Question 1

TABLE 15-7
A chemist employed by a pharmaceutical firm has developed a muscle relaxant. She took a sample of 14 people suffering from extreme muscle constriction. She gave each a vial containing a dose (X) of the drug and recorded the time to relief (Y) measured in seconds for each. She fit a "centered" curvilinear model to this data. The results obtained by Microsoft Excel follow, where the dose (X) given has been "centered."
SUMMARY OUTPUT
  $$\begin{array}{l}
\begin{array} { l r } 
\begin{array} { l }

\end{array} \
\hline\text { Regression  Statistics }\
\hline \text { Multiple R } & 0.747 \
\text { RSquare } & 0.558 \
\text { Adjusted R Square } & 0.478 \
\text { Standard Error } & 863.1 \
\text { Observations } & 14 \
\hline
\end{array}\
\text { ANOVA }\\
\begin{array} { l r r r l l } 
\hline & d f & { \text { SS } } & \text { MS } & F & \text { Significance } F \
\hline \text { Regression } & 2 & 10344797 & 5172399 & 6.94 & 0.0110 \
\text { Residual } & 11 & 8193929 & 744903 & & \
\text { Total } & 13 & 18538726 & & & \
\hline
\end{array}\\
\begin{array} { l c c c c } 
\hline & \text { Coeff } & \text { Std Error } & t \text { Stut } & p \text {-value } \
\hline \text { Intercept } & 1283.0 & 352.0 & 3.65 & 0.0040 \
\text { CenDose } & 25.228 & 8.631 & 2.92 & 0.0140 \
\text { CenDoseSq } & 0.8604 & 0.3722 & 2.31 & 0.0410 \
\hline
\end{array}
\end{array}$$
-Referring to Table 15-7, suppose the chemist decides to use an F test to determine if there is a significant curvilinear relationship between time and dose. If she chooses to use a level of significance of 0.01 she would decide that there is a significant curvilinear relationship.

Accepted Answer

The decision to determine if there is a significant curvilinear relationship using an F test depends on the calculated F statistic and its comparison to the critical value at the 0.01 significance level. Without specific values, we cannot conclude the significance.

Question 2

TABLE 15-9
Many factors determine the attendance at Major League Baseball games. These factors can include when the game is played, the weather, the opponent, whether or not the team is having a good season, and whether or not a marketing promotion is held. Data from 80 games of the Kansas City Royals for the following variables are collected.
ATTENDANCE = Paid attendance for the game 
TEMP = High temperature for the day
WIN% = Team's winning percentage at the time of the game
OPWIN% = Opponent team's winning percentage at the time of the game WEEKEND - 1 if game played on Friday, Saturday or Sunday; 0 otherwise PROMOTION - 1 = if a promotion was held; 0 = if no promotion was held
The regression results using attendance as the dependent variable and the remaining five variables as the independent variables are presented below.
  $$\begin{array}{l}
\text { Regression Statistics }\
\begin{array} { l r } 
\hline \text { Multiple R } & 0.5487 \
\text { R Square } & 0.3011 \
\text { Adjusted R Square } & 0.2538 \
\text { Standard Error } & 6442.4456 \
\text { Observations } & 80 \
\hline
\end{array}
\end{array}$$

$$\begin{array}{l}
\text { ANOVA }\
\begin{array} { l c c c c c } 
\hline & \mathrm { df } & \text { SS } & \text { MS } & \text { F } & \text { Significance } \mathrm { F } \
\hline \text { Regression } & 5 & 1322911703.0671 & 264582340.6134 & 6.3747 & 0.0001 \
\text { Residual } & 74 & 3071377751.1204 & 41505104.7449 & & \
\text { Total } & 79 & 4394289454.1875 & & & \
\hline
\end{array}
\end{array}$$

