Question 1

TABLE 15-3
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
Regression Statistics
$\begin{array} { l r } \text { Multiple R } & 0.747 \ \text { R Square } & 0.558 \ \text { Adjusted R Square } & 0.478 \ \text { Standard Error } & 863.1 \ \text { Observations } & 14 \end{array}$
ANOVA
$\begin{array} { l r r r r r } & d f & \text { SS } & \text { MS } & F & \text { Signif F } \ \text { Regression } & 2 & 10344797 & 5172399 & 6.94 & 0.0110 \ \text { Residual } & 11 & 8193929 & 744903 & & \ \text { Total } & 13 & 18538726 & & & \end{array}$
$\begin{array} { l c c c c } & \text { Coeff } & \text { StdFrror } & t \text { Stat } & p \text {-value } \ \text { Intercept } & 1283.0 & 352.0 & 3.65 & 0.0040 \ \text { CenDose } & 25.228 & 8.631 & 2.92 & 0.0140 \ \text { CenDoseSa } & 0.8604 & 0.3722 & 2.31 & 0.0410 \end{array}$
-Referring to Table 15-3, 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 ANOVA table shows that the regression is significant with an F value of 6.94 and a significance level of 0.011, which is less than the chosen level of significance of 0.05. This indicates that there is a significant relationship between time and dose, including a curvilinear component.

Question 2

Collinearity will result in excessively low standard errors of the parameter estimates reported in the regression output.

Accepted Answer

Collinearity typically results in excessively high standard errors of the parameter estimates, making it difficult to determine the individual effect of collinear variables on the dependent variable.

Question 3

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 4

TABLE 15-3
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
Regression Statistics
$\begin{array} { l r } \text { Multiple R } & 0.747 \ \text { R Square } & 0.558 \ \text { Adjusted R Square } & 0.478 \ \text { Standard Error } & 863.1 \ \text { Observations } & 14 \end{array}$
ANOVA
$\begin{array} { l r r r r r } & d f & \text { SS } & \text { MS } & F & \text { Signif F } \ \text { Regression } & 2 & 10344797 & 5172399 & 6.94 & 0.0110 \ \text { Residual } & 11 & 8193929 & 744903 & & \ \text { Total } & 13 & 18538726 & & & \end{array}$
$\begin{array} { l c c c c } & \text { Coeff } & \text { StdFrror } & t \text { Stat } & p \text {-value } \ \text { Intercept } & 1283.0 & 352.0 & 3.65 & 0.0040 \ \text { CenDose } & 25.228 & 8.631 & 2.92 & 0.0140 \ \text { CenDoseSa } & 0.8604 & 0.3722 & 2.31 & 0.0410 \end{array}$
-Referring to Table 15-3, 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. The p-value of the test statistic for the contribution of the curvilinear term is ________.

Accepted Answer

The answer of TABLE 15-3
A chemist employed by a pharmaceutical...

Question 5

TABLE 15-3
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
Regression Statistics
$\begin{array} { l r } \text { Multiple R } & 0.747 \ \text { R Square } & 0.558 \ \text { Adjusted R Square } & 0.478 \ \text { Standard Error } & 863.1 \ \text { Observations } & 14 \end{array}$
ANOVA
$\begin{array} { l r r r r r } & d f & \text { SS } & \text { MS } & F & \text { Signif F } \ \text { Regression } & 2 & 10344797 & 5172399 & 6.94 & 0.0110 \ \text { Residual } & 11 & 8193929 & 744903 & & \ \text { Total } & 13 & 18538726 & & & \end{array}$
$\begin{array} { l c c c c } & \text { Coeff } & \text { StdFrror } & t \text { Stat } & p \text {-value } \ \text { Intercept } & 1283.0 & 352.0 & 3.65 & 0.0040 \ \text { CenDose } & 25.228 & 8.631 & 2.92 & 0.0140 \ \text { CenDoseSa } & 0.8604 & 0.3722 & 2.31 & 0.0410 \end{array}$
-Referring to Table 15-3, suppose the chemist decides to use a t test to determine if the linear term is significant. The value of the test statistic is ________.

Accepted Answer

The answer of TABLE 15-3
A chemist employed by a pharmaceutical...

