Deck 18: Multiple Regression

Question 1

Interpret multiple regression coefficients.

-What affects flat panel LCD TV sales? Flat panel LCD TV's are sold through a
Variety of outlets. Sales figures (number of units) for the popular Sony Bravia were
Obtained for last quarter from a sample of 30 different stores. Also collected were data
On the selling price and amount spent on advertising the Sony Bravia (as a percentage of
Total advertising expenditure in the previous quarter) at each store. Based on the results
Shown below, the correct interpretation of the regression coefficient for Advertising is $$\begin{array}{l}\text { Dependent Variable is Sales }\\\begin{array} { l r r r r } \text { Predictor } & \text { Coef } & \text { SE Coef } & \text { T } & \text { P } \\\text { Constant } & 90.19 & 25.08 & 3.60 & 0.001 \\\text { Price } & - 0.03055 & 0.01005 & - 3.04 & 0.005 \\\text { Advertising } & 3.0926 & 0.3680 & 8.40 & 0.000\end{array}\end{array}$$

A) While price negatively affects the number of Sony Bravia LCD TV's sold, an increase in the amount of money spent on advertising will result in at least 3 additional TV's sold.
B) At a given price, increasing the amount spent on advertising the Sony Bravia over the previous quarter will increase sales by 3.0926 units, on average.
C) A one percent increase in the amount spent on advertising the Sony Bravia over the previous quarter will increase sales by 3.0926 units, on average.
D) At a given price, a one percent increase in the amount spent on advertising the Sony Bravia over the previous quarter is associated with an increase in sales of 3.0926 units, on
Average.
E) None of the above.

Answer 1

At a given price, a one percent increase in the amount spent on advertising the Sony Bravia over the previous quarter is associated with an increase in sales of 3.0926 units, on
Average.

Question 2

Interpret multiple regression coefficients.

-What affects flat panel LCD TV sales? Flat panel LCD TV's are sold through a
Variety of outlets. Sales figures (number of units) for the popular Sony Bravia were
Obtained for last quarter from a sample of 30 different stores. Also collected were data
On the selling price and amount spent on advertising the Sony Bravia (as a percentage of
Total advertising expenditure in the previous quarter) at each store. Based on the results
Shown below, the correct interpretation of the regression coefficient for Price is $$\begin{array}{l}\text { Dependent Variable is Sales }\\\begin{array} { l r r r r } \text { Predictor } & \text { Coef } & \text { SE Coef } & \text { T } & \text { P } \\\text { Constant } & 90.19 & 25.08 & 3.60 & 0.001 \\\text { Price } & - 0.03055 & 0.01005 & - 3.04 & 0.005 \\\text { Advertising } & 3.0926 & 0.3680 & 8.40 & 0.000\end{array}\end{array}$$

A) Increasing the price of the Sony Bravia by $100 will result in at least 3 fewer TV's sold.
B) For a given amount spent on advertising, a $100 increase in price of the Sony Bravia is associated with a decrease in sales of 3.055 units, on average.
C) Holding the amount spent on advertising constant, an increase of $100 in the price of the Sony Bravia will decrease sales by 3.055 units.
D) Holding the amount spent on advertising constant, an increase of $100 in the price of the Sony Bravia will decrease sales by .03%.
E) None of the above.

Answer 2

For a given amount spent on advertising, a $100 increase in price of the Sony Bravia is associated with a decrease in sales of 3.055 units, on average.

Question 3

Perform statistical inference for multiple regression.

