Question 1

What relationship between x and y is suggested by the scattergram?

Accepted Answer

A)   a quadratic relationship with downward concavity
B)   a linear relationship with negative slope
C)   a linear relationship with positive slope
D)   a quadratic relationship with upward concavity A

Question 2

An elections officer wants to model voter turnout (y) in a precinct as a function of type of election, national or state.

Write a model for mean voter turnout, E(y), as a function of type of election.

Accepted Answer

A)    E(y) =β0+β1xE ( y )  = \beta _ { 0 } + \beta 1 ^ { x }E(y) =β0​+β1x , where  x=1x = 1x=1  if national, 0 if state
B)    E(y) =β0+β1xE ( y )  = \beta _ { 0 } + \beta _ { 1 } xE(y) =β0​+β1​x , where  x=x =x=  voter turnout
C)    E(y) =β0+β1x1+β2x2E ( y )  = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 }E(y) =β0​+β1​x1​+β2​x2​ , where  x1=1x _ { 1 } = 1x1​=1  if national, 0 if not and  x2=1x _ { 2 } = 1x2​=1  if state, 0 if not
D)    E(y) =β0+β1x+β2x2E ( y )  = \beta _ { 0 } + \beta _ { 1 } x + \beta _ { 2 } x ^ { 2 }E(y) =β0​+β1​x+β2​x2 , where  x=x =x=  voter turnout  A

Question 3

Consider the model $y = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 } + \beta _ { 3 } x _ { 3 } + \varepsilon$ where $x _ { 1 }$ is a quantitative variable and $x _ { 2 }$ and $x _ { 3 }$ are dummy variables describing a qualitative variable at three levels using the coding scheme $x _ { 2 } = \left\{ \begin{array} { l l } 1 & \text { if level } 2 \ 0 & \text { otherwise } \end{array} \quad x _ { 3 } = \left\{ \begin{array} { l l } 1 & \text { if level } 3 \ 0 & \text { otherwise } \end{array} \right. \right.$

The resulting least squares prediction equation is $\hat { y } = 36.7 + 1.3 x _ { 1 } + 5.4 x _ { 2 } + 3.2 x _ { 3 }$. What is the least squares regression equation associated with level 2?

Accepted Answer

A)    y^=39.9+1.3x1\hat { y } = 39.9 + 1.3 x _ { 1 }y^​=39.9+1.3x1​ 
B)    y^=38.0+5.4x2\hat { y } = 38.0 + 5.4 x _ { 2 }y^​=38.0+5.4x2​ 
C)    y^=39.9+5.4x2\hat { y } = 39.9 + 5.4 x _ { 2 }y^​=39.9+5.4x2​ 
D)    y^=42.1+1.3x1\hat { y } = 42.1 + 1.3 x _ { 1 }y^​=42.1+1.3x1​ D

Question 4

The sum of squared errors (SSE) of a least squares regression model decreases when new terms are added to the model.

Accepted Answer

A) True 
 B)False

Question 5

In stepwise regression, the probability of making one or more Type I or Type II errors is quite small.

Accepted Answer

A) True 
 B)False

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

$\begin{array}{l}
\text { Predictor }\
\begin{array}{lcccccccc}
\text { Variables } & \text { Coefficient } & {\text { Std Error }} & \text { T } &  \text { P } & {\text { VIF }} \
\text { Constant } & -203.402 & 51.6573 & -3.94 & 0.0002 & 0.0 \
\text { Gmat } & 0.39412 & 0.09039 & 4.36 &  0.0000 & 2.0 & \
\text { Tuition } & 0.92012 & 0.17875 & 5.15 &  0.0000 & 2.0 &
\end{array}
\end{array}$

$\begin{array}{lccc}
\text { R-Squared } & 0.6857 & \text { Resid. Mean Square (MSE) } & 427.511 \
\text { Adjusted R-Squared } & 0.6769 & \text { Standard Deviation } & 20.6763
\end{array}$

