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Statistics for Business Economics

David R. Anderson, Dennis J. Sweeney, Thomas A. Williams

Chapter 15

Multiple Regression - all with Video Answers

Educators

Chapter Questions

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Problem 1

The estimated regression equation for a model involving two independent variables and 10 observations follows.
\[
\hat{y}=29.1270+.5906 x_{1}+.4980 x_{2}
\]
a. Interpret $b_{1}$ and $b_{2}$ in this estimated regression equation.
b. Estimate $y$ when $x_{1}=180$ and $x_{2}=310$.

Emily Himsel
Emily Himsel
Numerade Educator
15:32

Problem 2

Consider the following data for a dependent variable $y$ and two independent variables, $x_{1}$ and $x_{2}$.
a. Develop an estimated regression equation relating $y$ to $x_{1}$. Estimate $y$ if $x_{1}=45$
b. Develop an estimated regression equation relating $y$ to $x_{2}$. Estimate $y$ if $x_{2}=15$
c. Develop an estimated regression equation relating $y$ to $x_{1}$ and $x_{2}$. Estimate $y$ if $x_{1}=45$ and $x_{2}=15$.

Paul A.
Paul A.
California State Polytechnic University, Pomona
05:16

Problem 3

In a regression analysis involving 30 observations, the following estimated regression equation was obtained.
\[
\hat{y}=17.6+3.8 x_{1}-2.3 x_{2}+7.6 x_{3}+2.7 x_{4}
\]
a. Interpret $b_{1}, b_{2}, b_{3},$ and $b_{4}$ in this estimated regression equation.
b. Estimate $y$ when $x_{1}=10, x_{2}=5, x_{3}=1,$ and $x_{4}=2$.

Sneha Ravi
Sneha Ravi
Numerade Educator
04:14

Problem 4

A shoe store developed the following estimated regression equation relating sales to inventory investment and advertising expenditures.
\[
\hat{y}=25+10 x_{1}+8 x_{2}
\]
where
\[
\begin{aligned}
x_{1} &=\text { inventory investment }(\$ 1000 \mathrm{s}) \\
x_{2} &=\text { advertising expenditures }(\$ 1000 \mathrm{s}) \\
y &=\text { sales }(\$ 1000 \mathrm{s})
\end{aligned}
\]
a. Estimate sales resulting from a $\$ 15,000$ investment in inventory and an advertising budget of $\$ 10,000$
b. Interpret $b_{1}$ and $b_{2}$ in this estimated regression equation.

Sneha Ravi
Sneha Ravi
Numerade Educator
02:46

Problem 5

The owner of Showtime Movie Theaters, Inc., would like to estimate weekly gross revenue as a function of advertising expenditures. Historical data for a sample of eight weeks follow.
a. Develop an estimated regression equation with the amount of television advertising as the independent variable.
b. Develop an estimated regression equation with both television advertising and newspaper advertising as the independent variables.
c. Is the estimated regression equation coefficient for television advertising expenditures the same in part (a) and in part (b)? Interpret the coefficient in each case.
d. What is the estimate of the weekly gross revenue for a week when $\$ 3500$ is spent on television advertising and $\$ 1800$ is spent on newspaper advertising?

Shu Naito
Shu Naito
Numerade Educator
01:34

Problem 6

In baseball, a team's success is often thought to be a function of the team's hitting and pitching performance. One measure of hitting performance is the number of home runs the team hits, and one measure of pitching performance is the earned run average for the team's pitching staff. It is generally believed that teams that hit more home runs and have a lower earned run average will win a higher percentage of the games played. The following data show the proportion of games won, the number of team home runs (HR), and the earned run average (ERA) for the 16 teams in the National League for the 2003 Major League Baseball season (USA Today website, January 7,2004 ).
a. Determine the estimated regression equation that could be used to predict the proportion of games won given the number of team home runs.
b. Determine the estimated regression equation that could be used to predict the proportion of games won given the earned run average for the team's pitching staff.
c. Determine the estimated regression equation that could be used to predict the proportion of games won given the number of team home runs and the earned run average for the team's pitching staff.
d. For the 2003 season San Diego won only $39.5 \%$ of the games they played, the lowest in the National League. To improve next year's record, the team tried to acquire new players who would increase the number of team home runs to 180 and decrease the earned run average for the team's pitching staff to $4.0 .$ Use the estimated regression equation developed in part (c) to estimate the percentage of games San Diego will win if they have 180 team home runs and have an earned run average of 4.0.

