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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Would you like further details on any part of this process?

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.