Question 1

A graphing calculator was used to fit the model E(y) = ?0 + ?1x + ?2x2 to a set of data. The resulting screen is shown below.  Which number on the screen represents the estimator of $\beta _ { 2 }$ ?

Accepted Answer

A)  .9286 
B)  5.5 
C)  11 
D)  .9405

Question 2

It is desired to build a regression model to predict y = the sales price of a single family home, based on the neighborhood the home is located in. The goal is to compare the prices of homes that are located in four different neighborhoods. Which regression model should be built? A) $E ( y ) = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 } + \beta _ { 3 } x _ { 3 } + \beta _ { 4 } x _ { 4 }$, where $x _ { 1 } - x _ { 4 }$ are qualitative variables that describe the four neighborhoods.
B) $\mathrm { E } ( \mathrm { y } ) = \beta _ { 0 } + \beta _ { 1 } \mathrm { x } _ { 1 } + \beta _ { 2 } \mathrm { x } _ { 2 } + \beta _ { 3 } \mathrm { x } _ { 3 }$, where $\mathrm { x } _ { 1 } - \mathrm { x } _ { 3 }$ are qualitative variables that describe the four neighborhoods.
C) $\mathrm { E } ( \mathrm { y } ) = \beta _ { 0 } + \beta _ { 1 } \mathrm { x } _ { 1 }$, where $\mathrm { x } _ { 1 }$ is a qualitative variable that describes the four neighborhoods.
D) $\mathrm { E } ( \mathrm { y } ) = \beta _ { 0 } + \beta _ { 1 } \mathrm { x } _ { 1 } + \beta _ { 2 } \mathrm { x } _ { 1 } { } ^ { 2 }$, where $\mathrm { x } _ { 1 }$ is a qualitative variable that describes the four neighborhoods.

Accepted Answer

The answer of It is desired to build a regression...

Question 3

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

Question 4

A first-order model may include terms for both quantitative and qualitative independent variables.

Accepted Answer

A) True 
 B)False

Question 5

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 6

It is desired to build a regression model to predict $\mathrm { y } =$ the sales price of a single family home, based on the $\mathrm { x } _ { 1 } =$ size of the house and $\mathrm { x } _ { 2 } =$ 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:
$$E ( y ) = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 }$$
A regression model was fit and the following residual plot was observed.
 Which of the following assumptions appears violated based on this plot?

Accepted Answer

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

Question 7

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 Predictor

Cases Included 75 Missing Cases 0
The global-f test statistic is shown on the printout to be the value $\mathrm { F } = 100.42$. 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 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.
C) There is sufficient evidence, at $\alpha = 0.05$, to indicate that there is a curvilinear relationship between average starting salary of graduates of MBA programs and the tuition of the MBA program.
D) There is sufficient evidence, at $\alpha = 0.05$, to indicate that there is a linear relationship between average starting salary of graduates of MBA programs and the tuition of the MBA program.

Accepted Answer

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

Question 8

A certain type of rare gem serves as a status symbol for many of its owners. In theory, for low prices, the demand decreases as the price of the gem increases. However, experts hypothesize that when the gem is valued at very high prices, the demand increases with price due to the status the owners believe they gain by obtaining the gem. Thus, the model proposed to best explain the demand for the gem by its price is the quadratic model $$E ( y ) = \beta _ { 0 } + \beta _ { 1 } x + \beta _ { 2 } x ^ { 2 }$$
where $y =$ Demand (in thousands) and $x =$ Retail price per carat (dollars).
This model was fit to data collected for a sample of 12 rare gems.
$\begin{array} { l r r r r } \text { VARIABLES } & \begin{array} { r } \text { PARAMETER } \ \text { ESTIMATES } \end{array} & \text { STD. ERROR } & \begin{array} { r } \text { T for HO: } \ \text { PARAMETER } = 0 \end{array} & \text { PR } > | T | \ \text { INTERPCEP } & 286.42 & 9.66 & & \ \text { X } & - .31 & .06 & 29.64 & .0001 \ \text { X.X } & .000067 & .00007 & - 5.14 & .0006 \ & & & .95 & .3647 \end{array}$
Does there appear to be upward curvature in the response curve relating $y$ (demand) to $x$ (retail price)?
A) No, since the $p$-value for the test is greater than .10.
B) Yes, since the $p$-value for the test is less than .01.
C) No, since the value of $\beta _ { 2 }$ is near 0 .
D) Yes, since the value of $\beta _ { 2 }$ is positive.

Accepted Answer

The answer of A certain type of rare gem serves...

Question 9

When using the model E(y) = β0 + β1x for one qualitative independent variable with a 0-1 coding convention, β1 represents the difference between the mean responses for the level assigned the value 1 and the base level.

Accepted Answer

A) True 
 B)False

Question 10

The stepwise regression model should not be used as the final model for predicting y.

Accepted Answer

A) True 
 B)False

Question 11

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 restults can be predicted based on the first three tests. To test this claim, multiple regression will be used to model Test 4 score (y), as a function of Test1 score $\left( x _ { 1 } \right)$. Test 2 score $\left( x _ { 2 } \right)$, and Test 3 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.

A) $.33 \pm .19$
B) $33 \pm 105$
C). $.33 \pm .08$
D) $.33 \pm 4.04$

Accepted Answer

The answer of During its manufacture, a product is subjected...

Question 12

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. A) $E ( y ) = \beta _ { 0 } + \beta _ { 1 } x$, where $x = 1$ if national, 0 if state
B) $E ( y ) = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 }$, where $x _ { 1 } = 1$ if national, 0 if not and $x _ { 2 } = 1$ if state, 0 if not
C) $E ( y ) = \beta _ { 0 } + \beta _ { 1 } x$, where $x =$ voter turnout
D) $E ( y ) = \beta _ { 0 } + \beta _ { 1 } x + \beta _ { 2 } x ^ { 2 }$, where $x =$ voter turnout

Accepted Answer

The answer of An elections officer wants to model voter...