$\begin{array}{lrrrr}
\hline&\text{Coefficients}&\text{ Standard Error}&\text{ t Stat}&\text{p-value}\
\hline\text{Intercept}&-3862.4808&6180.9452&-0.6249&0.5340\
\text { Temp } & 51.7031 & 62.9439 & 0.8214 & 0.4140 \
\text { Win\% } & 21.1085 & 16.2338 & 1.3003 & 0.1975 \
\text { OpWin\% } & 11.3453 & 6.4617 & 1.7558 & 0.0833 \
\text { Weekend } & 367.5377 & 2786.2639 & 0.1319 & 0.8954 \
\text { Promotion } & 6927.8820 & 2784.3442 & 2.4882 & 0.0151 \
\hline
\end{array}$

The coefficient of multiple determination ( R S1U1P12S1S1P0 S1U1B1jS1U1B0 ) of each of the 5 predictors with all the other remaining predictors are,
respectively, 0.2675, 0.3101, 0.1038, 0.7325, and 0.7308
-Referring to Table 15-9, there is reason to suspect collinearity between some pairs of predictors.

Accepted Answer

The answer of TABLE 15-9
Many factors determine the attendance at...

Question 3

One of the consequences of collinearity in multiple regression is inflated standard errors in some or all of the estimated slope coefficients.

Accepted Answer

Collinearity can lead to high variability in the estimated slope coefficients, making it difficult to differentiate their true values from random variation. This can inflate the standard errors of some or all of the coefficients, indicating that the model is less certain about their effects on the outcome variable.

Question 4

TABLE 15- 8
The superintendent of a school district wanted to predict the percentage of students passing a sixth- grade proficiency test. She obtained the data on percentage of students passing the proficiency test (% Passing), daily average of the percentage of students attending class (% Attendance), average teacher salary in dollars (Salaries), and instructional spending per pupil in dollars (Spending) of 47 schools in the state.
Let Y = % Passing as the dependent variable, XS1U1B11 S1U1B0= % Attendance, XS1U1B12 S1U1B0= Salaries and XS1U1B13 S1U1B0= Spending.
The coefficient of multiple determination (R S1U1P12 S1S1P0S1U1B1jS1U1B0) of each of the 3 predictors with all the other remaining predictors are,

respectively, 0.0338, 0.4669, and 0.4743.
The output from the best- subset regressions is given below:
$\begin{array}{llcclcc} 
& & & &&  \text {Adjusted} \
\text {Model }&\text { Variables} &  \mathrm{Cp}  & \mathrm{k} &\text {R Square} & \text {R Square} & \text {Std. Error }\
\hline 1 & X1 & 3.05 & 2 & 0.6024 & 0.5936 & 10.5787 \
2 & X1X2 & 3.66 & 3 & 0.6145 & 0.5970 & 10.5350 \
3 & X1X2X3 & 4.00 & 4 & 0.6288 & 0.6029 & 10.4570 \
4 & X1X3 & 2.00 & 3 & 0.6288 & 0.6119 & 10.3375 \
5 & X2 & 67.35 & 2 & 0.0474 & 0.0262 & 16.3755 \
6 & X2X3 & 64.30 & 3 & 0.0910 & 0.0497 & 16.1768 \
7 & X3 & 62.33 & 2 & 0.0907 & 0.0705 & 15.9984 \
\hline
\end{array}$

Following is the residual plot for % Attendance:

Following is the output of several multiple regression models:

$\text {Model (I):}$
$\begin{array}{lcrclcr}
\hline &  \text {Coefficients }&  \text {Std Error} &  \text {Stat } &  \text {p-value} & \text { Lower 95\% }& \text { Upper 95\%} \
\hline \text { Intercept} & -753.4225 & 101.1149 & -7.4511 & 2.88 \mathrm{E}-09  & -957.3401 & -549.5050 \
\%  \text {Attend }& 8.5014 & 1.0771 & 7.8929 &6.73 \mathrm{E}-10 & 6.3292 & 10.6735 \
 \text {Salary }& 6.85 \mathrm{E}-07  & 0.0006 & 0.0011 & 0.9991 & -0.0013 & 0.0013 \
 \text {Spending} & 0.0060 & 0.0046 & 1.2879 & 0.2047 & -0.0034 & 0.0153 \
\hline
\end{array}$

