Question 1

We decide to conduct a multiple regression analysis to predict the attendance at a major league baseball game. We use the size of the stadium as a quantitative independent variable and the type
Of game as a qualitative variable (with two levels - day game or night game). We hypothesize the
Following model: $$\mathrm { E } ( \mathrm { y } ) = \beta _ { 0 } + \beta _ { 1 } \mathrm { x } _ { 1 } + \beta _ { 2 } \mathrm { x } _ { 2 } + \beta _ { 3 } \mathrm { x } _ { 3 }$$
Where $\mathrm { x } _ { 1 } =$ size of the stadium
$x _ { 2 } = 1$ if a day game, 0 if a night game
A plot of the $y - x$ relationship would show:
A) Two non-parallel lines
B) Two parallel curves
C) Two non-parallel curves
D) Two parallel lines

Accepted Answer

The model includes an interaction term between the size of the stadium and the type of game, leading to two non-parallel lines representing different slopes for day and night games.

Question 2

Consider the partial printout below. $$\begin{array}{l}
\begin{array} { l c l l l l l } 
\hline & \text { Coefficients } & \text { Standard Error } & t \text { Stat } & \text { P-value } & \text { Lower 95\% } & \text { Upper 95\% } \
\hline \text { Intercept } & - 63.14873931 & 25.09115112 & - 2.516773304 & 0.045484943 & - 124.5446192 & - 1.752859365 \
\text { X1 } & 14.72507864 & 8.113581741 & 1.814867849 & 0.119466699 & - 5.128155197 & 34.57831248 \
\text { X2 } & 12.48784546 & 4.686063743 & 2.664890224 & 0.037279879 & 1.021452165 & 23.95423875 \
\text { X1X2 } & - 1.886935135 & 1.344999834 & - 1.402925924 & 0.210210141 & - 5.178033575 & 1.404163305 \
\hline
\end{array}\
\text { Is there evidence (at } \alpha = .05 \text { ) that } x _ { 1 } \text { and } x _ { 2 } \text { interact? Explain. }
\end{array}$$

Accepted Answer

The answer of Consider the partial printout below. \[\begin{array}{l}
\begin{array} {...

Question 3

A study of the top MBA programs attempted to predict the average starting salary (in $1000ʹs)of graduates of the program based on the amount of tuition (in $1000ʹs)charged by the program and
The average GMAT score of the programʹs students. The results of a regression analysis based on a
Sample of 75 MBA programs is shown below: Least Squares Linear Regression of Salary

$\text { Predictor }$

The model was then used to create $95 \%$ confidence and prediction intervals for $y$ and for $E ( Y )$ when the tuition charged by the MBA program was $\$ 75,000$ and the GMAT score was 675 . The results are shown here:
$95 \%$ confidence interval for $\mathrm { E } ( \mathrm { Y } ) : ( \$ 126,610 , \$ 136,640 )$
$95 \%$ prediction interval for $\mathrm { Y } : ( \$ 90,113 , \$ 173,160 )$
Which of the following interpretations is correct if you want to use the model to estimate $E ( Y )$ for all MBA programs?
A) We are $95 \%$ confident that the average of all starting salaries for graduates of all MBA programs that charge $\$ 75,000$ in tuition and have an average GMAT score of 675 will fall between $\$ 126,610$ and $\$ 136,640$.
B) We are $95 \%$ confident that the average starting salary for graduates of a single MBA program that charges $\$ 75,000$ in tuition and has an average GMAT score of 675 will fall between $\$ 126,610$ and $\$ 136,640$.
C) We are $95 \%$ confident that the average starting salary for graduates of a single MBA program that charges $\$ 75,000$ in tuition and has an average GMAT score of 675 will fall between $\$ 90,113$ and $\$ 173,16,30$.
D) We are $95 \%$ confident that the average of all starting salaries for graduates of all MBA programs that charge $\$ 75,000$ in tuition and have an average GMAT score of 675 will fall between $\$ 90,113$ and $\$ 173,16,30$.

