Question 1

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 interaction terms between the two independent variables.

Question 2

A college admissions officer proposes to use regression to model a student's 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 student's 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 3

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. $$\begin{array} { c c c } 
\hline \text { Bottles Sold } & \text { Temperature } \left( { } ^ { \circ } \mathrm { F } \right) &\text { People } \
\hline 341 & 73 & 1625 \
425 & 79 & 2100 \
457 & 80 & 2125 \
485 & 80 & 2800 \
469 & 81 & 2550 \
395 & 82 & 1975 \
511 & 83 & 2675 \
549 & 83 & 2800 \
543 & 85 & 2850 \
537 & 88 & 2775 \
621 & 89 & 2800 \
897 & 91 & 3100 \
\hline
\end{array}$$ 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. 12.5 Interaction Models 1 Write Interaction Model

Accepted Answer

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

Question 4

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

Accepted Answer

A negative value of $\beta_1$ affects the linear component of the model, not the concavity. Downward concavity is indicated by a negative $\beta_2$.

Question 5

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:
$$\begin{array}{lllllllll}
\hline 
&&&\text { Dep Var }&\text {Predict }&\text {Std Err}&\text { Lower 95\%}&\text { Upper 95\% }\ 
\text {OBS }&\text {SPEED }&\text {CHIP }&\text {PRICE}&\text { Value }&\text {Predict }&\text {Predict}&\text { Predict }&\text {Residual}\
1 & 33 & 286 & 5099.0 & 4464.9 & 260.768 & 3942.7 & 4987.1 & 634.1\
\hline 
\end{array}$$
 Interpret the interval given in the printout.
A) 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.
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 average price of all hard drives with 33 megahertz speed and 386 CPU falls between $3,943 and $4,987.

Accepted Answer

The interval provided is a prediction interval for a single observation, indicating that we are 95% confident that the price of a single hard drive with the specified characteristics falls within this range.

Question 6

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  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 E(Y): ($126,610, $136,640) 95% prediction interval for 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 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.
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 $90,113 and $173,16,30.
C) 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.
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 7

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

This is true as a complete second-order model with two quantitative independent variables can result in three types of surfaces: a paraboloid that opens upward (convex), a paraboloid that opens downward (concave), or a saddle-shaped (non-convex) surface.

Question 8

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

In regression models, even if an interaction term is included, the first-order terms for the interacting variables should generally be retained to properly interpret the interaction effect.

Question 9

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 $x_1$ on $E(y)$ depends on the level of $x_2$, allowing for different slopes for different values of $x_2$.

Question 10

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) $10.8$
C) $11.4$
D) $4.2$

Accepted Answer

The answer of Consider the interaction model \(E ( y...

Question 11

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 atiction; the restilting 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 decrease $\$ .31$ for each additional bidder at the atiction.
C) We estimate the auction price will increase $\$ .31$ for each additional bidder at the auction.
D) We estimate the auction price will be $- \$ .31$ when there are no bidders at the atuction.

Accepted Answer

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

Question 12

A college admissions officer proposes to use regression to model a student's 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 student's SAT score. Write the regression model she should fit. 2 Test if Model is Useful for Predicting y

Accepted Answer

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

Question 13

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

Accepted Answer

The answer of Which equation represents a complete second-order model...

Question 14

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 $x _ { 2 } = 2$ when x2 = 2.
A) 13
B) 10
C) 1
D) 16

Accepted Answer

The slope of the line relating $E(y)$ and $x_1$ is given by the derivative of $E(y)$ with respect to $x_1$. When $x_2 = 2$, the interaction term becomes $5 \times 2 = 10$. Thus, the slope is $3 + 10 = 13$.

Question 15

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 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 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.
B) 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.
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 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.