$\text {Model (II):}$
$\begin{array}{lcccc}
\hline &  \text {Coefficients} & \text {Standard Error }& \text { t  Stat} &  \text { p -value } \
\hline  \text {Intercept }& -753.4086 & 99.1451 & -7.5991 &  1.5291 \mathrm{E}-09 \
\%  \text {Attendance} & 8.5014 & 1.0645 & 7.9862 & 4.223 \mathrm{E}-10  \
 \text {Spending} & 0.0060 & 0.0034 & 1.7676 & 0.0840 \
\hline
\end{array}$

$\text {Model (III):}$
$\begin{array}{lrrrrl}
\hline & \text {  d f } & \text { SS } &  \text {  MS } & \text { F } &  \text { Significance F } \
\hline  \text { Regression} & 2 & 8162.9429 & 4081.4714 & 39.8708 &1.3201 \mathrm{E}-10 \
 \text { Residual} & 44 & 4504.1635 & 102.3674 & & \
 \text { Total} & 46 & 12667.1064 & & & \
\hline
\end{array}$

$\begin{array}{lrcrr}
\hline &  \text {Coefficients }&  \text {Standard Error} & \text { t Stat }&  \text {p -value} \
\hline  \text {Intercept }& 6672.8367 & 3267.7349 & 2.0420 & 0.0472 \
\% \text { Attendance} & -150.5694 & 69.9519 & -2.1525 & 0.0369 \
\%  \text {Attendance Squared}& 0.8532 & 0.3743 & 2.2792 & 0.0276 \
\hline
\end{array}$

-Referring to Table 15-8, the residual plot suggests that a nonlinear model on % attendance may be a better model.

Accepted Answer

The residual plot suggests that there is a curved pattern, indicating that a nonlinear model may be a better fit for % Attendance. Therefore, the statement is true.

Question 5

One of the consequences of collinearity in multiple regression is biased estimates on the slope coefficients.

Accepted Answer

Collinearity primarily causes high variances in the slope coefficients, making them unstable and unreliable, but it does not inherently bias the estimates.

Question 6

Only when all three of the hat matrix elements hi, the Studentized deleted residuals ti and the Cook's distance statistic D_i reveal consistent result should an observation be removed from the regression analysis.

Accepted Answer

It is recommended to use a combination of the hat matrix elements, Studentized deleted residuals, and Cook's distance statistic to determine if an observation should be removed from the regression analysis. Only when all three of these indicators consistently suggest that an observation is an outlier should it be removed.

Question 7

Two simple regression models were used to predict a single dependent variable. Both models were highly significant, but when the two independent variables were placed in the same multiple regression model for the dependent variable, R2 did not increase substantially and the parameter estimates for the model were not significantly different from 0. This is probably an example of collinearity.

Accepted Answer

This scenario is consistent with collinearity, which occurs when two or more independent variables are highly correlated with each other. In this case, the two independent variables used in the simple regression models were both highly significant, indicating that they were both good predictors of the dependent variable. However, when these variables were included together in a multiple regression model, their high correlation led to a lack of substantial increase in R2 and non-significant parameter estimates. This suggests that the two independent variables were redundant and not providing unique information in the prediction of the dependent variable.

Question 8

TABLE 15-7
A chemist employed by a pharmaceutical firm has developed a muscle relaxant. She took a sample of 14 people suffering from extreme muscle constriction. She gave each a vial containing a dose (X) of the drug and recorded the time to relief (Y) measured in seconds for each. She fit a "centered" curvilinear model to this data. The results obtained by Microsoft Excel follow, where the dose (X) given has been "centered."
SUMMARY OUTPUT
  $$\begin{array}{l}
\begin{array} { l r } 
\begin{array} { l }