Dominador Tan
Dominador Tan
Numerade Educator
15:46

Problem 7

$P C$ World rated four component characteristics for 10 ultraportable laptop computers: features; performance; design; and price. Each characteristic was rated using a $0-100$ point scale. An overall rating, referred to as the $P C W$ World Rating, was then developed for each laptop. The following table shows the performance rating, features rating, and the $P C W$ World Rating for the 10 laptop computers (PC World website, February 5,2009 ).
a. Determine the estimated regression equation that can be used to predict the $P C W$ World Rating using the performance rating as the independent variable.
b. Determine the estimated regression equation that can be used to predict the $P C W$ World Rating using both the performance rating and the features rating.
c. Predict the $P C W$ World Rating for a laptop computer that has a performance rating of 80 and a features rating of 70.

nn
No Name
Numerade Educator
05:34

Problem 8

Would you expect more reliable and better performing cars to cost more? Consumer Reports provided reliability ratings, overall road-test scores, and prices for affordable family
sedans, midpriced family sedans, and large sedans (Consumer Reports, February 2008). A portion of the data follows. Reliability was rated on a 5 -point scale from poor (1) to excellent $(5) .$ The road-test score was rated on a 100 -point scale, with higher values indicating better performance. The complete data set is contained in the file named Sedans.
a. Develop an estimated regression equation that can be used to predict the price of the car given the reliability rating. Test for significance using $\alpha=.05$
b. Consider the addition of the independent variable overall road-test score. Develop the estimated regression equation that can be used to predict the price of the car given the road-test score and the reliability rating.
c. Estimate the price for a car with a road-test score of 80 and a reliability rating of 4.

Sneha Ravi
Sneha Ravi
Numerade Educator
02:37

Problem 9

Waterskiing and wakeboarding are two popular water-sports. Finding a model that best suits your intended needs, whether it is waterskiing, wakeboading, or general boating, can be a difficult task. WaterSki magazine did extensive testing for 88 boats and provided a wide variety of information to help consumers select the best boat. A portion of the data they reported for 20 boats with a length of between 20 and 22 feet follows (WaterSki, January/February 2006 ). Beam is the maximum width of the boat in inches, HP is the horsepower of the boat's engine, and TopSpeed is the top speed in miles per hour (mph).
a. Using these data, develop an estimated regression equation relating the top speed with the boat's beam and horsepower rating.
b. The Svfara SV609 has a beam of 85 inches and an engine with a 330 horsepower rating. Use the estimated regression equation developed in part (a) to estimate the top speed for the Svfara SV609.

Neel Faucher
Neel Faucher
Numerade Educator
09:25

Problem 10

The National Basketball Association (NBA) records a variety of statistics for each team. Four of these statistics are the proportion of games won (PCT), the proportion of field goals made by the team (FG\%), the proportion of three-point shots made by the team's opponent $(\mathrm{Opp} 3 \mathrm{Pt} \%),$ and the number of turnovers committed by the team's opponent (Opp TO). The following data show the values of these statistics for the 29 teams in the NBA for a portion of the 2004 season (NBA website, January 3,2004 ).
a. Determine the estimated regression equation that can be used to predict the proportion of games won given the proportion of field goals made by the team.
b. Provide an interpretation for the slope of the estimated regression equation developed in part (a).
c. Determine the estimated regression equation that can be used to predict the proportion of games won given the proportion of field goals made by the team, the proportion of three-point shots made by the team's opponent, and the number of turnovers committed by the team's opponent.
d. Discuss the practical implications of the estimated regression equation developed in part (c).
e. Estimate the proportion of games won for a team with the following values for the three independent variables: $\mathrm{FG} \%=.45, \mathrm{Opp} 3 \mathrm{Pt} \%=.34,$ and $\mathrm{Opp} \mathrm{TO}=17$.

Sneha Ravi
Sneha Ravi
Numerade Educator
02:11

Problem 11

In exercise $1,$ the following estimated regression equation based on 10 observations was presented.
\[
\hat{y}=29.1270+.5906 x_{1}+.4980 x_{2}
\]
The values of SST and SSR are 6724.125 and $6216.375,$ respectively.
a. Find SSE.
b. $\quad$ Compute $R^{2}$.
c. Compute $R_{\mathrm{a}}^{2}$
d. Comment on the goodness of fit.