$\begin{array}{lllcccc}
\text { Source } & \text { DF } & \text { SS } & \text { MS } & \text { F } & \text { P } \
\text { Regression } & 2  & 67140.9 & 33570.5 & 78.53 & 0.0000 \
\text { Residual } &  72 & 30780.8 & 427.5 & \
\text { Total } &  74 & 97921.7 & &
\end{array}$

Accepted Answer

A)   At  α=0.05\alpha = 0.05α=0.05 , there is insufficient evidence to indicate that something in the regression model is useful for predicting the average starting salary of the graduates of an MBA program.
B)   We expect most of the average starting salaries to fall within  $20,676\$ 20,676$20,676  of their least squares predicted values.
C)   We expect most of the average starting salaries to fall within  $41,353\$ 41,353$41,353  of their least squares predicted values.
D)   We can explain  68.57%68.57 \%68.57%  of the variation in the average starting salaries around their mean using the model that includes the average GMAT score and the tuition for the MBA program.

Question 7

The printout below shows part of the least squares regression analysis for the model $E ( y ) = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 }$ fit to a set of data. The model attempts to predict a score on the final exam in a statistics course based on the scores on the first two tests in the class.
ANOVA
$\begin{array} { l l l l l l } 
\hline & d f & S S & M S & F & \text { Significance F } \
\hline \text { Regression } & 2 & 1293.125328 & 646.5626641 & 21.27366772 & 2.35769 \mathrm { E } - 05 \
\text { Residual } & 17 & 516.6746719 & 30.39262776 & & \
\text { Total } & 19 & 1809.8 & & & \
\hline
\end{array}$

$\begin{array}{lllllll}
\hline & \text { Coefficients } & \text { Standard Error } & \text { t Stat } & \text { P-value } & \text { Lower 95\% } & \text { Upper 95\% } \
\hline \text { Intercept } & -4.409686163 & 16.72267106 & -0.263695085 & 0.795184685 & -39.69148734 & 30.87211502 \
\text { Test 1 } & 0.397435806 & 0.343012569 & 1.158662514 & 0.262611745 & -0.326258467 & 1.121130079 \
\text { Test 2 } & 0.638805278 & 0.224623383 & 2.843894834 & 0.011217936 & 0.164890704 & 1.112719852 \
\hline
\end{array}$
 Is there evidence of multicollinearity in the printout? Explain.

Accepted Answer

The answer of The printout below shows part of the...

Question 8

For a multiple regression model, we assume that the mean of the probability distribution of the
random error is 0.

Accepted Answer

A) True 
 B)False

Question 9

The model $E ( y ) = \beta _ { 0 } + \beta _ { 1 } x$ was fit to a set of data, and the following plot of residuals against $x$ values was obtained.

Interpret the residual plot.

Accepted Answer

The answer of The model \(E ( y ) =...

Question 10

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: $$\begin{array}{l}
\text { Least Squares Linear Regression of Salary }\
\begin{array} { l c c c c c } 
\text { Predictor } & & & & & \
\text { Variables } & \text { Coefficient } & \text { Std Error } & \text { T } & \text { P } \
\text { Constant } & - 687.851 & 165.406 & 4.16 & 0.0001 \
\text { Tuition } & - 11.3197 & 2.19724 & - 5.15 & 0.0000 \
\text { GMAT } & - 0.96727 & 0.25535 & - 3.79 & 0.0003
\end{array}
\end{array}$$

$ \begin{array}{lllll}\text { TxG } & 0.01850 & 0.00331 & 5.58 & 0.0000\end{array} $

$\begin{array}{lccc}
\text { R-Squared } & 0.7816 & \text { Resid. Mean Square (MSE) } & 301.251\
\text { Adjusted R-Squared } & 0.7723& \text { Standard Deviation } & 17.3566
\end{array}$

$\begin{array}{lllcccc}
\text { Source } & \text { DF } & \text { SS } & \text { MS } & \text { F } & \text { P } \
\text { Regression } & 3  &76523.8& 25510.9 & 84.68& 0.0000 \
\text { Residual } &  71 &21388.8& 301.3 & \
\text { Total } &  74 & 97921.7 & &
\end{array}$

Cases Included 75 Missing Cases 0

The global-f test statistic is shown on the printout to be the value $F = 84.68$. Interpret this value.

