A real estate agent wants to predict the selling price of single-family homes from the size of each house. A scatter plot created from a sample of houses shows an exponential relationship between price, in thousands of dollars, and size, in 100 square feet. To create a linear model, the natural logarithm of price was taken and the least-squares regression line was given as:

\[ \ln(\hat{\text{price}}) = 2.08 + 0.11(\text{size}) \]

Based on the model, which of the following is closest to the predicted selling price for a house with a size of 3,200 square feet?

A. $54,500
B. $270,000
C. $354,000
D. $398,000
E. $560,000

In this mathematical model, a house of 3200 square feet has a logarithmic predicted price of approximately 5.64. Taking the exponent of this gives us a predicted selling price of around $281,000. Hence the closest option is B, $270,000.

Explanation:

In this case, where a real estate agent wants to predict the selling price of single-family homes from the size of each house, we are working with a linear regression model which is based on a natural logarithm of price and size of the house. The given regression line equation is ln(price) = 2.08 + 0.11(size). To predict the selling price for a house with a size of 3200 square feet, first, we have to convert this into 100 square foot units, which gives us 32. Substituting this into the equation, we get ln(price) = 2.08 + 0.11(32). Solving this, the natural log of the price is approximately 5.64. To get the price in dollars, we take the exponent of this number, e^5.64, which is approximately $281,000. Thus, option B ($270,000) is the closest predicted selling price for a house with a size of 3200 square feet.

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