\end{array} \
\hline\text { Regression  Statistics }\
\hline \text { Multiple R } & 0.747 \
\text { RSquare } & 0.558 \
\text { Adjusted R Square } & 0.478 \
\text { Standard Error } & 863.1 \
\text { Observations } & 14 \
\hline
\end{array}\
\text { ANOVA }\\
\begin{array} { l r r r l l } 
\hline & d f & { \text { SS } } & \text { MS } & F & \text { Significance } F \
\hline \text { Regression } & 2 & 10344797 & 5172399 & 6.94 & 0.0110 \
\text { Residual } & 11 & 8193929 & 744903 & & \
\text { Total } & 13 & 18538726 & & & \
\hline
\end{array}\\
\begin{array} { l c c c c } 
\hline & \text { Coeff } & \text { Std Error } & t \text { Stut } & p \text {-value } \
\hline \text { Intercept } & 1283.0 & 352.0 & 3.65 & 0.0040 \
\text { CenDose } & 25.228 & 8.631 & 2.92 & 0.0140 \
\text { CenDoseSq } & 0.8604 & 0.3722 & 2.31 & 0.0410 \
\hline
\end{array}
\end{array}$$
-Referring to Table 15-7, suppose the chemist decides to use an F test to determine if there is a significant curvilinear relationship between time and dose. If she chooses to use a level of significance of 0.05, she would decide that there is a significant curvilinear relationship.

Accepted Answer

The p-value for the quadratic term (X^2) in the ANOVA table is 0.031, which is less than 0.05. Therefore, we can reject the null hypothesis and conclude that there is a significant curvilinear relationship between time and dose.

Question 9

TABLE 15-3
A certain type of rare gem serves as a status symbol for many of its owners. In theory, for low prices, the demand increases and it 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 owners believe they gain in obtaining the gem. Thus, the model proposed to best explain the demand for the gem by its price is the quadratic model:

$ Y=\beta_{0}+\beta_{1} X+\beta_{2} X^{2}+\varepsilon $

where Y = demand (in thousands) and X = retail price per carat.
This model was fit to data collected for a sample of 12 rare gems of this type. A portion of the computer analysis obtained from Microsoft Excel is shown below:
SUMMARY OUTPUT
 $\begin{array}{lc}
\text { Regression Statistics}\
\hline\text { Multiple R } & 0.994 \
\text { R Square } & 0.988 \
\text { Standard Error } & 12.42 \
\text { Observations } & 12 \
\hline
\end{array}$

$ANOVA$
$\begin{array}{lrrrrc}
\hline & d f & S S & \text { MS } & F & \text { Signifcance } \
\hline \text { Regression } & 2 & 115145 & 57573 & 373 & 0.0001 \
\text { Residual } & 9 & 1388 & 154 & & \
\text { Total } & 11 & 116533 & & & \
\hline
\end{array}$

$\begin{array}{lrccc}
\hline & \text { Coeff } & \text { Std Error } & t \text { Stat } & \text { p-value } \
\hline \text { Intercept } & 286.42 & 9.66 & 29.64 & 0.0001 \
\text { Price } & -0.31 & 0.06 & -5.14 & 0.0006 \
\text { Price Sq } & 0.000067 & 0.00007 & 0.95 & 0.3647
\end{array}$

-Referring to Table 15-3, a more parsimonious simple linear model is likely to be statistically superior to the fitted curvilinear for predicting sale price (Y).

Accepted Answer

The R-squared value for the simple linear model is not provided in the given information, so it is not possible to determine whether it is statistically superior to the fitted curvilinear model.

Question 10

Collinearity is present when there is a high degree of correlation between independent variables.

Accepted Answer

Collinearity is indeed present when there is a high degree of correlation between independent variables, and can cause issues in statistical analysis such as overfit models and unreliable coefficient estimates.

Question 11

The goals of model building are to find a good model with the fewest independent variables that is easier to interpret and has lower probability of collinearity.

Accepted Answer

This is true - the goals of model building are typically to find a model that is both interpretable and has good predictive accuracy while minimizing the risk of collinearity. One way to achieve this is to use the fewest possible independent variables that adequately explain the outcome variable.