Accepted Answer

The answer of A study of the top MBA programs...

Question 4

In an interaction model, the relationship between $E ( y )$ and $x _ { 1 }$ is linear for each fixed value of $x _ { 2 }$ but the slopes of the lines relating $E ( y )$ and $x _ { 1 }$ may be different for two different fixed values of $x _ { 2 }$.

Accepted Answer

In an interaction model, the effect of one explanatory variable ($x_1$) on the expected value of the outcome variable ($E(y)$) depends on the level of another explanatory variable ($x_2$). This means that while the relationship between $E(y)$ and $x_1$ remains linear for any fixed value of $x_2$, the slope of this relationship (i.e., how much $E(y)$ changes with a one-unit change in $x_1$) can vary depending on the value of $x_2$, indicating an interaction between $x_1$ and $x_2$.

Question 5

In the quadratic model $E ( y ) = \beta _ { 0 } + \beta _ { 1 } x + \beta _ { 2 } x ^ { 2 }$, a negative value of $\beta _ { 1 }$ indicates downward concavity.

Accepted Answer

The sign of $\beta_1$ in the quadratic model $E(y) = \beta_0 + \beta_1x + \beta_2x^2$ does not indicate the concavity of the curve; concavity is determined by the sign of $\beta_2$. If $\beta_2 > 0$, the curve is concave upwards, and if $\beta_2 < 0$, it is concave downwards.

Question 6

One of three surfaces is produced by a complete second-order model with two quantitative
independent variables: a paraboloid that opens upward, a paraboloid that opens downward, or a
saddle-shaped surface.

Accepted Answer

A complete second-order model with two quantitative independent variables can indeed produce one of three types of surfaces: a paraboloid that opens upward, a paraboloid that opens downward, or a saddle-shaped surface, depending on the signs and magnitudes of the coefficients in the model.

Question 7

Retail price data for n = 60 hard disk drives were recently reported in a computer magazine. Three variables were recorded for each hard disk drive: $y =$ Retail PRICE (measured in dollars)
$x _ { 1 } =$ Microprocessor SPEED (measured in megahertz)
(Values in sample range from 10 to 40 )
$x _ { 2 } =$ CHIP size (measured in computer processing units)
(Values in sample range from 286 to 486 )
a first-order regression model was fit to the data. Part of the printout follows:

Interpret the interval given in the printout.
A) We are $95 \%$ confident that the average price of all hard drives with 33 megahertz speed and 386 CPU falls between $\$ 3,943$ and $\$ 4,987$.
B) We are $95 \%$ confident that the price of a single hard drive falls between $\$ 3,943$ and $\$ 4,987$.
C) We are $95 \%$ confident that the average price of all hard drives falls between $\$ 3,943$ and $\$ 4,987 .$
D) We are $95 \%$ confident that the price of a single hard drive with 33 megahertz speed and 386 CPU falls between $\$ 3,943$ and $\$ 4,987$.

Accepted Answer

The answer of Retail price data for n = 60...

Question 8

A college admissions officer proposes to use regression to model a studentʹs college GPA
at graduation in terms of the following two variables: $$\begin{array} { l } 
x _ { 1 } = \text { high school GPA } \
x _ { 2 } = \text { SAT score }
\end{array}$$ The admissions officer believes the relationship between college GPA and high school
GPA is linear and the relationship between SAT score and college GPA is linear. She also
believes that the relationship between college GPA and high school GPA depends on the
studentʹs SAT score. Write the regression model she should fit.

Accepted Answer

The answer of A college admissions officer proposes to use...