Accepted Answer

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

Question 16

Consider the partial printout below. $\begin{array}{lclllll}
\hline & \text { Coefficients } & \text { Standard Error } & t \text { Stat } & \text { P-value } & \text { Lower 95\% } & \text { Upper 95\% } \
\hline \text { Intercept } & -63.14873931 & 25.09115112 & -2.516773304 & 0.045484943 & -124.5446192 & -1.752859365 \
X_{1} & 14.72507864 & 8.113581741 & 1.814867849 & 0.119466699 & -5.128155197 & 34.57831248 \
\text { X2 } & 12.48784546 & 4.686063743 & 2.664890224 & 0.037279879 & 1.021452165 & 23.95423875 \
\text { X1X2 } & -1.886935135 & 1.344999834 & -1.402925924 & 0.210210141 & -5.178033575 & 1.404163305 \
\hline
\end{array}$
 Is there evidence (at α = .05) that x1 and x2 interact? Explain. 12.6 Quadratic and Other Higher Order Models 1 Write and Interpret Second-Order Model

Accepted Answer

The answer of Consider the partial printout below. \(\begin{array}{lclllll}
\hline &...

Question 17

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 Missing Cases 0
The global-f test statistic is shown on the printout to be the value $\mathrm { F } = 84.68$. Interpret this value.
A) 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.
B) There is instufficient 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.
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 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.

Accepted Answer

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

Question 18

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: $$E ( y ) = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 } + \beta _ { 3 } x _ { 3 }$$
Where $\mathrm { x } _ { 1 } =$ size of the stadium
 $\mathrm { x } _ { 2 } = 1$ if a day game, 0 if a night game

A plot of the $y - x _ { 1 }$ relationship would show: :
A) Two parallel curves
B) Two non-parallel curves
C) Two non-parallel lines
D) Two parallel lines

Accepted Answer

The answer of We decide to conduct a multiple regression...

Question 19

Consider the partial printout for an interaction regression analysis of the relationship between a dependent variable y and two independent variables x1 and x2. $$\begin{array}{l}
\text { ANOVA }\
\begin{array} { l l l l l l } 
\hline & d f & \text { SS } & \text { MS } & F & \text { Significance } F \
\hline \text { Regression } & 3 & 3393.677324 & 1131.225775 & 9391.974782 & 2.11084 \mathrm { E } - 11 \
\text { Residual } & 6 & 0.722675987 & 0.120445998 & & \
\text { Total } & 9 & 3394.4 & & & \
\hline\
\end{array}\
\begin{array} { l l l l l l l } 
\hline & \text { Coefficients } & \text { Standard Error } & t \text { Stat } & \text { P-value } & \text { Lower 95\% } & \text { Upper 95\% } \
\hline \text { Intercept } & 16.72197014 & 8.283997219 & 2.018587126 & 0.09007654 & - 3.548255659 & 36.99219593 \
\text { X1 } & - 3.037317759 & 2.678748705 & - 1.133856921 & 0.300116382 & - 9.591984506 & 3.517348987 \
\text { X2 } & - 1.046522754 & 1.547132645 & - 0.676427297 & 0.523973988 & - 4.832222727 & 2.73917722 \
\text { X1X2 } & 4.071685147 & 0.444059933 & 9.169224345 & 9.47663 \mathrm { E } - 05 & 2.98510884 & 5.158261454 \
\hline
\end{array}
\end{array}$$ 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 $x _ { 2 } = 6$. 3 Test for Interaction Between Two Variables

Accepted Answer

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

Question 20

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

Accepted Answer

Interaction occurs when the effect of one independent variable on the dependent variable depends on the level of another independent variable.

Quiz 12: Multiple Regression and Model Building

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

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

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 $x _ { 2 }$ .

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 $x _ { 2 }$ .

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) $10.8$ C) $11.4$ D) $4.2$

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 $x _ { 2 } = 2$ when x2 = 2.

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

Quiz 12: Multiple Regression and Model Building

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

In the quadratic model E(y)=β0+β1x+β2x2, a negative value of β1E ( y ) = \beta _ { 0 } + \beta _ { 1 } x + \beta _ { 2 } x ^ { 2 } , \text { a negative value of } \beta _ { 1 }E(y)=β0​+β1​x+β2​x2, a negative value of 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.

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 _ { 1 }x1​ and x2x _ { 2 }x2​ .

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​ .

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 when x2 = 2.

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

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 $x _ { 2 }$ .

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 $x _ { 2 } = 2$ when x2 = 2.