Question 6

TABLE 15-3
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
Regression Statistics
$\begin{array} { l r } \text { Multiple R } & 0.747 \ \text { R Square } & 0.558 \ \text { Adjusted R Square } & 0.478 \ \text { Standard Error } & 863.1 \ \text { Observations } & 14 \end{array}$
ANOVA
$\begin{array} { l r r r r r } & d f & \text { SS } & \text { MS } & F & \text { Signif F } \ \text { Regression } & 2 & 10344797 & 5172399 & 6.94 & 0.0110 \ \text { Residual } & 11 & 8193929 & 744903 & & \ \text { Total } & 13 & 18538726 & & & \end{array}$
$\begin{array} { l c c c c } & \text { Coeff } & \text { StdFrror } & t \text { Stat } & p \text {-value } \ \text { Intercept } & 1283.0 & 352.0 & 3.65 & 0.0040 \ \text { CenDose } & 25.228 & 8.631 & 2.92 & 0.0140 \ \text { CenDoseSa } & 0.8604 & 0.3722 & 2.31 & 0.0410 \end{array}$
-Referring to Table 15-3, suppose the chemist decides to use an F test to determine if there is a significant curvilinear relationship between time and dose. The p-value of the test is ________.

Accepted Answer

The answer of TABLE 15-3
A chemist employed by a pharmaceutical...

Question 7

TABLE 15-3
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
Regression Statistics
$\begin{array} { l r } \text { Multiple R } & 0.747 \ \text { R Square } & 0.558 \ \text { Adjusted R Square } & 0.478 \ \text { Standard Error } & 863.1 \ \text { Observations } & 14 \end{array}$
ANOVA
$\begin{array} { l r r r r r } & d f & \text { SS } & \text { MS } & F & \text { Signif F } \ \text { Regression } & 2 & 10344797 & 5172399 & 6.94 & 0.0110 \ \text { Residual } & 11 & 8193929 & 744903 & & \ \text { Total } & 13 & 18538726 & & & \end{array}$
$\begin{array} { l c c c c } & \text { Coeff } & \text { StdFrror } & t \text { Stat } & p \text {-value } \ \text { Intercept } & 1283.0 & 352.0 & 3.65 & 0.0040 \ \text { CenDose } & 25.228 & 8.631 & 2.92 & 0.0140 \ \text { CenDoseSa } & 0.8604 & 0.3722 & 2.31 & 0.0410 \end{array}$
-Referring to Table 15-3, suppose the chemist decides to use an F test to determine if there is a significant curvilinear relationship between time and dose. The value of the test statistic is ________.

Accepted Answer

The answer of TABLE 15-3
A chemist employed by a pharmaceutical...

Question 8

TABLE 15-3
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
Regression Statistics
$\begin{array} { l r } \text { Multiple R } & 0.747 \ \text { R Square } & 0.558 \ \text { Adjusted R Square } & 0.478 \ \text { Standard Error } & 863.1 \ \text { Observations } & 14 \end{array}$
ANOVA
$\begin{array} { l r r r r r } & d f & \text { SS } & \text { MS } & F & \text { Signif F } \ \text { Regression } & 2 & 10344797 & 5172399 & 6.94 & 0.0110 \ \text { Residual } & 11 & 8193929 & 744903 & & \ \text { Total } & 13 & 18538726 & & & \end{array}$
$\begin{array} { l c c c c } & \text { Coeff } & \text { StdFrror } & t \text { Stat } & p \text {-value } \ \text { Intercept } & 1283.0 & 352.0 & 3.65 & 0.0040 \ \text { CenDose } & 25.228 & 8.631 & 2.92 & 0.0140 \ \text { CenDoseSa } & 0.8604 & 0.3722 & 2.31 & 0.0410 \end{array}$
-Referring to Table 15-3, the prediction of time to relief for a person receiving a dose of the drug 10 units above the mean dose (i.e., the prediction of Y for X = 10)is ________.

Accepted Answer

The answer of TABLE 15-3
A chemist employed by a pharmaceutical...

Question 9

In multiple regression, the ________ procedure permits variables to enter and leave the model at different stages of its development.

