Question 1

Which of the following best describes the purpose of multiple regression analysis in business statistics?

Accepted Answer

To examine the linear relationship between one dependent variable and two or more independent variables.

Question 2

Given the multiple regression equation $$\hat{Y} = 306.526 - 24.975X_1 + 74.131X_2$$, where $$X_1 is $$price (in $$) and X_2$ is advertising (in 100s$$), what is the predicted sales when price is $$6.00 and advertising is 400$$?

Accepted Answer

$$\hat{Y} = 306.526 - 24.975(6.00) + 74.131(4.0) = 428.6216$$

Question 3

In the context of multiple regression, what does the coefficient $$b_1$$ represent in the equation $$\hat{Y} = b_0$ + b_1X_1 + b_2X_2$$?

Accepted Answer

The average change in $$Y$$ for a one-unit increase in $$X_1$$, holding $$X_2$$ constant.

Question 4

Which of the following is the correct formula for the coefficient of multiple determination ($$r^2) in $$multiple regression?

Accepted Answer

$$r^2$$ = \frac{SSR}{SST}$$

Question 5

A business analyst adds a new independent variable to a regression model, but the $$r^2$$ value increases only slightly. What is the main reason to use adjusted $$r^2$$ instead of $$r^2 in $$this situation?

Accepted Answer

Adjusted $$r^2$$ penalizes the addition of unimportant variables by accounting for the number of predictors and sample size.

Question 6

Which of the following is NOT an assumption of the multiple regression model?

Accepted Answer

The independent variables must be uncorrelated with each other.

Question 7

Suppose you are testing the overall significance of a multiple regression model with $$k$$ independent variables and $$n$$ observations. What are the null and alternative hypotheses for the F-test?

Accepted Answer

$$H_0$$: \$$beta_1$$ = \$$beta_2$$ = ... = \beta_k = 0$$; H_1$: at least one $$\beta_j 
eq 0$$

Question 8

A regression model for predicting house prices includes a dummy variable for 'fireplace' (1 if present, 0 if not). The coefficient for 'fireplace' is 15,000. How should this coefficient be interpreted?

Accepted Answer

On average, houses with a fireplace are predicted to sell for $15,000 more than those without, holding other variables constant.

Question 9

Which of the following residual plots would indicate a violation of the constant variance (homoscedasticity) assumption in multiple regression?

Accepted Answer

A plot of residuals vs. predicted values that shows a funnel shape (widening or narrowing spread).

Question 10

In a multiple regression model, the F-test for overall significance yields a p-value of 0.012. If $$\alpha = 0.05$$, what is the correct conclusion?

Accepted Answer

Reject the null hypothesis; there is evidence that at least one independent variable affects the dependent variable.

Question 11

Which formula correctly expresses the adjusted $$r^2$$ for a multiple regression model with $$n$$ observations and $$k$$ independent variables?

Accepted Answer

$$r^2$$_{adj} = 1 - (1 - $$r^2$$) \left( \frac{n-1}{n-k-1} \right)$$

Question 12

A company wants to know if the effect of advertising on sales depends on whether it is a holiday week. Which regression model would allow you to test for this interaction?

Accepted Answer

$$Y = b_0$ + b_1$ X_1$ + b_2$ X_2$ + b_3$ X_1$ X_2$

Question 13

If a regression model includes a dummy variable for 'holiday' (1 if holiday, 0 if not), and the coefficient is positive and significant, what does this imply?

Accepted Answer

Sales are higher on average during holiday weeks, holding other variables constant.

Question 14

Which of the following best describes the meaning of the residual $$e_i in a $$multiple regression model?

Accepted Answer

$$e_i = Y_i - \hat{Y}_i$$, the difference between the observed and predicted value for observation $$i.$$

Question 15

A regression model for pie sales includes price and advertising as independent variables. The estimated coefficient for price is $$-24.975. $$What does this mean in practical terms?

Accepted Answer

For each $1 increase in price, pie sales are expected to decrease by 24.975 units, holding advertising constant.

Question 16

Which of the following is a correct interpretation of $$r^2 = 0.52$ in a multiple regression model predicting pie sales?

Accepted Answer

52% of the variation in pie sales is explained by the variation in the independent variables included in the model.

Question 17

Which of the following statements about dummy variables in regression is TRUE?

Accepted Answer

A dummy variable with two levels (e.g., yes/no) is coded as 0 or 1.

Question 18

Which of the following is the correct test statistic for testing the significance of an individual regression coefficient $$b_j$$?

Accepted Answer

$$t_{STAT} = \frac{b_j - 0}{S_{b_j}}$$

Chapter 13: Introduction to Multiple Regression (Business Statistics)

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