Dominador Tan
Dominador Tan
Numerade Educator
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Problem 12

In exercise 2,10 observations were provided for a dependent variable $y$ and two independent variables $x_{1}$ and $x_{2} ;$ for these data $\mathrm{SST}=15,182.9,$ and $\mathrm{SSR}=14,052.2$
a. $\quad$ Compute $R^{2}$.
b. Compute $R_{\mathrm{a}}^{2}$
c. Does the estimated regression equation explain a large amount of the variability in the data? Explain.

Rashmi Sinha
Rashmi Sinha
Numerade Educator
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Problem 13

In exercise $3,$ the following estimated regression equation based on 30 observations was presented.
\[
\hat{y}=17.6+3.8 x_{1}-2.3 x_{2}+7.6 x_{3}+2.7 x_{4}
\]
The values of SST and SSR are 1805 and $1760,$ respectively.
a. Compute $R^{2}$.
b. Compute $R_{\mathrm{a}}^{2}$
c. Comment on the goodness of fit.

Rashmi Sinha
Rashmi Sinha
Numerade Educator
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Problem 14

In exercise $4,$ the following estimated regression equation relating sales to inventory investment and advertising expenditures was given.
\[
\hat{y}=25+10 x_{1}+8 x_{2}
\]
The data used to develop the model came from a survey of 10 stores; for those data,
\[
\mathrm{SST}=16,000 \text { and } \mathrm{SSR}=12,000
\]
a. For the estimated regression equation given, compute $R^{2}$
b. Compute $R_{\mathrm{a}^{*}}^{2}$
c. Does the model appear to explain a large amount of variability in the data? Explain.

Rashmi Sinha
Rashmi Sinha
Numerade Educator
01:46

Problem 15

In exercise $5,$ the owner of Showtime Movie Theaters, Inc., used multiple regression analysis to predict gross revenue ( $y$ ) as a function of television advertising ( $x_{1}$ ) and newspaper advertising $\left(x_{2}\right) .$ The estimated regression equation was
\[
\hat{y}=83.2+2.29 x_{1}+1.30 x_{2}
\]
The computer solution provided $\mathrm{SST}=25.5$ and $\mathrm{SSR}=23.435$
a. Compute and interpret $R^{2}$ and $R_{\mathrm{a}}^{2}$
b. When television advertising was the only independent variable, $R^{2}=.653$ and $R_{\mathrm{a}}^{2}=$ $.595 .$ Do you prefer the multiple regression results? Explain.

Dominador Tan
Dominador Tan
Numerade Educator
01:31

Problem 16

In exercise $6,$ data were given on the proportion of games won, the number of team home runs, and the earned run average for the team's pitching staff for the 16 teams in the National League for the 2003 Major League Baseball season (USA Today website, January 7,2004 ).
a. Did the estimated regression equation that uses only the number of home runs as the independent variable to predict the proportion of games won provide a good fit? Explain.
b. Discuss the benefits of using both the number of home runs and the earned run average to predict the proportion of games won.

Dominador Tan
Dominador Tan
Numerade Educator
03:17

Problem 17

In exercise $9,$ an estimated regression equation was developed relating the top speed for a boat to the boat's beam and horsepower rating.
a. Compute and interpret and $R^{2}$ and $R_{\mathrm{a}}^{2}$
b. Does the estimated regression equation provide a good fit to the data? Explain.

Sneha Ravi
Sneha Ravi
Numerade Educator
01:43

Problem 18

Refer to exercise $10,$ where data were reported on a variety of statistics for the 29 teams in the National Basketball Association for a portion of the 2004 season (NBA website, January 3,2004 ).
a. In part (c) of exercise $10,$ an estimated regression equation was developed relating the proportion of games won given the percentage of field goals made by the team, the proportion of three-point shots made by the team's opponent, and the number of turnovers committed by the team's opponent. What are the values of $R^{2}$ and $R_{\mathrm{a}}^{2} ?$
b. Does the estimated regression equation provide a good fit to the data? Explain.

Dominador Tan
Dominador Tan
Numerade Educator
05:28

Problem 19

In exercise $1,$ the following estimated regression equation based on 10 observations was presented.
\[
\begin{aligned}
\hat{y} &=29.1270+.5906 x_{1}+.4980 x_{2} \\
\text { Here SST }=6724.125, \mathrm{SSR} &=6216.375, s_{b_{1}}=.0813, \text { and } s_{b_{2}}=.0567
\end{aligned}
\]
a. Compute MSR and MSE.
b. Compute $F$ and perform the appropriate $F$ test. Use $\alpha=.05$
c. Perform a $t$ test for the significance of $\beta_{1} .$ Use $\alpha=.05$
d. Perform a $t$ test for the significance of $\beta_{2} .$ Use $\alpha=.05$.