Accepted Answer

The answer of In multiple regression, the ________ procedure permits...

Question 10

The parameter estimates are biased when collinearity is present in a multiple regression equation.

Accepted Answer

The answer of The parameter estimates are biased when collinearity...

Question 11

So that we can fit curves as well as lines by regression, we often use mathematical manipulations for converting one variable into a different form. These manipulations are called dummy variables.

Accepted Answer

Dummy variables are used to encode categorical data into numeric form so that they can be included in regression models, not for converting one variable into a different form to fit curves. Mathematical manipulations for fitting curves often involve transforming the variables themselves, such as through polynomial regression or logarithmic transformation.

Question 12

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 13

TABLE 15-3
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
Regression Statistics
$\begin{array} { l r } \text { Multiple R } & 0.747 \ \text { R Square } & 0.558 \ \text { Adjusted R Square } & 0.478 \ \text { Standard Error } & 863.1 \ \text { Observations } & 14 \end{array}$
ANOVA
$\begin{array} { l r r r r r } & d f & \text { SS } & \text { MS } & F & \text { Signif F } \ \text { Regression } & 2 & 10344797 & 5172399 & 6.94 & 0.0110 \ \text { Residual } & 11 & 8193929 & 744903 & & \ \text { Total } & 13 & 18538726 & & & \end{array}$
$\begin{array} { l c c c c } & \text { Coeff } & \text { StdFrror } & t \text { Stat } & p \text {-value } \ \text { Intercept } & 1283.0 & 352.0 & 3.65 & 0.0040 \ \text { CenDose } & 25.228 & 8.631 & 2.92 & 0.0140 \ \text { CenDoseSa } & 0.8604 & 0.3722 & 2.31 & 0.0410 \end{array}$
-Referring to Table 15-3, 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 0.05, she would decide that the linear model is sufficient.

Accepted Answer

The answer of TABLE 15-3
A chemist employed by a pharmaceutical...

Question 14

TABLE 15-3
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
Regression Statistics
$\begin{array} { l r } \text { Multiple R } & 0.747 \ \text { R Square } & 0.558 \ \text { Adjusted R Square } & 0.478 \ \text { Standard Error } & 863.1 \ \text { Observations } & 14 \end{array}$
ANOVA
$\begin{array} { l r r r r r } & d f & \text { SS } & \text { MS } & F & \text { Signif F } \ \text { Regression } & 2 & 10344797 & 5172399 & 6.94 & 0.0110 \ \text { Residual } & 11 & 8193929 & 744903 & & \ \text { Total } & 13 & 18538726 & & & \end{array}$
$\begin{array} { l c c c c } & \text { Coeff } & \text { StdFrror } & t \text { Stat } & p \text {-value } \ \text { Intercept } & 1283.0 & 352.0 & 3.65 & 0.0040 \ \text { CenDose } & 25.228 & 8.631 & 2.92 & 0.0140 \ \text { CenDoseSa } & 0.8604 & 0.3722 & 2.31 & 0.0410 \end{array}$
-Referring to Table 15-3, suppose the chemist decides to use a t test to determine if the linear term is significant. Using a level of significance of 0.05, she would decide that the linear term is significant.

Accepted Answer

Since the p-value associated with the linear term (CenDose) is 0.0140 which is less than the level of significance (0.05), the chemist would decide that the linear term is significant. Therefore, the statement is true.

Question 15

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

The fact that the parameter estimates were not significantly different from 0 suggests that the independent variables are highly correlated with each other, indicating collinearity.

Question 16

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 cause the estimated coefficients to become unstable and their standard errors to increase, leading to difficulties in interpreting the model and performing hypothesis testing.

