Sale Price of Homes The average sale price of new one-family houses in the United States for a recent year was \( \$ 242,900 \) Find the range of values in which at least \( 75 \% \) of the sale prices will lie if the standard deviation is \( \$ 47,500 \). Round your \( k \) to the nearest whole number. The range of values is between \( \$ \square \) and \( \$ \square \)

To determine the range of sale prices in which at least **75%** of the homes fall, we'll use **Chebyshev's Inequality**. This inequality is helpful when we don't know the exact distribution of the data. ### Chebyshev's Inequality Chebyshev's Inequality states that for any dataset (regardless of its distribution), the proportion of values that lie within **\( k \)** standard deviations of the mean is at least **\( 1 - \frac{1}{k^2} \)**. Mathematically: \[ P\left(|X - \mu| < k\sigma\right) \geq 1 - \frac{1}{k^2} \] ### Step-by-Step Calculation 1. **Given:** - **Mean (\( \mu \))**: \$242,900 - **Standard Deviation (\( \sigma \))**: \$47,500 - **Desired proportion**: At least 75% (or 0.75) 2. **Set up the inequality:** \[ 1 - \frac{1}{k^2} \geq 0.75 \] 3. **Solve for \( k \):** \[ \frac{1}{k^2} \leq 0.25 \\ k^2 \geq 4 \\ k \geq 2 \] We round \( k \) to the nearest whole number, so \( k = 2 \). 4. **Calculate the range:** \[ \text{Lower Bound} = \mu - k\sigma = 242,900 - 2 \times 47,500 = 242,900 - 95,000 = 147,900 \\ \text{Upper Bound} = \mu + k\sigma = 242,900 + 2 \times 47,500 = 242,900 + 95,000 = 337,900 \] ### **Conclusion** At least **75%** of the sale prices of new one-family houses lie between **\$147,900** and **\$337,900**. \[ \boxed{\$147,\!900 \text{ and } \$337,\!900} \]