Question 9

A study of the top MBA programs attempted to predict the average starting salary (in $1000ʹs)of graduates of the program based on the amount of tuition (in $1000ʹs)charged by the program and
The average GMAT score of the programʹs students. The results of a regression analysis based on a
Sample of 75 MBA programs is shown below: Least Squares Linear Regression of Salary
Predictor

Cases Included $75 \quad$ Missing Cases 0
The global- $\mathrm { f }$ test statistic is shown on the printout to be the value $\mathrm { F } = 84.68$. Interpret this value.
A) There is insufficient evidence, at $\alpha = 0.05$, to indicate that the interaction between average tuition and average GMAT score is a useful predictor of the average starting salary of graduates of MBA programs.
B) There is sufficient evidence, at $\alpha = 0.05$, to indicate that the interaction between average tuition and average GMAT score is a useful predictor of the average starting salary of graduates of MBA programs.
C) There is insufficient evidence, at $\alpha = 0.05$, to indicate that at least one of the variables proposed in the interaction model is useful at predicting the average starting salary of graduates of MBA programs.
D) There is sufficient evidence, at $\alpha = 0.05$, to indicate that at least one of the variables proposed in the interaction model is useful at predicting the average starting salary of graduates of MBA programs.

Accepted Answer

The answer of A study of the top MBA programs...

Question 10

A college admissions officer proposes to use regression to model a studentʹs college GPA
at graduation in terms of the following two variables:
x1 = high school GPA
x2 = SAT score
The admissions officer believes the relationship between college GPA and high school
GPA is linear and the relationship between SAT score and college GPA is linear. She also
believes that the relationship between college GPA and high school GPA depends on the
studentʹs SAT score. She proposes the regression model: $$E ( y ) = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 } + \beta _ { 3 } x _ { 1 } x _ { 2 }$$ Explain how to determine if the relationship between college GPA and SAT score depends
on the high school GPA.

Accepted Answer

The answer of A college admissions officer proposes to use...

Question 11

Consider the partial printout for an interaction regression analysis of the relationship between a dependent variable $y$ and two independent variables $x _ { 1 }$ and $x _ { 2 }$.
ANOVA
$\begin{array}{llllll}
\hline & d f & \text { SS } & \text { MS } & F & \text { Significance F } \
\hline \text { Regression } & 3 & 3393.677324 & 1131.225775 & 9391.974782 & 2.11084 \mathrm{E}-11 \
\text { Residual } & 6 & 0.722675987 & 0.120445998 & & \
\text { Total } & 9 & 3394.4 & & & \
\hline
\end{array}$
$\begin{array}{lllllll}
\hline & \text { Coefficients } & \text { Standard Error } & t \text { Stat } & \text { P-value } & \text { Lower 95\% } & \text { Upper 95\% } \
\hline \text { Intercept } & 16.72197014 & 8.283997219 & 2.018587126 & 0.09007654 & -3.548255659 & 36.99219593 \
\mathrm{X} 1 & -3.037317759 & 2.678748705 & -1.133856921 & 0.300116382 & -9.591984506 & 3.517348987 \
\mathrm{X} 2 & -1.046522754 & 1.547132645 & -0.676427297 & 0.523973988 & -4.832222727 & 2.73917722 \
\mathrm{X} 1 \mathrm{X} 2 & 4.071685147 & 0.444059933 & 9.169224345 & 9.47663 \mathrm{E}-05 & 2.98510884 & 5.158261454 \
\hline
\end{array}$

a. Write the prediction equation for the interaction model.
b. Test the overall utility of the interaction model using the global $F$-test at $\alpha = .05$.
c. Test the hypothesis (at $\alpha = .05 )$ that $x _ { 1 }$ and $x _ { 2 }$ interact positively.
d. Estimate the change in $y$ for each additional 1 -unit increase in $x _ { 1 }$ when $x _ { 2 } = 6$.

Accepted Answer

The answer of Consider the partial printout for an interaction...

Question 12

The complete second-order model with two quantitative independent variables does not allow for
interaction between the two independent variables.

Accepted Answer

The complete second-order model with two quantitative independent variables includes terms for the interaction between the two independent variables, allowing for the assessment of how the effect of one variable changes at different levels of the other variable.

