^2}$:
$$\frac{1}{k^2}=1-0.9375=0.0625$$
Take reciprocal and square root:
$$k^2=\frac{1}{0.0625}=16 \implies k=\sqrt{16}=4$$

Step3: Calculate lower bound

Subtract $4$ standard deviations from the mean:
$$251900 - 4\times52900=251900-211600$$

Step4: Calculate upper bound

Add $4$ standard deviations to the mean:
$$251900 + 4\times52900=251900+211600$$

Answer:

The range of values is between $\$40,300$ and $\$463,500$.