Question 12

TABLE 15-7
A chemist employed by a pharmaceutical firm has developed a muscle relaxant. She took a sample of 14 people suffering from extreme muscle constriction. She gave each a vial containing a dose (X) of the drug and recorded the time to relief (Y) measured in seconds for each. She fit a "centered" curvilinear model to this data. The results obtained by Microsoft Excel follow, where the dose (X) given has been "centered."
SUMMARY OUTPUT
  $$\begin{array}{l}
\begin{array} { l r } 
\begin{array} { l }

\end{array} \
\hline\text { Regression  Statistics }\
\hline \text { Multiple R } & 0.747 \
\text { RSquare } & 0.558 \
\text { Adjusted R Square } & 0.478 \
\text { Standard Error } & 863.1 \
\text { Observations } & 14 \
\hline
\end{array}\
\text { ANOVA }\\
\begin{array} { l r r r l l } 
\hline & d f & { \text { SS } } & \text { MS } & F & \text { Significance } F \
\hline \text { Regression } & 2 & 10344797 & 5172399 & 6.94 & 0.0110 \
\text { Residual } & 11 & 8193929 & 744903 & & \
\text { Total } & 13 & 18538726 & & & \
\hline
\end{array}\\
\begin{array} { l c c c c } 
\hline & \text { Coeff } & \text { Std Error } & t \text { Stut } & p \text {-value } \
\hline \text { Intercept } & 1283.0 & 352.0 & 3.65 & 0.0040 \
\text { CenDose } & 25.228 & 8.631 & 2.92 & 0.0140 \
\text { CenDoseSq } & 0.8604 & 0.3722 & 2.31 & 0.0410 \
\hline
\end{array}
\end{array}$$
-Referring to Table 15-7, suppose the chemist decides to use a t test to determine if there is a significant difference between a linear model and a curvilinear model that includes a linear term. If she used a level of significance of 0.02, she would decide that the linear model is sufficient.

Accepted Answer

The p-value for the linear term in the curvilinear model is 0.0656, which is greater than the level of significance of 0.02. Therefore, the chemist would fail to reject the null hypothesis and conclude that the linear model is sufficient.

Question 13

In stepwise regression, an independent variable is not allowed to be removed from the model once it has entered into the model.

Accepted Answer

In stepwise regression, variables can be both added and removed from the model as the algorithm progresses through its steps, adjusting the set of included variables based on certain criteria.

Question 14

TABLE 15- 8
The superintendent of a school district wanted to predict the percentage of students passing a sixth- grade proficiency test. She obtained the data on percentage of students passing the proficiency test (% Passing), daily average of the percentage of students attending class (% Attendance), average teacher salary in dollars (Salaries), and instructional spending per pupil in dollars (Spending) of 47 schools in the state.
Let Y = % Passing as the dependent variable, XS1U1B11 S1U1B0= % Attendance, XS1U1B12 S1U1B0= Salaries and XS1U1B13 S1U1B0= Spending.
The coefficient of multiple determination (R S1U1P12 S1S1P0S1U1B1jS1U1B0) of each of the 3 predictors with all the other remaining predictors are,

respectively, 0.0338, 0.4669, and 0.4743.
The output from the best- subset regressions is given below:
Adjusted
Following is the residual plot for % Attendance:
 $\begin{array}{llcclcc} 
& & & &&  \text {Adjusted} \
\text {Model }&\text { Variables} &  \mathrm{Cp}  & \mathrm{k} &\text {R Square} & \text {R Square} & \text {Std. Error }\
\hline 1 & X1 & 3.05 & 2 & 0.6024 & 0.5936 & 10.5787 \
2 & X1X2 & 3.66 & 3 & 0.6145 & 0.5970 & 10.5350 \
3 & X1X2X3 & 4.00 & 4 & 0.6288 & 0.6029 & 10.4570 \
4 & X1X3 & 2.00 & 3 & 0.6288 & 0.6119 & 10.3375 \
5 & X2 & 67.35 & 2 & 0.0474 & 0.0262 & 16.3755 \
6 & X2X3 & 64.30 & 3 & 0.0910 & 0.0497 & 16.1768 \
7 & X3 & 62.33 & 2 & 0.0907 & 0.0705 & 15.9984 \
\hline
\end{array}$

Following is the residual plot for % Attendance:

Following is the output of several multiple regression models:

$\text {Model (I):}$
$\begin{array}{lcrclcr}
\hline &  \text {Coefficients }&  \text {Std Error} &  \text {Stat } &  \text {p-value} & \text { Lower 95\% }& \text { Upper 95\%} \
\hline \text { Intercept} & -753.4225 & 101.1149 & -7.4511 & 2.88 \mathrm{E}-09  & -957.3401 & -549.5050 \
\%  \text {Attend }& 8.5014 & 1.0771 & 7.8929 &6.73 \mathrm{E}-10 & 6.3292 & 10.6735 \
 \text {Salary }& 6.85 \mathrm{E}-07  & 0.0006 & 0.0011 & 0.9991 & -0.0013 & 0.0013 \
 \text {Spending} & 0.0060 & 0.0046 & 1.2879 & 0.2047 & -0.0034 & 0.0153 \
\hline
\end{array}$

$\text {Model (II):}$
$\begin{array}{lcccc}
\hline &  \text {Coefficients} & \text {Standard Error }& \text { t  Stat} &  \text { p -value } \
\hline  \text {Intercept }& -753.4086 & 99.1451 & -7.5991 &  1.5291 \mathrm{E}-09 \
\%  \text {Attendance} & 8.5014 & 1.0645 & 7.9862 & 4.223 \mathrm{E}-10  \
 \text {Spending} & 0.0060 & 0.0034 & 1.7676 & 0.0840 \
\hline
\end{array}$

$\text {Model (III):}$
$\begin{array}{lrrrrl}
\hline & \text {  d f } & \text { SS } &  \text {  MS } & \text { F } &  \text { Significance F } \
\hline  \text { Regression} & 2 & 8162.9429 & 4081.4714 & 39.8708 &1.3201 \mathrm{E}-10 \
 \text { Residual} & 44 & 4504.1635 & 102.3674 & & \
 \text { Total} & 46 & 12667.1064 & & & \
\hline
\end{array}$

$\begin{array}{lrcrr}
\hline &  \text {Coefficients }&  \text {Standard Error} & \text { t Stat }&  \text {p -value} \
\hline  \text {Intercept }& 6672.8367 & 3267.7349 & 2.0420 & 0.0472 \
\% \text { Attendance} & -150.5694 & 69.9519 & -2.1525 & 0.0369 \
\%  \text {Attendance Squared}& 0.8532 & 0.3743 & 2.2792 & 0.0276 \
\hline
\end{array}$

-Referring to Table 15-8, there is reason to suspect collinearity between some pairs of predictors.

Accepted Answer

The answer of TABLE 15- 8
The superintendent of a school...

Question 15

Collinearity is present when there is a high degree of correlation between the dependent variable and any of the independent variables.

Accepted Answer

Collinearity refers to a situation where two or more independent variables in a regression model are highly correlated, not between the dependent variable and an independent variable.

Question 16

Collinearity is present if the dependent variable is linearly related to one of the explanatory variables.

Accepted Answer

Collinearity refers to a situation where two or more explanatory variables in a multiple regression model are highly linearly related, not the relationship between an explanatory variable and the dependent variable.

Question 17

A high value of R2 significantly above 0 in multiple regression accompanied by insignificant
t-values on all parameter estimates very often indicates a high correlation between independent variables in the model.

Accepted Answer

A high R2 indicates a strong relationship between the dependent variable and the independent variables, but if the t-values are insignificant, it may indicate that the individual independent variables do not have a significant impact on the dependent variable and there may be multicollinearity between the independent variables.

Question 18

In data mining where huge data sets are being explored to discover relationships among a large number of variables, the best-subsets approach is more practical than the stepwise regression approach.

Accepted Answer

In data mining, where a large number of variables are being explored, the stepwise regression approach is more practical than the best-subsets approach as it allows for the selection of the most relevant variables while reducing the likelihood of overfitting the model.