Question 13

Consider the interaction model $E ( y ) = 3.6 + 1.2 x _ { 1 } + 2.4 x _ { 2 } + .2 x _ { 1 } x _ { 2 }$. Determine the change in $E ( y )$ when $x _ { 1 }$ is changed from 6 to 7 and $x _ { 2 }$ is held fixed at 3 .
A) $1.8$
B) $4.2$
C) $10.8$
D) $11.4$

Accepted Answer

The change in $E(y)$ due to an increase in $x_1$ from 6 to 7, while $x_2$ is fixed at 3, can be calculated by substituting the values into the interaction term and the $x_1$ term. The change is $1.2 + 0.2 \times 3 = 1.8$.

Question 14

Which equation represents a complete second-order model for two quantitative independent variables? 
A) $E ( y ) = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } ^ { 2 } + \beta _ { 2 } x _ { 2 } ^ { 2 } + \beta _ { 3 } x _ { 1 } ^ { 2 } x _ { 2 } + \beta _ { 4 } x _ { 1 } x _ { 2 } ^ { 2 } + \beta _ { 5 } x _ { 1 } ^ { 2 } x _ { 2 } ^ { 2 }$
B) $E ( y ) = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 } + \beta _ { 3 } x _ { 1 } ^ { 2 } + \beta _ { 4 } x _ { 2 } ^ { 2 }$
C) $E ( y ) = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 } + \beta _ { 3 } x _ { 1 } x _ { 2 } + \beta _ { 4 } x _ { 1 } ^ { 2 } + \beta _ { 5 } x _ { 2 } ^ { 2 }$
D) $E ( y ) = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } x _ { 2 } + \beta _ { 2 } x _ { 1 } ^ { 2 } + \beta _ { 3 } x _ { 2 } ^ { 2 }$

Accepted Answer

A complete second-order model for two quantitative independent variables includes not only the linear terms ($\beta_1 x_1$ and $\beta_2 x_2$), the squared terms ($\beta_4 x_1^2$ and $\beta_5 x_2^2$), but also the interaction term ($\beta_3 x_1 x_2$). Option C includes all these components, making it a complete second-order model.

Question 15

A collector of grandfather clocks believes that the price received for the clocks at an auction increases with the number of bidders, but at an increasing (rather than a constant)rate. Thus, the
Model proposed to best explain auction price (y, in dollars)by number of bidders (x)is the
Quadratic model $$E ( y ) = \beta _ { 0 } + \beta _ { 1 } x + \beta _ { 2 } x ^ { 2 }$$
This model was fit to data collected for a sample of 32 clocks sold at auction; the resulting estimate of $\beta _ { 1 }$ was $- .31$.
Interpret this estimate of $\beta _ { 1 }$.
A) $\beta _ { 1 }$ is a shift parameter that has no practical interpretation.
B) We estimate the auction price will increase $\$ .31$ for each additional bidder at the auction.
C) We estimate the auction price will decrease $\$ .31$ for each additional bidder at the auction.
D) We estimate the auction price will be $- \$ .31$ when there are no bidders at the auction.

Accepted Answer

The answer of A collector of grandfather clocks believes that...

Question 16

A study of the top MBA programs attempted to predict the average starting salary (in $\$ 1000$ 's) of graduates of the program based on the amount of tuition (in $\$ 1000$ 's) charged by the program and the average GMAT score of the program's students. The results of a regression analysis based on a sample of 75 MBA programs is shown below:
Least Squares Linear Regression of Salary

Cases Included $75 \quad$ Missing Cases 0
One of the $t$-test test statistics is shown on the printout to be the value $t = 5.58$. Interpret this value.
A) There is insufficient evidence, at $\alpha = 0.05$, to indicate that at least one of the variables proposed in the interaction model is useful at predicting the average starting salary of graduates of MBA programs.
B) There is insufficient evidence, at $\alpha = 0.05$, to indicate that the interaction between average tuition and average GMAT score is a useful predictor of the average starting salary of graduates of MBA programs.
C) There is sufficient evidence, at $\alpha = 0.05$, to indicate that the interaction between average tuition and average GMAT score is a useful predictor of the average starting salary of graduates of MBA programs.
D) There is sufficient evidence, at $\alpha = 0.05$, to indicate that at least one of the variables proposed in the interaction model is useful at predicting the average starting salary of graduates of MBA programs.

