Question 1

Which of the following best describes the purpose of a multiple regression equation?

Accepted Answer

To express a linear relationship between a response variable $$y$$ and two or more predictor variables $$(x_1$, x_2$, \ldots, x_k).$$

Question 2

Given the multiple regression equation $$\hat{y} = b_0$ + b_1x_1 + b_2x_2$$, what does $$b_2$$ represent?

Accepted Answer

The estimated change in $$y$$ for a one-unit increase in $$x_2$$, holding $$x_1$$ constant.

Question 3

Which of the following is a requirement for the errors in a multiple regression model?

Accepted Answer

The errors must be normally distributed with mean 0 and constant standard deviation $$\sigma.$$

Question 4

Suppose you are using a multiple regression model to predict ice cream sales based on temperature and price. If the coefficient for temperature is positive and significant, what does this imply?

Accepted Answer

Higher temperatures are associated with higher ice cream sales, holding price constant.

Question 5

What is the adjusted coefficient of determination ($$	ext{Adjusted } R^2$) used for in multiple regression?

Accepted Answer

To account for the number of predictor variables and sample size when assessing model fit.

Question 6

Given the formula for adjusted $$R^2$$:

$$	ext{Adjusted } R^2$ = 1 - \frac{(n-1)}{[n-(k+1)]}(1-R^2$)$$

where $$n is $$the sample size and $$k is $$the number of predictor variables, what happens to adjusted $$R^2 if $$you add a predictor variable that does not improve the model?

Accepted Answer

Adjusted $$R^2$$ will decrease.

Question 7

In the context of regression, what is a dummy variable?

Accepted Answer

A variable coded as 0 or 1 to represent two categories of a qualitative variable.

Question 8

A regression equation for predicting a child's height is:

$$	ext{Height of child} = 36.5 - 0.0336(	ext{Height of father}) + 0.461(	ext{Height of mother}) + 6.14(	ext{Sex})$$

where Sex = 0 for daughter and 1 for son. What does the coefficient 6.14 represent?

Accepted Answer

The predicted difference in height between a son and a daughter, given the same parental heights.

Question 9

Which of the following best describes multicollinearity in multiple regression?

Accepted Answer

Two predictor variables are highly linearly correlated with each other.

Question 10

You are building a multiple regression model to predict weight using height and waist circumference. If you find that height and waist circumference are highly correlated, what should you do?

Accepted Answer

Consider removing one of the variables to avoid multicollinearity.

Question 11

Which of the following is the correct formula for the total variation in $$y in $$regression analysis?

Accepted Answer

$$	ext{Total variation} = 	ext{Explained variation} + 	ext{Unexplained variation}$$

Question 12

If the linear correlation coefficient $$r = 0.947$$ for a data set, what is the coefficient of determination $$r^2$$?

Accepted Answer

$$r^2 = 0.897$

Question 13

A 95% prediction interval for the number of tickets sold in a lottery is $$73.7 \u003c y \u003c 122$$ million. What does this interval mean?

Accepted Answer

We are 95% confident that the actual number of tickets sold will fall between 73.7 and 122 million for a given jackpot.

Question 14

Which of the following is the correct formula for the margin of error $$E in a $$prediction interval for regression?

Accepted Answer

$$E = t_{\alpha/2} s_e \sqrt{1 + \frac{1}{n} + \frac{(x_0$ - \bar{x})^2$}{S_{xx}}}$$

Question 15

In regression analysis, what is the unexplained deviation for a data point $$(x, y)$$?

Accepted Answer

The vertical distance $$y - \hat{y}$$ between the observed $$y$$ and the predicted $$\hat{y}.$$

Question 16

A researcher is considering adding a new predictor variable to a multiple regression model. According to the guidelines, when should the new variable be included?

Accepted Answer

If the adjusted $$R^2$$ increases when the variable is added.

Question 17

Which of the following statements about prediction intervals and confidence intervals in regression is correct?

Accepted Answer

A prediction interval estimates the range for an individual predicted $$y$$ value, while a confidence interval estimates the range for the mean $$y$$ value.

Question 18

In the context of regression, what does the coefficient of determination $$R^2$$ represent?

Accepted Answer

The proportion of the variation in $$y$$ that is explained by the regression line.

Question 19

A regression model for predicting height from shoe print length in a forensic investigation is $$	ext{Height} = 80.9 + 3.22 	imes 	ext{Shoe Print Length}. If a $$shoe print length is 30 cm, what is the predicted height?

Accepted Answer

$$80.9 + 3.22 	imes 30 = 177.5 cm$$

Chapter 10: Multiple Regression, Dummy Variables, and Model Selection in Statistics

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