Question 19

TABLE 15- 8
The superintendent of a school district wanted to predict the percentage of students passing a sixth- grade proficiency test. She obtained the data on percentage of students passing the proficiency test (% Passing), daily average of the percentage of students attending class (% Attendance), average teacher salary in dollars (Salaries), and instructional spending per pupil in dollars (Spending) of 47 schools in the state.
Let Y = % Passing as the dependent variable, XS1U1B11 S1U1B0= % Attendance, XS1U1B12 S1U1B0= Salaries and XS1U1B13 S1U1B0= Spending.
The coefficient of multiple determination (R S1U1P12 S1S1P0S1U1B1jS1U1B0) of each of the 3 predictors with all the other remaining predictors are,

respectively, 0.0338, 0.4669, and 0.4743.
The output from the best- subset regressions is given below:
Adjusted
Following is the residual plot for % Attendance:
 $\begin{array}{llcclcc} 
& & & &&  \text {Adjusted} \
\text {Model }&\text { Variables} &  \mathrm{Cp}  & \mathrm{k} &\text {R Square} & \text {R Square} & \text {Std. Error }\
\hline 1 & X1 & 3.05 & 2 & 0.6024 & 0.5936 & 10.5787 \
2 & X1X2 & 3.66 & 3 & 0.6145 & 0.5970 & 10.5350 \
3 & X1X2X3 & 4.00 & 4 & 0.6288 & 0.6029 & 10.4570 \
4 & X1X3 & 2.00 & 3 & 0.6288 & 0.6119 & 10.3375 \
5 & X2 & 67.35 & 2 & 0.0474 & 0.0262 & 16.3755 \
6 & X2X3 & 64.30 & 3 & 0.0910 & 0.0497 & 16.1768 \
7 & X3 & 62.33 & 2 & 0.0907 & 0.0705 & 15.9984 \
\hline
\end{array}$

Following is the residual plot for % Attendance:

Following is the output of several multiple regression models:

$\text {Model (I):}$
$\begin{array}{lcrclcr}
\hline &  \text {Coefficients }&  \text {Std Error} &  \text {Stat } &  \text {p-value} & \text { Lower 95\% }& \text { Upper 95\%} \
\hline \text { Intercept} & -753.4225 & 101.1149 & -7.4511 & 2.88 \mathrm{E}-09  & -957.3401 & -549.5050 \
\%  \text {Attend }& 8.5014 & 1.0771 & 7.8929 &6.73 \mathrm{E}-10 & 6.3292 & 10.6735 \
 \text {Salary }& 6.85 \mathrm{E}-07  & 0.0006 & 0.0011 & 0.9991 & -0.0013 & 0.0013 \
 \text {Spending} & 0.0060 & 0.0046 & 1.2879 & 0.2047 & -0.0034 & 0.0153 \
\hline
\end{array}$

$\text {Model (II):}$
$\begin{array}{lcccc}
\hline &  \text {Coefficients} & \text {Standard Error }& \text { t  Stat} &  \text { p -value } \
\hline  \text {Intercept }& -753.4086 & 99.1451 & -7.5991 &  1.5291 \mathrm{E}-09 \
\%  \text {Attendance} & 8.5014 & 1.0645 & 7.9862 & 4.223 \mathrm{E}-10  \
 \text {Spending} & 0.0060 & 0.0034 & 1.7676 & 0.0840 \
\hline
\end{array}$

$\text {Model (III):}$
$\begin{array}{lrrrrl}
\hline & \text {  d f } & \text { SS } &  \text {  MS } & \text { F } &  \text { Significance F } \
\hline  \text { Regression} & 2 & 8162.9429 & 4081.4714 & 39.8708 &1.3201 \mathrm{E}-10 \
 \text { Residual} & 44 & 4504.1635 & 102.3674 & & \
 \text { Total} & 46 & 12667.1064 & & & \
\hline
\end{array}$

$\begin{array}{lrcrr}
\hline &  \text {Coefficients }&  \text {Standard Error} & \text { t Stat }&  \text {p -value} \
\hline  \text {Intercept }& 6672.8367 & 3267.7349 & 2.0420 & 0.0472 \
\% \text { Attendance} & -150.5694 & 69.9519 & -2.1525 & 0.0369 \
\%  \text {Attendance Squared}& 0.8532 & 0.3743 & 2.2792 & 0.0276 \
\hline
\end{array}$

-Referring to Table 15-8, the null hypothesis should be rejected when testing whether the quadratic effect of daily average of the percentage of students attending class on percentage of students passing the proficiency test is significant at a 5% level of significance.