Neel Faucher
Neel Faucher
Numerade Educator
04:24

Problem 20

Refer to the data presented in exercise $2 .$ The estimated regression equation for these data is
\[
\begin{array}{c}
\hat{y}=-18.37+2.01 x_{1}+4.74 x_{2} \\
\text { Here SST }=15,182.9, \text { SSR }=14,052.2, s_{b_{1}}=.2471, \text { and } s_{b_{2}}=.9484
\end{array}
\]
a. Test for a significant relationship among $x_{1}, x_{2},$ and $y .$ Use $\alpha=.05$.
b. Is $\beta_{1}$ significant? Use $\alpha=.05$.
c. $\quad$ Is $\beta_{2}$ significant? Use $\alpha=.05$.

Dominador Tan
Dominador Tan
Numerade Educator
01:56

Problem 21

The following estimated regression equation was developed for a model involving two independent variables.
\[
\hat{y}=40.7+8.63 x_{1}+2.71 x_{2}
\]
After $x_{2}$ was dropped from the model, the least squares method was used to obtain an estimated regression equation involving only $x_{1}$ as an independent variable.
\[
\hat{y}=42.0+9.01 x_{1}
\]
a. Give an interpretation of the coefficient of $x_{1}$ in both models.
b. Could multicollinearity explain why the coefficient of $x_{1}$ differs in the two models? If so, how?

NV
Nicholas Vasconcellos
Numerade Educator
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Problem 22

In exercise $4,$ the following estimated regression equation relating sales to inventory investment and advertising expenditures was given.
\[
\hat{y}=25+10 x_{1}+8 x_{2}
\]
The data used to develop the model came from a survey of 10 stores; for these data
\[
\mathrm{SST}=16,000 \text { and } \mathrm{SSR}=12,000
\]
a. Compute SSE, MSE, and MSR.
b. Use an $F$ test and a .05 level of significance to determine whether there is a relationship among the variables.

Rashmi Sinha
Rashmi Sinha
Numerade Educator
05:04

Problem 23

Refer to exercise 5
a. Use $\alpha=.01$ to test the hypotheses
\[
H_{0}: \beta_{1}=\beta_{2}=0
\]
$H_{\mathrm{a}}: \beta_{1}$ and/or $\beta_{2}$ is not equal to zero
for the model $y=\beta_{0}+\beta_{1} x_{1}+\beta_{2} x_{2}+\epsilon,$ where
\[
\begin{array}{l}
x_{1}=\text { television advertising }(\$ 1000 \mathrm{s}) \\
x_{2}=\text { newspaper advertising }(\$ 1000 \mathrm{s})
\end{array}
\]
b. Use $\alpha=.05$ to test the significance of $\beta_{1}$. Should $x_{1}$, be dropped from the model?
c. Use $\alpha=.05$ to test the significance of $\beta_{2} .$ Should $x_{2}$ be dropped from the model?

Sneha Ravi
Sneha Ravi
Numerade Educator
02:16

Problem 24

The Wall Street Journal conducted a study of basketball spending at top colleges. A portion of the data showing the revenue ( $\$$ millions), percentage of wins, and the coach's salary ( $\$$ millions) for 39 of the country's top basketball programs follows (The Wall Street Journal, March $11-12,2006$ ).
a. Develop the estimated regression equation that can be used to predict the coach's salary given the revenue generated by the program and the percentage of wins.
b. Use the $F$ test to determine the overall significance of the relationship. What is your conclusion at the .05 level of significance?
c. Use the $t$ test to determine the significance of each independent variable. What is your conclusion at the .05 level of significance?