Accepted Answer

A)   There is sufficient evidence, at  α=0.05\alpha = 0.05α=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  α=0.05\alpha = 0.05α=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 insufficient evidence, at  α=0.05\alpha = 0.05α=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  α=0.05\alpha = 0.05α=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.

Question 11

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 consecutiveSaturdays. The data are shown below.

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

a. Fit the model $E ( y ) = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 } + \beta _ { 3 } x _ { 1 } x _ { 2 }$ 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. Identify at least two indicators of multicollinearity in the model.
c. Comment on the usefulness of the model to predict the number of bottles of water sold on a Saturday when the high temperature is $103 ^ { \circ } \mathrm { F }$ and there are 3500 people at the park.

Accepted Answer

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

Question 12

It is desired to build a regression model to predict $y =$ the sales price of a single family home, based on the $x _ { 1 } =$ size of the house and $x _ { 2 } =$ the neighborhood the home is located in. The goal is to compare the prices of homes that are located in two different neighborhoods. The following model is proposed:
$$\mathrm { E } ( \mathrm { y } ) = \beta _ { 0 } + \beta _ { 1 } \mathrm { x } _ { 1 } + \beta _ { 2 } \mathrm { x } _ { 2 }$$
A regression model was fit and the following residual plot was observed.
Residual

Which of the following assumptions appears violated based on this plot?

Accepted Answer

A)   The variance of the errors is constant
B)   The errors are independent
C)   The mean of the errors is zero
D)   The errors are normally distributed

Question 13

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
$\begin{array}{llllll}
\hline & \text { df } & \text { SS } & \text { MS } & F & \text { Significance F } \
\hline \text { Regression } & 3 & 3393.677324 & 1131.225775 & 9391.974782 & 2.11084 \mathrm{E}-11 \
\text { Residual } & 6 & 0.722675987 & 0.120445998 & & \
\text { Total } & 9 & 3394.4 & & & \
\hline
\end{array}$

$\begin{array}{lllllll} 
& \text { Coefficients } & \text { Standard Error } & t \text { Stat } & \text { P-value } & \text { Lower 95\% } & \text { Upper 95\% } \
\hline \text { Intercept } & 16.72197014 & 8.283997219 & 2.018587126 & 0.09007654 & -3.548255659 & 36.99219593 \
\text { X1 }_{1} & -3.037317759 & 2.678748705 & -1.133856921 & 0.300116382 & -9.591984506 & 3.517348987 \
\text { X2 }_{2} & -1.046522754 & 1.547132645 & -0.676427297 & 0.523973988 & -4.832222727 & 2.73917722 \
\text { X1X2 }_{1} & 4.071685147 & 0.444059933 & 9.169224345 & 9.47663 \mathrm{E}-05 & 2.98510884 & 5.158261454
\end{array}$

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 14

Suppose that the following model was fit to a set of data.
$$E ( y ) = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 }$$
The corresponding plot if residuals against predicted values $\hat { y }$ is shown. Interpret the plot.

Accepted Answer

A)   The residuals appear to be randomly scattered so that no model modifications are necessary.
B)   It appears that the variance ofε is not constant.
C)   It appears that the data contain an outlier.
D)   It appears that a quadratic model would be a better fit.