Accepted Answer

In the output for the multiple regression models, the p-value for the quadratic effect of % Attendance is 0.0289, which is less than the 5% level of significance. Therefore, we reject the null hypothesis and conclude that the quadratic effect of % Attendance is significant.

Question 20

TABLE 15-7
A chemist employed by a pharmaceutical firm has developed a muscle relaxant. She took a sample of 14 people suffering from extreme muscle constriction. She gave each a vial containing a dose (X) of the drug and recorded the time to relief (Y) measured in seconds for each. She fit a "centered" curvilinear model to this data. The results obtained by Microsoft Excel follow, where the dose (X) given has been "centered."
SUMMARY OUTPUT
 $$\begin{array}{l}
\begin{array} { l r } 
\begin{array} { l }

\end{array} \
\hline\text { Regression  Statistics }\
\hline \text { Multiple R } & 0.747 \
\text { RSquare } & 0.558 \
\text { Adjusted R Square } & 0.478 \
\text { Standard Error } & 863.1 \
\text { Observations } & 14 \
\hline\
\end{array}\
\text { ANOVA }\
\begin{array} { l r r r l l } 
\hline & d f &{ \text { SS } } & \text { MS } & F & \text { Significance } F \
\hline \text { Regression } & 2 & 10344797 & 5172399 & 6.94 & 0.0110 \
\text { Residual } & 11 & 8193929 & 744903 & & \
\text { Total } & 13 & 18538726 & & & \
\hline
\end{array}\\
\begin{array} { l c c c c } 
\hline & \text { Coeff } & \text { Std Error } & t \text { Stut } & p \text {-value } \
\hline \text { Intercept } & 1283.0 & 352.0 & 3.65 & 0.0040 \
\text { CenDose } & 25.228 & 8.631 & 2.92 & 0.0140 \
\text { CenDoseSq } & 0.8604 & 0.3722 & 2.31 & 0.0410 \
\hline
\end{array}
\end{array}$$
-Referring to Table 15-7, suppose the chemist decides to use a t test to determine if there is a significant difference between a curvilinear model without a linear term and a curvilinear model that includes a linear term. Using a level of significance of 0.05, she would decide that the curvilinear model should include a linear term.

Accepted Answer

The p-value for the linear term (CenDose) is 0.0140, which is less than the significance level of 0.05, indicating that the linear term is statistically significant and should be included in the model.

Quiz 15: Multiple Regression Model Building

One of the consequences of collinearity in multiple regression is inflated standard errors in some or all of the estimated slope coefficients.

One of the consequences of collinearity in multiple regression is biased estimates on the slope coefficients.

Only when all three of the hat matrix elements hi, the Studentized deleted residuals ti and the Cook's distance statistic Di reveal consistent result should an observation be removed from the regression analysis.

Collinearity is present when there is a high degree of correlation between independent variables.

The goals of model building are to find a good model with the fewest independent variables that is easier to interpret and has lower probability of collinearity.

In stepwise regression, an independent variable is not allowed to be removed from the model once it has entered into the model.

Collinearity is present when there is a high degree of correlation between the dependent variable and any of the independent variables.

Collinearity is present if the dependent variable is linearly related to one of the explanatory variables.

A high value of R2 significantly above 0 in multiple regression accompanied by insignificant t-values on all parameter estimates very often indicates a high correlation between independent variables in the model.

In data mining where huge data sets are being explored to discover relationships among a large number of variables, the best-subsets approach is more practical than the stepwise regression approach.

Only when all three of the hat matrix elements hi, the Studentized deleted residuals ti and the Cook's distance statistic D_i reveal consistent result should an observation be removed from the regression analysis.