Chapter 9, Problem 1E

Housing prices The following regression model was found for the houses in upstate New York considered in the chapter:

$\widehat{Price}=20,986.09-7483.10Bedrooms+93.84LivingArea$

1. a) Find the predicted price of a 2 bedroom, 1000-sq-ft house from this model.
2. b) The house just sold for $135,000. Find the residual corresponding to this house.
3. c) What does that residual say about this transaction?

To determine

Find the predicted price of a 2 bedroom, 1,000 sq-ft house.

Answer to Problem 1E

The predicted price of a 2 bedroom, 1,000 sq-ft house in upstate New York is likely to be $99,859.89.

Explanation of Solution

Given info:

The regression model for the price of houses in upstate New York with respect to number of bedrooms and living area is given as:

$\widehat{\text{Price}}=20,986.09-7483.10\text{ Bedrooms}+93.84\text{ Living Area}$.

Calculation:

The given regression equation is:

$\widehat{\text{Price}}=20,986.09-7483.10\text{ Bedrooms}+93.84\text{ Living Area}$

Substitute $\text{Bedrooms}=2$ and $\text{Living Area}=1,000$ in the equation:

$\begin{array}{c}\widehat{\text{Price}}=20,986.09-\left(7483.10\times 2\right)+\left(93.84\times 1,000\right)\\ =20,986.09-14,966.2+93,840\\ =99,\mathrm{859.89.}\end{array}$

Thus, the predicted price of a 2 bedroom, 1,000 sq-ft house in upstate New York is likely to be $99,859.89.

To determine

Find the residual corresponding to a house that sold for $135,000.

Answer to Problem 1E

The residual corresponding to a house that sold for $135,000 is $35,140.11.

Explanation of Solution

Calculation:

Residual:

The residual corresponding to a predictor variable is given as the difference between actual value of the response variable and the predicted value. That is, $e=y-\widehat{y}$, where y be the actual value of the response variable and $\widehat{y}$ be the predicted value of the response variable for same predictor variable.

Put the actual value of ‘price’ as $y=135,000$ and the predicted value as $\widehat{y}=99,859.89$.

Then,

$\begin{array}{c}e=135,000-99,859.89\\ =35,\mathrm{140.11.}\end{array}$

Thus, the residual corresponding to a house that sold for $135,000 is $35,140.11.

To determine

Explain what the residual says about the transaction.

Explanation of Solution

Justification:

The real price, at which the house is sold, is $135,000. The regression model predicted that the price of the house would be sold at $99,859.89.

The residual $35,140.11 means that the house is sold at a price of $35,140.11 more than what was predicted.