Question 15

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)
$\mathrm { x } _ { 1 } =$ Microprocessor SPEED (measured in megahertz)
(Values in sample range from 10 to 40 )
$\mathrm { x } _ { 2 } = \mathrm { CHIP }$ 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:

$\begin{array} { r r r r r r r r r } 
\hline& & & \text { Dep Var } & \text { Predict } & \text { Std Err } & \text { Lower 95\% } & \text { Upper 95\% } & \
\text { OBS } & \text { SPEED } & \text { CHIP } & \text { PRICE } & \text { Value } & \text { Predict } & \text { Predict } & \text { Predict } & \text { Residual } \
& & & & & & & & \
1 & 33 & 386 & 5099.0 & 4464.9 & 260.768 & 3942.7 & 4987.1 & 634.1\
\hline
\end{array}$

Interpret the $95 \%$ prediction interval for $y$ when $x _ { 1 } = 33$ and $x _ { 2 } = 386$.

Accepted Answer

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

Question 16

Consider the second-order model $$\hat { y } = - 3.24 + 1.12 x _ { 1 } + 2.57 x _ { 2 } - 3.22 x _ { 1 } x _ { 2 } + 5.78 x _ { 1 } ^ { 2 } = 4.69 x _ { 2 } ^ { 2 }$$
If $x _ { 2 }$ is held fixed at $x _ { 2 } = 3$, describe the relationship between $\hat { y }$ and $x _ { 1 }$.

Accepted Answer

A)   The relationship between  y^\hat { y }y^​  and  x1x _ { 1 }x1​  is linear with negative slope.
B)   The relationship between  yyy  and  x1x _ { 1 }x1​  is quadratic with upward concavity.
C)   The relationship between  y^\hat { y }y^​  and  x1x _ { 1 }x1​  is linear with positive slope.
D)   The relationship between  y^\hat { y }y^​  and  x1x _ { 1 }x1​  is quadratic with downward concavity.

Question 17

During its manufacture, a product is subjected to four different tests in sequential order. An efficiency expert claims that the fourth (and last) test is unnecessary since its results can be predicted based on the first three tests. To test this claim, multiple regression will be used to model Test4 score $( y )$, as a function of Test1 score $\left( x _ { 1 } \right)$ ), Test 2 score $\left( x _ { 2 } \right)$, and Test3 score $\left( x _ { 3 } \right)$. [Note: All test scores range from 200 to 800 , with higher scores indicative of a higher quality product.] Consider the model:
$$E ( y ) = \beta _ { 1 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 } + \beta _ { 3 } x _ { 3 }$$
The first-order model was fit to the data for each of 12 units sampled from the production line. The results are summarized in the printout.

$\begin{array}{lrrrrr}
\text { SOURCE } & \text { DF } & \text { SS } & \text { MS } & \text { FVALUE } & \text { PROB > F } \
\text { MODEL } & 3 & 151417 & 50472 & 18.16 & .0075 \
\text { ERROR } & 8 & 22231 & 2779 & & \
\text { TOTAL } & 12 & 173648 & & &
\end{array}$

$\begin{array}{llll}
\text { ROOT MSE } & 52.72 & \text { R-SQUARE } & 0.872 \
\text { DEP MEAN } & 645.8 & \text { ADJ R-SQ } & 0.824
\end{array}$

$\begin{array} { l l l l } 
& \text {PARAMETER }& \text {STANDARD}& \text {T FOR 0:}\
 \text {VARIABLES}&\text { ESTIMATE } & \text { ERROR } & \text { PARAMETER } = 0 & \text { PROB } > | T | \end{array}$

$\begin{array} { l r r r r } 
\text { INTERCEPT } & 11.98 & 80.50 & 0.15 & 0.885 \
\text { X1(TEST1) } & 0.2745 & 0.1111 & 2.47 & 0.039 \
\text { X2(TEST2) } & 0.3762 & 0.0986 & 3.82 & 0.005 \
\text { X3(TEST3) } & 0.3265 & 0.0808 & 4.04 & 0.004 \
\hline
\end{array}$

Compute a $95 \%$ confidence interval for $\beta _ { 3 }$.

