Question 1

Which of the following best describes the purpose of using a multiple regression model in business decision-making?

Accepted Answer

To separate the effects of each explanatory variable on the response and reveal which variables really matter.

Question 2

Given the multiple regression equation for women’s apparel stores: $$\hat{y} = 60.359 + 7.966 	imes 	ext{Income} - 24.165 	imes 	ext{Competitors}$$, what is the interpretation of the coefficient for Income?

Accepted Answer

For each $$1,000 increase in median household income, sales increase by 7.966 $$per square foot, holding the number of competitors constant.

Question 3

Which of the following is a key assumption for the residuals in a multiple regression model?

Accepted Answer

Residuals must have equal variance, be independent, and be normally distributed around the regression equation.

Question 4

In the context of multiple regression, what does the $$R^2$$ value represent?

Accepted Answer

The proportion of the total variation in the response variable explained by the fitted regression model.

Question 5

Why is the adjusted $$R^2$$ ($$\overline{R}^2) $$used instead of $$R^2 in $$multiple regression analysis?

Accepted Answer

Because it adjusts for both the sample size and the number of explanatory variables, providing a more accurate measure of model fit.

Question 6

A company wants to predict sales using median household income and number of competitors. The partial slope for competitors is negative, but the marginal slope is positive. What does this indicate?

Accepted Answer

The effect of competitors on sales is confounded with income in the simple regression, but separated in the multiple regression.

Question 7

Which of the following residual plots would indicate a violation of the equal variance assumption in multiple regression?

Accepted Answer

A plot where residuals fan out or contract as fitted values increase.

Question 8

What is the null hypothesis tested by the F-statistic in multiple regression?

Accepted Answer

$$H_0$$: All regression coefficients (except the intercept) are equal to zero.

Question 9

Suppose the F-statistic for a regression model is 45.4940 with a p-value \u003c 0.0001. What does this imply?

Accepted Answer

At least one explanatory variable is significantly related to the response variable.

Question 10

Given the regression output: $$b_1 = 7.966$, SE = 0.8383$, t = 9.50$, p$$ \u003c 0.0001$$ for Income, what can you conclude?

Accepted Answer

There is strong evidence that Income is positively associated with Sales, controlling for Competitors.

Question 11

Which of the following best describes collinearity in multiple regression?

Accepted Answer

A situation where explanatory variables are highly correlated with each other, making estimates unreliable.

Question 12

A 95% prediction interval for sales at a location with median income of $$70,000 and 3 competitors is 545.48$$ \pm 137.29$. What does this interval represent?

Accepted Answer

The range in which the sales for a single new location with these characteristics will fall with 95% confidence.

Question 13

Which step should be performed first when fitting a multiple regression model to new business data?

Accepted Answer

Check the scatterplots of the response versus each explanatory variable.

Question 14

If a residual plot versus fitted values shows a clear pattern, what does this suggest?

Accepted Answer

The model may be missing an important explanatory variable or the relationship is not linear.

Question 15

Which of the following is true about partial and marginal slopes in multiple regression?

Accepted Answer

Partial slopes account for the effects of other variables, while marginal slopes do not.

Question 16

What is the main purpose of a calibration plot in regression analysis?

Accepted Answer

To visualize how closely the observed values cluster around the fitted values.

Question 17

Which of the following is NOT a condition that should be checked before making inferences from a multiple regression model?

Accepted Answer

Residuals are perfectly correlated with fitted values.

Question 18

If the residual degrees of freedom in a regression model is $$n - k - 1$$, what does $$k$$ represent?

Accepted Answer

The number of explanatory variables in the model.

Question 19

Which of the following best describes the effect of adding an explanatory variable to a multiple regression model?

Accepted Answer

Adjusted $$R^2$$ may increase or decrease, but $$s_e$$ and adjusted $$R^2$$ move in opposite directions.

Chapter 23: Multiple Regression – Business Statistics Study Notes

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