Question 1

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. A portion of the printout is given below:

$\begin{array} { l r r r r r } 
\text { SOURCE}&\text {  DF}&\text {  SS }&\text { MS}& \text {  F }&\text {  PR $>$ F}\
\
\text { Model } & 2 & 115145 & 57573 & 373 & .0001 \ \text { Error } & 9 & 1388 & 154 & & \ \text { TOTAL } & 11 & 116533 & & & \end{array}$

Root MSE $\quad 12.42 \quad$ R-Square $\quad .988$

PARAMETER T for $\mathrm { HO }$ :

$\begin{array} { l r r r r }
& \text {PARAMETER}&& \text {$ \mathrm{T} $ for $ \mathrm{HO} $ }:\
\text {VARIABLES}& \text {ESTIMATES }& \text {STD. ERROR}& \text { PARAMETER = 0 }& \text {PR $> | T |$}\
 \text { INTERPCEP } & 286.42 & 9.66 & 29.64 & .0001 \ \mathrm { X } & - .31 & .06 & - 5.14 & .0006 \ \mathrm { X } \cdot \mathrm { X } & .000067 & .00007 & .95 & .3647 \end{array}$

Is there sufficient evidence to indicate the model is tuseful for predicting the demand for the gem? Use $\alpha = .01$.

Accepted Answer

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

Question 2

When testing the utility of the quadratic model $E ( y ) = \beta _ { 0 } + \beta _ { 1 } x + \beta _ { 2 } x ^ { 2 }$, the most important tests involve the null hypotheses $H _ { 0 } : \beta 0 = 0$ and $H _ { 0 } : \beta _ { 1 } = 0$.

Accepted Answer

The most important test for the utility of a quadratic model is $H_0: \beta_2 = 0$, as it determines whether the quadratic term $x^2$ contributes significantly to the model.

Question 3

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; a portion of the printout follows:
$\begin{array}{lrrrr}
\hline &\text { PARAMETER }& \text {STANDARD }& \text { T FOR 0: }\
\text { VARIABLES } & \text { ESTIMATE } & \text { ERROR } & \text { PARAMETER }=0 & \text { PROB }>|T|\
\text { INTERCEPT } & 286.42 & 9.66 & 29.64 & .0001 \
\mathrm{X} & -.31 & .06 & -5.14 & .0016 \
\mathrm{X} \cdot \mathrm{X} & .000067 & .00007 & .95 & .3600 \
\hline
\end{array}$

Give the $p$-value for testing $H _ { 0 } : \beta _ { 2 } = 0$ against $H _ { a } : \beta 2 \neq 0$.

Accepted Answer

The p-value for testing $H_0: \beta_2 = 0$ is given in the row for $X \cdot X$, which is .3600.

Question 4

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 5

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

In a model with a qualitative independent variable coded as 0-1, β1 represents the change in the mean response when the variable changes from the base level (coded as 0) to the level coded as 1.

Question 6

The complete second-order model $$E ( y ) = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 } + \beta _ { 3 } x _ { 1 } x _ { 2 } + \beta _ { 4 } x _ { 1 } ^ { 2 } + \beta _ { 5 } x _ { 2 } ^ { 2 } \text { was fit to } n = 25$$ data points. The printout is shown below. $\text { ANOVA }$
$\begin{array}{llllll} 
\hline& d f & \text { SS } & M S & F & \text { Significance F } \
\hline \text { Regression } & 5 & 22812.46538 & 4562.493077 & 56487.98 & 6.12671 \mathrm{E}-39 \
\text { Residual } & 19 & 1.534616187 & 0.080769273 & & \
\text { Total } & 24 & 22814 & & & \
\hline 
\end{array}$