-What affects flat panel LCD TV sales? Flat panel LCD TV's are sold through a
Variety of outlets. Sales figures (number of units) for the popular Sony Bravia were
Obtained for last quarter from a sample of 30 different stores. Also collected were data
On the selling price and amount spent on advertising the Sony Bravia (as a percentage of
Total advertising expenditure in the previous quarter) at each store. Output is shown
Below. The calculated F statistic to determine the overall significance of the estimated
Multiple regression model is $$\begin{array}{l}\text { Analysis of Variance }\\\begin{array} { l r r r } \text { Source } & \text { DF } & \text { SS } & \text { MS } \\\text { Regression } & 2 & 16477.3 & 8238.7 \\\text { Residual Error } & 27 & 3038.0 & 112.5 \\\text { Total } & 29 & 19515.4 &\end{array}\end{array}$$

A) 10.61
B) 73.23
C) 112.5
D) 3.60
E) None of the above

Answer 3

73.23

Question 4

Perform statistical inference for multiple regression.

-In determining the best companies to work for, a number of variables are considered,
Including size, average annual pay, and turnover rate, etc. Employee surveys are often
Conducted in order to assess aspects of the organization's culture, such as trust and
Openness to change. A sample of 33 companies was randomly selected and data
Collected on the average annual bonus and turnover rate (%). A questionnaire was also
Administered to the employees of each company to arrive at a trust index (measured on a
Scale of 0 - 100). Based on the output shown below, the correct interpretation of the
Regression coefficient associated with Average Bonus is Dependent Variable is Turnover Rate
$\begin{array} { l r r r r } \text { Predictor } & \text { Coef } & \text { SE Coef } & \text { T } & \text { P } \\ \text { Constant } & 12.1005 & 0.7826 & 15.46 & 0.000 \\ \text { Trust Index } & - 0.07149 & 0.01966 & - 3.64 & 0.001 \\ \text { Average Bonus } & - 0.0007216 & 0.0001481 & - 4.87 & 0.000 \end{array}$
$$S = 1.49746 \quad \mathrm { R } - \mathrm { Sq } = 79.6 \% \quad \mathrm { R } - \mathrm { Sq } ( \mathrm { adj } \mathrm { j } ) = 78.3 \%$$

A) Holding the trust index constant, increasing the average annual bonus by $100 will decrease the turnover rate by 7.15%.
B) For companies that have the same score on the trust index, increasing the average annual bonus by $100 is associated with a decrease in turnover rate of 7.15%, on average.
C) For companies that have the same score on the trust index, increasing the average annual bonus by $100 is associated with an increase in turnover rate of 7.15%, on
Average.
D) An increase of $100 in the average annual bonus decreases the turnover rate by 7.15%.
E) None of the above.

Question 5

8.3. Check assumptions and conditions for the multiple regression model.
What affects flat panel LCD TV sales? Flat panel LCD TV's are sold through a
Variety of outlets. Sales figures (number of units) for the popular Sony Bravia were
Obtained for last quarter from a sample of 30 different stores. Also collected were data
On the selling price and amount spent on advertising the Sony Bravia (as a percentage of
Total advertising expenditure in the previous quarter) at each store. A multiple regression
Model was fit to the data and the plot of residuals versus predicted values is shown
Below. What does the residual plot suggest?

A) The Linearity condition is not satisfied.
B) There is an extreme departure from normality.
C) The variance is not constant.
D) The presence of a couple of outliers.
E) The plot thickens from left to right.

Question 6

Interpret multiple regression output.

-What affects flat panel LCD TV sales? Flat panel LCD TV's are sold through a
Variety of outlets. Sales figures (number of units) for the popular Sony Bravia were
Obtained for last quarter from a sample of 30 different stores. Also collected were data
On the selling price and amount spent on advertising the Sony Bravia (as a percentage of
Total advertising expenditure in the previous quarter) at each store. Output is shown
Below. Based on the output shown below, how much of the variability in Sales is
Explained by the estimated multiple regression model? $$\begin{array}{l}\text { Analysis of Variance }\\\begin{array} { l r r r } \text { Source } & \text { DF } & \text { SS } & \text { MS } \\\text { Regression } & 2 & 16477.3 & 8238.7 \\\text { Residual Error } & 27 & 3038.0 & 112.5 \\\text { Total } & 29 & 19515.4 &\end{array}\end{array}$$

A) 15.57%
B) 6.90%
C) 84.43%
D) 29%
E) None of the above.