Accepted Answer

The answer of A study of the top MBA programs...

Question 17

Once interaction has been established between $x _ { 1 }$ and $x _ { 2 }$, the first-order terms for $x _ { 1 }$ and $x _ { 2 }$ may be deleted from the regression model leaving the higher-order term containing the product of $x 1$ and $x _ { 2 }$.

Accepted Answer

The first-order terms for $x_1$ and $x_2$ should not be deleted from the regression model when an interaction term is included because the main effects (first-order terms) of $x_1$ and $x_2$ are still important for interpreting the independent effect of each variable on the outcome, aside from their interaction.

Question 18

The concessions manager at a beachside park recorded the high temperature, the number
of people at the park, and the number of bottles of water sold for each of 12 consecutive
Saturdays. The data are shown below.

a. Fit the model $E ( y ) = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 }$ to the data, letting $y$ represent the number of bottles of water sold, $x _ { 1 }$ the temperature, and $x _ { 2 }$ the number of people at the park.
b. Find the $95 \%$ confidence interval for the mean number of bottles of water sold when the temperature is $84 ^ { \circ } \mathrm { F }$ and there are 2700 people at the park.
c. Find the $95 \%$ prediction interval for the number of bottles of water sold when the temperature is $84 ^ { \circ } \mathrm { F }$ and there are 2700 people at the park.

Accepted Answer

The answer of The concessions manager at a beachside park...

Question 19

The independent variables $x _ { 1 }$ and $x _ { 2 }$ interact when the effect on $E ( y )$ of a change in $x _ { 1 }$ depends on $x _ { 2 }$.

Accepted Answer

Interaction between independent variables occurs when the effect of one variable on the dependent variable is different at different levels of another variable, meaning the impact of $x_1$ on $E(y)$ changes depending on the value of $x_2$.

Question 20

Consider the interaction model $E ( y ) = 7 + 3 x _ { 1 } - 4 x _ { 2 } + 5 x _ { 1 } x _ { 2 }$. Find the slope of the line relating $E ( y )$ and $x _ { 1 }$ when $x _ { 2 } = 2$.
A) 16
B) 10
C) 13
D) 1

Accepted Answer

The slope of the line relating $E(y)$ and $x_1$ when $x_2 = 2$ can be found by differentiating $E(y)$ with respect to $x_1$, which gives $3 + 5x_2$. Substituting $x_2 = 2$ into this expression gives $3 + 5(2) = 13$.

Quiz 12: Multiple Regression and Model Building

In an interaction model, the relationship between $E ( y )$ and $x _ { 1 }$ is linear for each fixed value of $x _ { 2 }$ but the slopes of the lines relating $E ( y )$ and $x _ { 1 }$ may be different for two different fixed values of $x _ { 2 }$ .

In the quadratic model $E ( y ) = \beta _ { 0 } + \beta _ { 1 } x + \beta _ { 2 } x ^ { 2 }$ , a negative value of $\beta _ { 1 }$ indicates downward concavity.

One of three surfaces is produced by a complete second-order model with two quantitative independent variables: a paraboloid that opens upward, a paraboloid that opens downward, or a saddle-shaped surface.

The complete second-order model with two quantitative independent variables does not allow for interaction between the two independent variables.

Consider the interaction model $E ( y ) = 3.6 + 1.2 x _ { 1 } + 2.4 x _ { 2 } + .2 x _ { 1 } x _ { 2 }$ . Determine the change in $E ( y )$ when $x _ { 1 }$ is changed from 6 to 7 and $x _ { 2 }$ is held fixed at 3 .

Which equation represents a complete second-order model for two quantitative independent variables?