Accepted Answer

A)    .33±105.33 \pm 105.33±105 
B)    .33±4.04.33 \pm 4.04.33±4.04 
C)    .33±.08.33 \pm .08.33±.08 
D)    .33±.19.33 \pm .19.33±.19

Question 18

The rejection of the null hypothesis in a global F-test means that the model is the best model for providing reliable estimates and predictions.

Accepted Answer

A) True 
 B)False

Question 19

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.
Suppose the $p$-value for the test of $H _ { 0 } : \beta _ { 2 } = 0$ vs. $H _ { \mathrm { a } } : \beta _ { 2 } > 0$ is $.02$. What is the proper conclusion?

Accepted Answer

A)   There is evidence (at  α=.05\alpha = .05α=.05  )  of downward curvature in the relationship between auction price  (y) ( y ) (y)   and number of bidders  (x) ( x ) (x)  .
B)   Reject  H0H _ { 0 }H0​  at  α=.05\alpha = .05α=.05 ; the model is not useful for predicting auction price  (y) ( y ) (y)  .
C)   There is evidence (at  α=.05\alpha = .05α=.05  )  of upward curvature in the relationship between auction price (y)  and number of bidders  (x) ( x ) (x)  .
D)   There is no evidence (at  α=.05\alpha = .05α=.05  )  of upward curvature in the relationship between auction price  (y) ( y ) (y)   and number of bidders  (x) ( x ) (x)  .

Question 20

Consider the data given in the table below. $$\begin{array} { c c } 
\hline \mathrm { X } & \mathrm { Y } \
\hline 1 & 7 \
2 & 6 \
2 & 5 \
3 & 5 \
3 & 4 \
4 & 4 \
4 & 3 \
4 & 2 \
5 & 4 \
5 & 5 \
6 & 6 \
\hline
\end{array}$$ Plot the data on a scattergram. Does a second-order model seem to be a good fit for the data? Explain.

Accepted Answer

The answer of Consider the data given in the table...

Exam 12: Multiple Regression and Model Building

What relationship between x and y is suggested by the scattergram?

An elections officer wants to model voter turnout (y) in a precinct as a function of type of election, national or state. Write a model for mean voter turnout, E(y) , as a function of type of election.

The sum of squared errors (SSE) of a least squares regression model decreases when new terms are added to the model.

In stepwise regression, the probability of making one or more Type I or Type II errors is quite small.

For a multiple regression model, we assume that the mean of the probability distribution of the random error is 0.

Consider the second-order model $\hat { y } = - 3.24 + 1.12 x _ { 1 } + 2.57 x _ { 2 } - 3.22 x _ { 1 } x _ { 2 } + 5.78 x _ { 1 } ^ { 2 } = 4.69 x _ { 2 } ^ { 2 }$ If $x _ { 2 }$ is held fixed at $x _ { 2 } = 3$ , describe the relationship between $\hat { y }$ and $x _ { 1 }$ .

The rejection of the null hypothesis in a global F-test means that the model is the best model for providing reliable estimates and predictions.

Exam 12: Multiple Regression and Model Building

What relationship between x and y is suggested by the scattergram?

An elections officer wants to model voter turnout (y) in a precinct as a function of type of election, national or state. Write a model for mean voter turnout, E(y) , as a function of type of election.

The sum of squared errors (SSE) of a least squares regression model decreases when new terms are added to the model.

In stepwise regression, the probability of making one or more Type I or Type II errors is quite small.

For a multiple regression model, we assume that the mean of the probability distribution of the random error is 0.

The model E(y)=β0+β1xE ( y ) = \beta _ { 0 } + \beta _ { 1 } xE(y)=β0​+β1​x was fit to a set of data, and the following plot of residuals against xxx values was obtained. Interpret the residual plot.

The rejection of the null hypothesis in a global F-test means that the model is the best model for providing reliable estimates and predictions.