Question 7

Interpret multiple regression output.

-In determining the best companies to work for, a number of variables are considered,
Including size, average annual pay, and turnover rate, etc. Employee surveys are often
Conducted in order to assess aspects of the organization's culture, such as trust and
Openness to change. A sample of 33 companies was randomly selected and data
Collected on the average annual bonus and turnover rate (%). A questionnaire was also
Administered to the employees of each company to arrive at a trust index (measured on a
Scale of 0 - 100). Based on the output shown below, how much of the variability in
Turnover Rate is explained by the estimated multiple regression model? $\text {Dependent Variable is Turnover Rate}$
$\begin{array} { l r r r r } \text { Predictor } & \text { Coef } & \text { SE Coef } & \text { T } & \text { P } \\ \text { Constant } & 12.1005 & 0.7826 & 15.46 & 0.000 \\ \text { Trust Index } & - 0.07149 & 0.01966 & - 3.64 & 0.001 \\ \text { Average Bonus } & - 0.0007216 & 0.0001481 & - 4.87 & 0.000 \end{array}$
$$S = 1.49746 \quad \mathrm { R } - \mathrm { Sq } = 79.68 \quad \mathrm { R } - \mathrm { Sq } ( \mathrm { adj } ) = 78.3\%$$

A) 2.24%
B) 79.6%
C) 12.1%
D) 95.4%
E) None of the above.

Question 8

Perform statistical inference for multiple regression.
What affects flat panel LCD TV sales? Flat panel LCD TV's are sold through a
Variety of outlets. Sales figures (number of units) for the popular Sony Bravia were
Obtained for last quarter from a sample of 30 different stores. Also collected were data
On the selling price and amount spent on advertising the Sony Bravia (as a percentage of
Total advertising expenditure in the previous quarter) at each store. Output is shown
Below. The correct null and alternative hypotheses for testing the regression coefficient
Of Price is

Question 9

Perform statistical inference for multiple regression.

-What affects flat panel LCD TV sales? Flat panel LCD TV's are sold through a
Variety of outlets. Sales figures (number of units) for the popular Sony Bravia were
Obtained for last quarter from a sample of 30 different stores. Also collected were data
On the selling price and amount spent on advertising the Sony Bravia (as a percentage of
Total advertising expenditure in the previous quarter) at each store. Output is shown
Below. Which of the following statements is / are true? Dependent Variable is Sales
$\begin{array} { l r r r r } \text { Predictor } & \text { Coef } & \text { SE Coef } & \text { T } & \text { P } \\ \text { Constant } & 90.19 & 25.08 & 3.60 & 0.001 \\ \text { Price } & - 0.03055 & 0.01005 & - 3.04 & 0.005 \\ \text { Advertising } & 3.0926 & 0.3680 & 8.40 & 0.000 \end{array}$
$$S = 10.6075 \quad \mathrm { R } - \mathrm { Sq } = 84.48 \quad \mathrm { R } - \mathrm { Sq } ( \operatorname { adj } \mathrm { j } ) = 83.3\%$$
Analysis of Variance
$\begin{array} { l r r r } \text { Source } & \text { DF } & \text { SS } & \text { MS } \\ \text { Regression } & 2 & 16477.3 & 8238.7 \\ \text { Residual Error } & 27 & 3038.0 & 112.5 \\ \text { Total } & 29 & 19515.4 & \end{array}$

A) The multiple regression model is significant overall.
B) Selling Price is a significant independent variable in explaining Bravia sales.
C) Amount Spent on Advertising is a significant independent variable in explaining Bravia sales.
D) Only A and B
E) A, B and C

Question 10

Interpret multiple regression coefficients.

