Question

sale price of homes the average sale price of new one - family houses in the united states for a recent year was $237,100. find the range of values in which at least 75% of the sale prices will lie if the standard deviation is $44,600. round your k to the nearest whole number. the range of values is between $\square$ and $\square$.

Explanation:

Step1: Recall Chebyshev's Theorem

Chebyshev's Theorem states that for any dataset, the proportion of data within \( k \) standard deviations of the mean is at least \( 1 - \frac{1}{k^2} \). We want at least 75% (or 0.75) of the data, so we set \( 1 - \frac{1}{k^2} = 0.75 \).

Step2: Solve for \( k \)

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Step3: Calculate the lower bound

The mean \( \mu = 237100 \) and standard deviation \( \sigma = 44600 \). The lower bound is \( \mu - k\sigma \):

$$ 237100 - 2\times44600 = 237100 - 89200 = 147900 $$

Step4: Calculate the upper bound

The upper bound is \( \mu + k\sigma \):

$$ 237100 + 2\times44600 = 237100 + 89200 = 326300 $$

Answer:

The range of values is between $147900 and $326300.