$\begin{array}{lllllll}
\hline & \text { Coefficients } & \text { Standard Error } & t \text { Stat } & \text { P-value } & \text { Lower 95\% } & \text { Upper 95\% } \
\hline \text { Intercept } & -0.202274307 & 0.377603882 & -0.535678569 & 0.598396064 & -0.99260856 & 0.588059946 \
\mathrm{X} 1 & 0.57956491 & 0.184697537 & 3.137913578 & 0.005416889 & 0.192988402 & 0.966141418 \
\mathrm{X} 2 & 0.502983937 & 0.130940123 & 3.841327815 & 0.001100855 & 0.228923024 & 0.777044849 \
\mathrm{X}1^{*} \mathrm{X} 2 & 1.976110807 & 0.022011043 & 89.77815357 & 1.92982 \mathrm{E}-26 & 1.93004115 & 2.022180464 \
\mathrm{X}1^{\wedge} 2 & -0.026825292 & 0.025350994 & -1.058155454 & 0.303252905 & -0.079885548 & 0.026234964 \
\mathrm{X}2^{\wedge} 2 & 0.012944358 & 0.015088978 & 0.857868446 & 0.401657492 & -0.018637245 & 0.044525961 \
\hline
\end{array}$ a. Write the complete second-order model for the data.
b. Is there sufficient evidence to indicate that at least one of the parameters $\beta _ { 1 } , \beta _ { 2 } , \beta _ { 3 } , \beta _ { 4 }$, and $\beta _ { 5 }$ is nonzero?
Test using $\alpha = .05$.
c. Test $H _ { 0 } : \beta _ { 3 } = 0$ against $H _ { \mathrm { a } } : \beta _ { 3 } \neq 0$. Use $\alpha = .01$.
d. Test $H _ { 0 } : \beta _ { 4 } = 0$ against $H _ { \mathrm { a } } : \beta _ { 4 } \neq 0$. Use $\alpha = .01$. 3 Test if Model is Useful for Predicting y

Accepted Answer

The answer of The complete second-order model \[E ( y...

Question 7

A study of the top MBA programs attempted to predict y = the average starting salary (in $1000's) of graduates of the program based on x = the amount of tuition (in $1000's) charged by the program. After first considering a simple linear model, it was decided that a quadratic model should be proposed. Which of the following models proposes a 2nd-order quadratic relationship between x and y? A) $\mathrm { E } ( \mathrm { y } ) = \beta _ { 0 } + \beta _ { 1 } \mathrm { x } _ { 1 } + \beta _ { 2 } \mathrm { x } _ { 1 } { } ^ { 2 }$
B) $E ( y ) = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 1 } ^ { 2 } + \beta _ { 3 } x _ { 1 } ^ { 3 }$
C) $\mathrm { E } ( \mathrm { y } ) = \beta _ { 0 } + \beta _ { 1 } \mathrm { x } _ { 1 }$
D) $\mathrm { E } ( \mathrm { y } ) = \beta _ { 0 } + \beta _ { 1 } \mathrm { x } _ { 1 } + \beta _ { 2 } \mathrm { x } _ { 2 } + \beta _ { 3 } \mathrm { x } _ { 1 } \mathrm { x } _ { 2 }$

Accepted Answer

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

Question 8

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

Accepted Answer

The answer of What relationship between x and y is...

Question 9

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 10

When modeling E(y) with a single qualitative independent variable, the number of 0-1 dummy variables in the model is equal to the number of levels of the qualitative variable.

Accepted Answer

When modeling $E(y)$ with a single qualitative independent variable, the number of 0-1 dummy variables required is equal to the number of levels of the qualitative variable minus one, to avoid the dummy variable trap and ensure the model is identifiable.

Question 11

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 $x _ { 1 }$.
A) The relationship between $\hat{y}$ and $x _ { 1 }$ is quadratic with upward concavity.
B) The relationship between $\hat{y}$ and $x _ { 1 }$ is quadratic with downward concavity.
C) The relationship between $\hat{y}$ and $x _ { 1 }$ is linear with positive slope.
D) The relationship between $\hat{y}$ and $x _ { 1 }$ is linear with negative slope.

Accepted Answer

The answer of Consider the second-order model \[\hat { y...