-In determining the best companies to work for, a number of variables are considered,
Including size, average annual pay, and turnover rate, etc. Employee surveys are often
Conducted in order to assess aspects of the organization's culture, such as trust and
Openness to change. A sample of 33 companies was randomly selected and data
Collected on the average annual bonus and turnover rate (%). A questionnaire was also
Administered to the employees of each company to arrive at a trust index (measured on a
Scale of 0 - 100). Based on the results shown below, the correct interpretation of the
Regression coefficient for the variable Trust Index is $$\begin{array}{l}\text { Dependent Variable is Turnover Rate }\\\begin{array} { l r r r r } \text { Predictor } & \text { Coef } & \text { SE Coef } & \mathbf { T } & \mathbf { P } \\\text { Constant } & 12.1005 & 0.7826 & 15.46 & 0.000 \\\text { Trust Index } & - 0.07149 & 0.01966 & - 3.64 & 0.001 \\\text { Average Bonus } & - 0.0007216 & 0.0001481 & - 4.87 & 0.000\end{array}\end{array}$$

A) For companies that give the same average annual bonus, an increase of 10 points on the trust index is associated with a decrease of 0.72% in turnover rate, on average.
B) An increase of 10 points on the trust index results in a decrease of 7.2% in turnover rate.
C) Holding average annual bonus constant, increasing the trust index by 10 points will decrease the turnover rate by 7.2%.
D) For companies that give the same average annual bonus, an increase of 10 points on the trust index is associated with an increase of 0.72% in turnover rate, on average.
E) None of the above.

Question 11

Perform statistical inference for multiple regression.

-What affects flat panel LCD TV sales? Flat panel LCD TV's are sold through a
Variety of outlets. Sales figures (number of units) for the popular Sony Bravia were
Obtained for last quarter from a sample of 30 different stores. Also collected were data
On the selling price and amount spent on advertising the Sony Bravia (as a percentage of
Total advertising expenditure in the previous quarter) at each store. Output is shown
Below. The calculated t-statistic to determine if amount spent on advertising is a
Significant independent variable in explaining Sony Bravia sales is $\text {Dependent Variable is Sales}$
$\begin{array} { l r r r r } \text { Predictor } & \text { Coef } & \text { SE Coef } & \text { T } & \text { P } \\ \text { Constant } & 90.19 & 25.08 & 3.60 & 0.001 \\ \text { Price } & - 0.03055 & 0.01005 & - 3.04 & 0.005 \\ \text { Advertising } & 3.0926 & 0.3680 & 8.40 & 0.000 \end{array}$
$$S = 10.6075 \quad \mathrm { R } - \mathrm { Sq } = 84.48 \quad \mathrm { R } - \mathrm { Sq } ( \mathrm { adj } \mathrm { j } ) = 83.3 \%$$

A) 3.60
B) -3.04
C) 8.40
D) 10.61
E) None of the above

Question 12

Write out the multiple regression model.

-In determining the best companies to work for, a number of variables are considered,
Including size, average annual pay, and turnover rate, etc. Employee surveys are often
Conducted in order to assess aspects of the organization's culture, such as trust and
Openness to change. A sample of 33 companies was randomly selected and data
Collected on the average annual bonus and turnover rate (%). A questionnaire was also
Administered to the employees of each company to arrive at a trust index (measured on a
Scale of 0 - 100). Based on the output shown below, the estimated multiple regression
Model is $\text {Dependent Variable is Turnover Rate}$
$\begin{array} { l r r r r } \text { Predictor } & \text { Coef } & \text { SE Coef } & \text { T } & \text { P } \\ \text { Constant } & 12.1005 & 0.7826 & 15.46 & 0.000 \\ \text { Trust Index } & - 0.07149 & 0.01966 & - 3.64 & 0.001 \\ \text { Average Bonus } & - 0.0007216 & 0.0001481 & - 4.87 & 0.000 \end{array}$
$$S = 1.49746 \quad R - S q = 79.6 \% \quad R - S q ( a d j ) = 78.3 \%$$

A) Turnover Rate = 0.01966 Trust Index + 0.0001481 Average Bonus
B) Turnover Rate = 0.7826 + 0.01966 Trust Index + 0.0001481 Average Bonus
C) Turnover Rate = 12.1005 - 0.07149 Trust Index - 0.0007216 Average Bonus
D) Turnover Rate = 12.1005 + 0.07149 Trust Index + 0.0007216 Average Bonus
E) Turnover Rate = 15.46 - 3.64 Trust Index - 4.87 Average Bonus

Question 13

Interpret multiple regression output.