Question 17

TABLE 15-3
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
Regression Statistics
$\begin{array} { l r } \text { Multiple R } & 0.747 \ \text { R Square } & 0.558 \ \text { Adjusted R Square } & 0.478 \ \text { Standard Error } & 863.1 \ \text { Observations } & 14 \end{array}$
ANOVA
$\begin{array} { l r r r r r } & d f & \text { SS } & \text { MS } & F & \text { Signif F } \ \text { Regression } & 2 & 10344797 & 5172399 & 6.94 & 0.0110 \ \text { Residual } & 11 & 8193929 & 744903 & & \ \text { Total } & 13 & 18538726 & & & \end{array}$
$\begin{array} { l c c c c } & \text { Coeff } & \text { StdFrror } & t \text { Stat } & p \text {-value } \ \text { Intercept } & 1283.0 & 352.0 & 3.65 & 0.0040 \ \text { CenDose } & 25.228 & 8.631 & 2.92 & 0.0140 \ \text { CenDoseSa } & 0.8604 & 0.3722 & 2.31 & 0.0410 \end{array}$
-Referring to Table 15-3, suppose the chemist decides to use a t test to determine if the linear term is significant. The p-value of the test is ________.

Accepted Answer

The answer of TABLE 15-3
A chemist employed by a pharmaceutical...

Question 18

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 value of R² indicates that the model explains a large proportion of the variability in the dependent variable. However, when all parameter estimates have insignificant t-values, it may indicate that the independent variables are highly correlated and therefore, their individual effects cannot be estimated accurately. This is known as multicollinearity, which can lead to unreliable model results.

Question 19

TABLE 15-3
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
Regression Statistics
$\begin{array} { l r } \text { Multiple R } & 0.747 \ \text { R Square } & 0.558 \ \text { Adjusted R Square } & 0.478 \ \text { Standard Error } & 863.1 \ \text { Observations } & 14 \end{array}$
ANOVA
$\begin{array} { l r r r r r } & d f & \text { SS } & \text { MS } & F & \text { Signif F } \ \text { Regression } & 2 & 10344797 & 5172399 & 6.94 & 0.0110 \ \text { Residual } & 11 & 8193929 & 744903 & & \ \text { Total } & 13 & 18538726 & & & \end{array}$
$\begin{array} { l c c c c } & \text { Coeff } & \text { StdFrror } & t \text { Stat } & p \text {-value } \ \text { Intercept } & 1283.0 & 352.0 & 3.65 & 0.0040 \ \text { CenDose } & 25.228 & 8.631 & 2.92 & 0.0140 \ \text { CenDoseSa } & 0.8604 & 0.3722 & 2.31 & 0.0410 \end{array}$
-Referring to Table 15-3, 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 answer of TABLE 15-3
A chemist employed by a pharmaceutical...

Question 20

TABLE 15-3
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
Regression Statistics
$\begin{array} { l r } \text { Multiple R } & 0.747 \ \text { R Square } & 0.558 \ \text { Adjusted R Square } & 0.478 \ \text { Standard Error } & 863.1 \ \text { Observations } & 14 \end{array}$
ANOVA
$\begin{array} { l r r r r r } & d f & \text { SS } & \text { MS } & F & \text { Signif F } \ \text { Regression } & 2 & 10344797 & 5172399 & 6.94 & 0.0110 \ \text { Residual } & 11 & 8193929 & 744903 & & \ \text { Total } & 13 & 18538726 & & & \end{array}$
$\begin{array} { l c c c c } & \text { Coeff } & \text { StdFrror } & t \text { Stat } & p \text {-value } \ \text { Intercept } & 1283.0 & 352.0 & 3.65 & 0.0040 \ \text { CenDose } & 25.228 & 8.631 & 2.92 & 0.0140 \ \text { CenDoseSa } & 0.8604 & 0.3722 & 2.31 & 0.0410 \end{array}$
-Referring to Table 15-3, 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 0.01, she would decide that the linear model is sufficient.

Accepted Answer

The ANOVA table shows that the regression model (which includes both linear and curvilinear terms) has a significant F-statistic with a p-value of 0.0110. However, the t-test for the linear term (CenDose) has a p-value of 0.0140, which is not significant at the 0.01 level. Therefore, the chemist would conclude that the linear model is sufficient.

Quiz 15: Multiple Regression Model Building

Collinearity will result in excessively low standard errors of the parameter estimates reported in the regression output.

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

In multiple regression, the ________ procedure permits variables to enter and leave the model at different stages of its development.

The parameter estimates are biased when collinearity is present in a multiple regression equation.

So that we can fit curves as well as lines by regression, we often use mathematical manipulations for converting one variable into a different form. These manipulations are called dummy variables.

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

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

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.