Question 12

The table shows the profit y (in thousands of dollars) that a company made during a month when the price of its product was x dollars per unit. $\begin{array}{cc}
\hline \text { Profit, } y & \text { Price, } x \
\hline 12 & 1.20 \
17 & 1.25 \
20 & 1.29 \
21 & 1.30 \
24 & 1.35 \
26 & 1.39 \
27 & 1.40 \
23 & 1.45 \
21 & 1.49 \
20 & 1.50 \
15 & 1.55 \
11 & 1.59 \
10 & 1.60 \
5 & 1.65 \
\hline
\end{array}$

a. Fit the model $y = \beta _ { 0 } + \beta _ { 1 } x + \beta _ { 2 } x 2 + \varepsilon$ to the data and give the least squares prediction equation.
b. Plot the fitted equation on a scattergram of the data.
c. Is there sufficient evidence of downward curvature in the relationship between profit and price? Use $\alpha = .05$. 4 Perform Quadratic Regression and Make Predictions

Accepted Answer

The answer of The table shows the profit y (in...

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

Cases Included 75 Missing Cases 0
One of the $t$-test test statistics is shown on the printout to be the value $t = 6.03$. 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 14

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; a portion of the printout follows:
$\begin{array}{lrrrr}
\hline &\text { PARAMETER }& \text {STANDARD }& \text { T FOR 0: }\
\text { VARIABLES } & \text { ESTIMATE } & \text { ERROR } & \text { PARAMETER }=0 & \text { PROB }>|T|\
\text { INTERCEPT } & 286.42 & 9.66 & 29.64 & .0001 \
\mathrm{X} & -.31 & .06 & -5.14 & .0016 \
\mathrm{X} \cdot \mathrm{X} & .000067 & .00007 & .95 & .3600 \
\hline
\end{array}$

Find the $p$-value for testing $H _ { 0 } : \beta _ { 2 } = 0$ against $H _ { \mathbf { a } } : \beta _ { 2 } > 0$.

Accepted Answer

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

Question 15

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

The answer of A graphing calculator was used to fit...

Question 16

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. A portion of the printout is given below:

$\begin{array}{lrrrrr}
\text { SOURCE } & \text { DF } & 55 & \text { M5 } & \text { F } & \text { PR }>\text { F }\
\
\text { Model } & 2 & 115145 & 57573 & 373 & , 0001 \
\text { Error } & 9 & 1388 & 154 & & \
\text { TOTAL } & 11 & 116533 & &
\end{array}$

$\text { Root MSE } \quad 12.42 \quad \text { R-Square } \quad 988$

$\begin{array}{lrrrr}
\text { VARIABLES }& \text { ESTIMATES  }& \text { STD, ERROR } &\text { PARAMETER =0 } & \text {  P R>|T|}\
\
\text { INTERPCEP } & 286.42 & 9.66 & 29.64 & .0001 \
\mathrm{X} & -.31 & .06 & -5.14 & .0006 \
\mathrm{X} \cdot \mathrm{X} & .000067 & .00007 & .95 & .3647
\end{array}$

Does the quadratic term contribute useful information for predicting the demand for the gem? Use $\alpha = .10$.

Accepted Answer

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

Question 17

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 18

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...

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

Accepted Answer

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

Question 20

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. If the experts are correct in their assumptions about the relationship between price and demand, which of the following should be true? A) $\beta _ { 2 } > 0$ B) $\beta _ { 2 } < 0$ C) $\beta _ { 1 } > 0$ D) $\beta _ { 1 } < 0$

Accepted Answer

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

Quiz 12: Multiple Regression and Model Building

When testing the utility of the quadratic model $E ( y ) = \beta _ { 0 } + \beta _ { 1 } x + \beta _ { 2 } x ^ { 2 }$ , the most important tests involve the null hypotheses $H _ { 0 } : \beta 0 = 0$ and $H _ { 0 } : \beta _ { 1 } = 0$ .

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.

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

When modeling E(y) with a single qualitative independent variable, the number of 0-1 dummy variables in the model is equal to the number of levels of the qualitative variable.

Quiz 12: Multiple Regression and Model Building

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.

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

When modeling E(y) with a single qualitative independent variable, the number of 0-1 dummy variables in the model is equal to the number of levels of the qualitative variable.

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​ ?