Question 13

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  Interpret the coefficient of determination value shown in the printout.

Accepted Answer

A)  We can explain 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. 
B)  We expect most of the average starting salaries to fall within $41,353 of their least squares predicted values. 
C)  We expect most of the average starting salaries to fall within $20,676 of their least squares predicted values. 
D)  At α = 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.

Question 14

Consider the data given in the table below. $$\begin{array} { c c } 
\hline \mathrm { X } & \mathrm { Y } \
\hline 1 & 4 \
2 & 6 \
2 & 5 \
3 & 7 \
4 & 7 \
4 & 6 \
5 & 4 \
5 & 5 \
6 & 3 \
\hline
\end{array}$$ a. Plot the data on a scattergram. Does a quadratic model seem to be a good fit for the data? Explain. b. Use the method of least squares to find a quadratic prediction equation. c. Graph the prediction equation on your scattergram. 12.7 Qualitative (Dummy) Variable Models 1 Write and Interpret Model with Qualitative Variables

Accepted Answer

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

Question 15

The staff of a test kitchen is attempting to determine the baking time, y, of a roast, i.e., the time it takes the internal temperature of the roast to reach 165°F, using two variables, the temperature setting of the oven, x1, and the weight of the roast, x2, in pounds. The data for 24 roasts are shown below. $\text { Baking Times of Roasts }$
$\begin{array}{lll|lll|lll|llll}
\mathrm{X} 1\left({ }^{\circ} \mathrm{F}\right) & \mathrm{X} 2(\mathrm{lb}) & \mathrm{Y}(\mathrm{hr}) & \mathrm{X} 1\left({ }^{\circ} \mathrm{F}\right) & \mathrm{X} 2(\mathrm{lb}) & \mathrm{Y}(\mathrm{hr}) & \mathrm{X} 1\left({ }^{\circ} \mathrm{F}\right) & \mathrm{X} 2(\mathrm{lb}) & \mathrm{Y}(\mathrm{hr}) & \mathrm{X} 1\left({ }^{\circ} \mathrm{F}\right) & \mathrm{X} 2(\mathrm{lb}) &\mathrm{Y}(\mathrm{hr}) \
\hline 300 & 2.2 & 2.6 & 325 & 2.1 & 2.3 & 350 & 2.3 & 2.2 & 375 & 2.2 & 1.9 \
300 & 2.7 & 2.8 & 325 & 2.4 & 2.4 & 350 & 2.5 & 2.3 & 375 & 2.6 & 2.2 \
300 & 2.9 & 3.1 & 325 & 2.9 & 2.6 & 350 & 2.8 & 2.5 & 375 & 2.9 & 2.4 \
300 & 3.1 & 3.2 & 325 & 3.0 & 2.7 & 350 & 3.2 & 2.7 & 375 & 3.1 & 2.6 \
300 & 3.2 & 3.2 & 325 & 3.2 & 2.9 & 350 & 3.4 & 2.8 & 375 & 3.3 & 2.7 \
300 & 3.5 & 3.3 & 325 & 3.6 & 3.1 & 350 & 3.5 & 2.8 & 375 & 3.4 & 2.7
\end{array}$
 a. Fit a complete second-order model to the data. b. Do the data provide sufficient evidence to indicate that the second-order terms contribute information for the prediction of y? State the null and alternative hypotheses and the test statistic. Use α = .05. 12.10 Stepwise Regression (Optional) 1 Understand and Interpret Stepwise Regression Model

Accepted Answer

The answer of The staff of a test kitchen is...

Question 16

The number of levels of observed x-values must be equal to the order of the polynomial in x that you want to fit.

Accepted Answer

A) True 
 B)False

Question 17

Which of the following is not a possible indicator of multicollinearity?

Accepted Answer

A)  non-random patterns in the plot of the residuals versus the fitted values 
B)  significant correlations between pairs of independent variables 
C)  non-significant t-tests for individual β parameters when the F-test for overall model adequacy is significant 
D)  signs opposite from what is expected in the estimated β parameters

Question 18

A nested model F-test can only be used to determine whether second-order terms should be included in the model.

Accepted Answer

A) True 
 B)False

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

If when using the model $E ( y ) = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 } + \beta _ { 3 } x _ { 1 } x _ { 2 }$ we determine that interaction between $x _ { 1 }$ and $x _ { 2 }$ is not significant, we can drop the $x _ { 1 } x _ { 2 }$ term from the model and use the simpler model $E ( y ) = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 }$

Accepted Answer

A) True 
 B)False

Exam 12: Multiple Regression and Model Building

A graphing calculator was used to fit the model E(y) = ?0 + ?1x + ?2x2 to a set of data. The resulting screen is shown below. Which number on the screen represents the estimator of β2\beta _ { 2 }β2​ ?

Suppose that the following model was fit to a set of data. E(y) =β0+β1x1+β2x2E ( y ) = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 }E(y) =β0​+β1​x1​+β2​x2​ The corresponding plot if residuals against predicted values y^\hat{y}y^​ is shown. Interpret the plot.

A first-order model may include terms for both quantitative and qualitative independent variables.

When using the model E(y) = β0 + β1x for one qualitative independent variable with a 0-1 coding convention, β1 represents the difference between the mean responses for the level assigned the value 1 and the base level.

The stepwise regression model should not be used as the final model for predicting y.

The number of levels of observed x-values must be equal to the order of the polynomial in x that you want to fit.

Which of the following is not a possible indicator of multicollinearity?

A nested model F-test can only be used to determine whether second-order terms should be included in the model.