Once interaction has been established between $x _ { 1 }$ and $x _ { 2 }$ , the first-order terms for $x _ { 1 }$ and $x _ { 2 }$ may be deleted from the regression model leaving the higher-order term containing the product of $x 1$ and $x _ { 2 }$ .

The independent variables $x _ { 1 }$ and $x _ { 2 }$ interact when the effect on $E ( y )$ of a change in $x _ { 1 }$ depends on $x _ { 2 }$ .

Consider the interaction model $E ( y ) = 7 + 3 x _ { 1 } - 4 x _ { 2 } + 5 x _ { 1 } x _ { 2 }$ . Find the slope of the line relating $E ( y )$ and $x _ { 1 }$ when $x _ { 2 } = 2$ .

Quiz 12: Multiple Regression and Model Building

In an interaction model, the relationship between E(y)E ( y )E(y) and x1x _ { 1 }x1​ is linear for each fixed value of x2x _ { 2 }x2​ but the slopes of the lines relating E(y)E ( y )E(y) and x1x _ { 1 }x1​ may be different for two different fixed values of x2x _ { 2 }x2​ .

In the quadratic model E(y)=β0+β1x+β2x2E ( y ) = \beta _ { 0 } + \beta _ { 1 } x + \beta _ { 2 } x ^ { 2 }E(y)=β0​+β1​x+β2​x2 , a negative value of β1\beta _ { 1 }β1​ indicates downward concavity.

One of three surfaces is produced by a complete second-order model with two quantitative independent variables: a paraboloid that opens upward, a paraboloid that opens downward, or a saddle-shaped surface.

The complete second-order model with two quantitative independent variables does not allow for interaction between the two independent variables.

Which equation represents a complete second-order model for two quantitative independent variables?

Once interaction has been established between x1x _ { 1 }x1​ and x2x _ { 2 }x2​ , the first-order terms for x1x _ { 1 }x1​ and x2x _ { 2 }x2​ may be deleted from the regression model leaving the higher-order term containing the product of x1x 1x1 and x2x _ { 2 }x2​ .

The independent variables x1x _ { 1 }x1​ and x2x _ { 2 }x2​ interact when the effect on E(y)E ( y )E(y) of a change in x1x _ { 1 }x1​ depends on x2x _ { 2 }x2​ .

Consider the interaction model E(y) =7+3x1−4x2+5x1x2E ( y ) = 7 + 3 x _ { 1 } - 4 x _ { 2 } + 5 x _ { 1 } x _ { 2 }E(y) =7+3x1​−4x2​+5x1​x2​ . Find the slope of the line relating E(y) E ( y ) E(y) and x1x _ { 1 }x1​ when x2=2x _ { 2 } = 2x2​=2 .

In an interaction model, the relationship between $E ( y )$ and $x _ { 1 }$ is linear for each fixed value of $x _ { 2 }$ but the slopes of the lines relating $E ( y )$ and $x _ { 1 }$ may be different for two different fixed values of $x _ { 2 }$ .

In the quadratic model $E ( y ) = \beta _ { 0 } + \beta _ { 1 } x + \beta _ { 2 } x ^ { 2 }$ , a negative value of $\beta _ { 1 }$ indicates downward concavity.

Once interaction has been established between $x _ { 1 }$ and $x _ { 2 }$ , the first-order terms for $x _ { 1 }$ and $x _ { 2 }$ may be deleted from the regression model leaving the higher-order term containing the product of $x 1$ and $x _ { 2 }$ .

The independent variables $x _ { 1 }$ and $x _ { 2 }$ interact when the effect on $E ( y )$ of a change in $x _ { 1 }$ depends on $x _ { 2 }$ .

Consider the interaction model $E ( y ) = 7 + 3 x _ { 1 } - 4 x _ { 2 } + 5 x _ { 1 } x _ { 2 }$ . Find the slope of the line relating $E ( y )$ and $x _ { 1 }$ when $x _ { 2 } = 2$ .