A real estate agent wants to predict the selling price of single-family homes from the size of each house. A scatterplot 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( price ˆ )=2.08+0.11(size) ln⁡(price^)=2.08+0.11(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?

explain and solve

To find the predicted selling price for a house with a size of 3,200 square feet, we'll use the given regression model and follow these steps:

### Step 1: Convert the Size
The regression model is given in terms of "size" measured in hundreds of square feet. So, we first need to convert 3,200 square feet into hundreds of square feet:
\[
\text{Size in hundreds of square feet} = \frac{3200}{100} = 32
\]

### Step 2: Substitute the Size into the Model
The model provided is:
\[
\ln(\hat{\text{price}}) = 2.08 + 0.11 \times (\text{size})
\]
Now, substitute $\text{size} = 32$:
\[
\ln(\hat{\text{price}}) = 2.08 + 0.11 \times 32
\]
\[
\ln(\hat{\text{price}}) = 2.08 + 3.52 = 5.6
\]

### Step 3: Solve for $\hat{\text{price}}$
Since the model gives $\ln(\hat{\text{price}})$, we need to exponentiate both sides to find $\hat{\text{price}}$:
\[
\hat{\text{price}} = e^{5.6}
\]

### Step 4: Calculate $e^{5.6}$
Using a calculator:
\[
e^{5.6} \approx 270.43
\]

### Step 5: Interpret the Result
The predicted price of the house is approximately 270.43 thousand dollars, or:
\[
\hat{\text{price}} \approx 270,430 \text{ dollars}
\]

### Final Answer
The predicted selling price for a house with a size of 3,200 square feet is approximately **$270,430**.

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#### Related Questions:
1. How would this prediction change if the size were 2,500 square feet?
2. What is the interpretation of the slope (0.11) in the regression model?
3. Why is taking the natural logarithm of price helpful in creating a linear model?
4. How would the model change if price were in dollars instead of thousands of dollars?
5. What would the selling price be for a house twice the size, at 6,400 square feet?

#### Tip:
In exponential models, the rate of increase accelerates as the variable increases. Converting data to a logarithmic scale often helps create a linear relationship, making predictions easier to calculate and interpret.

A real estate agent wants to predict the selling price of single-family homes from the size of each house. A scatterplot 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(price^) = 2.08 + 0.11(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?

Discover how to use an exponential regression model to predict the selling price of a house based on its size. This example shows the steps to transform the model linearly using logarithms and calculate the predicted price for a 3,200 square foot house. Suitable for students in Grades 10-12.

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