Q: Suppose you want to predict the asking price for a home with $$2500$$ square feet, $$3$$ bedrooms, and $$2$$ bathrooms. Using the regression equation $$\hat{y} = 25.6 + 0.0719X_1 - 0.8X_2 + 55.3X_3$$, what is the predicted price (in thousands of dollars)?

$$25.6 + 0.0719 \times 2500 - 0.8 \times 3 + 55.3 \times 2 = 25.6 + 179.75 - 2.4 + 110.6 = 313.55$$

Question 1

Which of the following best describes the meaning of the coefficient $$b_1 = 0.0719$ in the multiple regression equation $$\hat{y} = 25.6 + 0.0719X_1 - 0.8X_2 + 55.3X_3$$ for predicting the asking price of homes?

Accepted Answer

For each additional square foot, the predicted asking price increases by $$0.0719$$ thousand dollars, holding the number of bedrooms and bathrooms constant.

Question 2

Given the regression equation $$\hat{y} = 25.6 + 0.0719X_1 - 0.8X_2 + 55.3X_3$$, what is the predicted asking price (in thousands of dollars) for a home with $$3000$$ square feet, $$4$$ bedrooms, and $$2.5$$ bathrooms?

Accepted Answer

$$376.54$$

Question 3

What does the coefficient of determination $$R^2 = 65.6$$\%$$ indicate in the context of this multiple regression model?

Accepted Answer

65.6\% of the variability in asking price among the 13 homes is explained by the square footage, number of bedrooms, and number of bathrooms.

Question 4

Suppose a home has an actual asking price of $$400$$ thousand dollars, and the predicted price from the regression equation is $$405.62$$ thousand dollars. What is the residual for this home?

Accepted Answer

$$-5.62$$

Question 5

Which of the following statements about the intercept $$b_0 = 25.6$ in the regression equation is correct?

Accepted Answer

It represents the predicted asking price when square footage, number of bedrooms, and number of bathrooms are all zero, but this interpretation is not meaningful in this context.

Question 6

If the sum of squared residuals (SSE) for the model is $$19247$$, what does this value represent?

Accepted Answer

The total squared difference between the observed and predicted asking prices for all homes in the sample.

Question 7

Why might the regression coefficient for square footage in the multiple regression model ($$b_1 = 0.0719$) differ from the coefficient in a simple regression of asking price on square footage alone (b_1$ = 0.0895$$)?

Accepted Answer

Because the multiple regression coefficient adjusts for the effects of the other explanatory variables (bedrooms and bathrooms), while the simple regression does not.

Question 8

A student claims that increasing the number of bedrooms always increases the predicted asking price, based on the regression equation. Is this claim correct?

Accepted Answer

No, because the coefficient for bedrooms is negative ($$-0.8$$), so increasing bedrooms actually decreases the predicted price, holding other variables constant.

Question 9

Which of the following best describes the 'least-squares' principle in the context of multiple regression?

Accepted Answer

The regression equation is chosen so that the sum of the squared residuals is minimized.

Question 10

If the correlation between asking price and square footage is $$0.665$$, what is the interpretation of this value?

Accepted Answer

There is a moderate positive linear relationship between asking price and square footage.

Question 11

Suppose you want to predict the asking price for a home with $$2500$$ square feet, $$3$$ bedrooms, and $$2$$ bathrooms. Using the regression equation $$\hat{y} = 25.6 + 0.0719X_1 - 0.8X_2 + 55.3X_3$$, what is the predicted price (in thousands of dollars)?

Accepted Answer

$$25.6 + 0.0719 	imes 2500 - 0.8 	imes 3 + 55.3 	imes 2 = 25.6 + 179.75 - 2.4 + 110.6 = 313.55$$

Question 12

Which of the following is a correct interpretation of the residual for a particular home in this regression analysis?

Accepted Answer

It is the difference between the actual asking price and the predicted asking price for that home.

Question 13

If the $$R^2$$ value for the model is $$65.6\%$$, what percentage of the variability in asking price is NOT explained by the model?

Accepted Answer

$$34.4\%$$

Question 14

A real estate agent wants to know how much the predicted asking price will change if the number of bathrooms increases by $$1$$, holding all other variables constant. What is the answer according to the regression equation?

Accepted Answer

The predicted asking price increases by $$55.3$$ thousand dollars.

Question 15

Which of the following best explains why the residuals are important in regression analysis?

Accepted Answer

They measure the discrepancies between observed and predicted values, indicating how well the model fits the data.

Question 16

If the regression equation is $$\hat{y} = 25.6 + 0.0719X_1 - 0.8X_2 + 55.3X_3$$, what is the predicted asking price for a home with $$2000$$ square feet, $$4$$ bedrooms, and $$3$$ bathrooms?

Accepted Answer

$$25.6 + 0.0719 	imes 2000 - 0.8 	imes 4 + 55.3 	imes 3 = 25.6 + 143.8 - 3.2 + 165.9 = 332.1$$

Question 17

Which of the following is NOT a valid interpretation of the $$R^2$$ value in multiple regression?

Accepted Answer

It is the proportion of the variance in the response variable explained by each explanatory variable individually.

Question 18

A home in the data set has a residual of $$-46.79. $$What does this mean?

Accepted Answer

The actual asking price is $$46.79$$ thousand dollars less than the predicted price for that home.

Multiple Regression: Estimation, Interpretation, and Application

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