Answer :
To solve the problem of predicting the selling price of a house based on its size, we have a model that relates the natural logarithm of the price to the size of the house. Here's how you can find the predicted selling price for a house with a size of 3,200 square feet:
1. Understand the Model: The given regression model is:
[tex]\[
\ln (\widehat{\text{price}}) = 2.08 + 0.11 \times (\text{size})
\][/tex]
This model predicts the natural logarithm of the price in thousands of dollars.
2. Convert Size: Since the size in the model is specified in hundreds of square feet, first convert the house size from square feet. For a house of 3,200 square feet, divide it by 100 to convert:
[tex]\[
\text{Size} = \frac{3200}{100} = 32
\][/tex]
3. Calculate Natural Logarithm of Price: Substitute the size into the model:
[tex]\[
\ln (\widehat{\text{price}}) = 2.08 + 0.11 \times 32
\][/tex]
Perform the multiplication and addition:
[tex]\[
\ln (\widehat{\text{price}}) = 2.08 + 3.52 = 5.60
\][/tex]
4. Convert from Logarithm to Actual Price: To find the predicted price, convert from the natural logarithm back to the actual price using the exponential function:
[tex]\[
\widehat{\text{price}} = e^{5.60}
\][/tex]
This means the predicted price is approximately 270.43 thousand dollars.
5. Determine the Selling Price in Dollars: Since the price is in thousands of dollars, multiply by 1,000 to convert to dollars:
[tex]\[
\widehat{\text{price}} = 270.426 \times 1000 = 270,426
\][/tex]
Based on this model and calculations, the closest answer to the predicted selling price of a house with a size of 3,200 square feet is:
- (B) \$270,000.
1. Understand the Model: The given regression model is:
[tex]\[
\ln (\widehat{\text{price}}) = 2.08 + 0.11 \times (\text{size})
\][/tex]
This model predicts the natural logarithm of the price in thousands of dollars.
2. Convert Size: Since the size in the model is specified in hundreds of square feet, first convert the house size from square feet. For a house of 3,200 square feet, divide it by 100 to convert:
[tex]\[
\text{Size} = \frac{3200}{100} = 32
\][/tex]
3. Calculate Natural Logarithm of Price: Substitute the size into the model:
[tex]\[
\ln (\widehat{\text{price}}) = 2.08 + 0.11 \times 32
\][/tex]
Perform the multiplication and addition:
[tex]\[
\ln (\widehat{\text{price}}) = 2.08 + 3.52 = 5.60
\][/tex]
4. Convert from Logarithm to Actual Price: To find the predicted price, convert from the natural logarithm back to the actual price using the exponential function:
[tex]\[
\widehat{\text{price}} = e^{5.60}
\][/tex]
This means the predicted price is approximately 270.43 thousand dollars.
5. Determine the Selling Price in Dollars: Since the price is in thousands of dollars, multiply by 1,000 to convert to dollars:
[tex]\[
\widehat{\text{price}} = 270.426 \times 1000 = 270,426
\][/tex]
Based on this model and calculations, the closest answer to the predicted selling price of a house with a size of 3,200 square feet is:
- (B) \$270,000.