-In determining the best companies to work for, a number of variables are considered,
Including size, average annual pay, and turnover rate, etc. Employee surveys are often
Conducted in order to assess aspects of the organization's culture, such as trust and
Openness to change. A sample of 33 companies was randomly selected and data
Collected on the average annual bonus and turnover rate (%). A questionnaire was also
Administered to the employees of each company to arrive at a trust index (measured on a
Scale of 0 - 100). Based on the output shown below, a company having a trust index
Score of 70 and an average annual bonus of $6500 has a predicted turnover rate of $\text {Dependent Variable is Turnover Rate}$
$\begin{array} { l r r r r } \text { Predictor } & \text { Coef } & \text { SE Coef } & \text { T } & \text { P } \\ \text { Constant } & 12.1005 & 0.7826 & 15.46 & 0.000 \\ \text { Trust Index } & - 0.07149 & 0.01966 & - 3.64 & 0.001 \\ \text { Average Bonus } & - 0.0007216 & 0.0001481 & - 4.87 & 0.000 \end{array}$
$$S = 1.49746 \quad R - S q = 79.6 \% \quad R - S q ( a d j ) = 78.3 \%$$

A) 3.5%
B) 4.2%
C) 1.9%
D) 2.4 %
E) None of the above.

Question 14

Interpret multiple regression output.

-What affects flat panel LCD TV sales? Flat panel LCD TV's are sold through a
Variety of outlets. Sales figures (number of units) for the popular Sony Bravia were
Obtained for last quarter from a sample of 30 different stores. Also collected were data
On the selling price and amount spent on advertising the Sony Bravia (as a percentage of
Total advertising expenditure in the previous quarter) at each store. Output is shown
Below. Using the estimated multiple regression model, the number of units sold on
Average at a store that sells the Sony Bravia for $2199 and spends 10% of its advertising
Budget on the product is $\text {Dependent Variable is Sales}$
$\begin{array} { l r r r r } \text { Predictor } & \text { Coef } & \text { SE Coef } & \text { T } & \text { P } \\ \text { Constant } & 90.19 & 25.08 & 3.60 & 0.001 \\ \text { Price } & - 0.03055 & 0.01005 & - 3.04 & 0.005 \\ \text { Advertising } & 3.0926 & 0.3680 & 8.40 & 0.000 \end{array}$
$$S = 10.6075 \quad R - S q = 84.4 \% \quad R - S q ( a d j ) = 83.3 \%$$

A) 53.94 units
B) 120 units
C) 66.54 units
D) 90.34 units
E) None of the above.

Question 15

What affects flat panel LCD TV sales? Flat panel LCD TV's are sold through a
variety of outlets such as large and small electronics stores, department stores, large
discount chains and online. Sales figures (number of units) for the popular Sony Bravia
were obtained for last quarter from a sample of 30 different stores. Also collected were
data on the selling price and amount spent on advertising the Sony Bravia (as a
percentage of total advertising expenditure in the previous quarter) at each store. Below
are the multiple regression results.
a. Write out the estimated regression equation.
b. Is the regression equation significant overall? Explain.
c. How much of the variability in Sales is explained by the regression equation?
d. State the hypotheses for testing the regression coefficient of Price. Based on the
results, what do you conclude?
e. State the hypotheses for testing the regression coefficient of Advertising Expenditure.
Based on the results, what do you conclude?
f. Predict the sales for a store that sells the Sony Bravia for $2199 and spends 10% of its
advertising budget on the product.
g. Comment on whether the conditions for multiple regression are satisfied based on the
plots shown below.