Neel Faucher
Neel Faucher
Numerade Educator
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Problem 25

Barron's conducts an annual review of online brokers, including both brokers who can be accessed via a Web browser, as well as direct-access brokers who connect customers directly with the broker's network server. Each broker's offerings and performance are evaluated in six areas, using a point value of $0-5$ in each category. The results are weighted to obtain an overall score, and a final star rating, ranging from zero to five stars, is assigned to each broker. Trade execution, ease of use, and range of offerings are three of the areas evaluated. A point value of 5 in the trade execution area means the order entry and execution process flowed easily from one step to the next. A value of 5 in the ease of use area means that the site was easy to use and can be tailored to show what the user wants to see. A value of 5 in the range offerings area means that all the investment transactions can be executed online. The following data show the point values for trade execution, ease of use, range of offerings, and the star rating for a sample of 10 of the online brokers that Barron's evaluated ( Barron's, March 10,2003 ).
a. Determine the estimated regression equation that can be used to predict the star rating given the point values for execution, ease of use, and range of offerings.
b. Use the $F$ test to determine the overall significance of the relationship. What is the conclusion at the .05 level of significance?
c. Use the $t$ test to determine the significance of each independent variable. What is your conclusion at the .05 level of significance?
d. Remove any independent variable that is not significant from the estimated regression equation. What is your recommended estimated regression equation? Compare the $R^{2}$ with the value of $R^{2}$ from part (a). Discuss the differences.

Victor Salazar
Victor Salazar
Numerade Educator
05:31

Problem 26

In exercise 10 an estimated regression equation was developed relating the proportion of games won given the proportion of field goals made by the team, the proportion of three-point shots made by the team's opponent, and the number of turnovers committed by the team's opponent.
a. Use the $F$ test to determine the overall significance of the relationship. What is your conclusion at the .05 level of significance?
b. Use the $t$ test to determine the significance of each independent variable. What is your conclusion at the .05 level of significance?

Neel Faucher
Neel Faucher
Numerade Educator
02:12

Problem 27

In exercise $1,$ the following estimated regression equation based on 10 observations was presented.
\[
\hat{y}=29.1270+.5906 x_{1}+.4980 x_{2}
\]
a. Develop a point estimate of the mean value of $y$ when $x_{1}=180$ and $x_{2}=310$
b. Develop a point estimate for an individual value of $y$ when $x_{1}=180$ and $x_{2}=310$.

Sneha Ravi
Sneha Ravi
Numerade Educator
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Problem 28

Refer to the data in exercise $2 .$ The estimated regression equation for those data is
\[
\hat{y}=-18.4+2.01 x_{1}+4.74 x_{2}
\]
a. Develop a $95 \%$ confidence interval for the mean value of $y$ when $x_{1}=45$ and $x_{2}=15$.
b. Develop a $95 \%$ prediction interval for $y$ when $x_{1}=45$ and $x_{2}=15$.

Rashmi Sinha
Rashmi Sinha
Numerade Educator
02:28

Problem 29

In exercise 5 , the owner of Showtime Movie Theaters, Inc., used multiple regression analysis to predict gross revenue ( $y$ ) as a function of television advertising $\left(x_{1}\right)$ and newspaper advertising $\left(x_{2}\right) .$ The estimated regression equation was
\[
\hat{y}=83.2+2.29 x_{1}+1.30 x_{2}
\]
a. What is the gross revenue expected for a week when $\$ 3500$ is spent on television advertising $\left(x_{1}=3.5\right)$ and $\$ 1800$ is spent on newspaper advertising $\left(x_{2}=1.8\right) ?$
b. Provide a $95 \%$ confidence interval for the mean revenue of all weeks with the expenditures listed in part (a).
c. Provide a $95 \%$ prediction interval for next week's revenue, assuming that the advertising expenditures will be allocated as in part (a).

Shu Naito
Shu Naito
Numerade Educator
04:02

Problem 30

In exercise 9 an estimated regression equation was developed relating the top speed for a boat to the boat's beam and horsepower rating.
a. Develop a $95 \%$ confidence interval for the mean top speed of a boat with a beam of 85 inches and an engine with a 330 horsepower rating.
b. The Svfara SV609 has a beam of 85 inches and an engine with a 330 horsepower rating. Develop a $95 \%$ confidence interval for the mean top speed for the Svfara SV609.

Robin Corrigan
Robin Corrigan
Numerade Educator
01:19

Problem 31

The Buyer's Guide section of the Web site for Car and Driver magazine provides reviews and road tests for cars, trucks, SUVs, and vans. The average ratings of overall quality, vehicle styling, braking, handling, fuel economy, interior comfort, acceleration, dependability, fit and finish, transmission, and ride are summarized for each vehicle using a scale ranging from 1 (worst) to 10 (best). A portion of the data for 14 Sports/GT cars is shown here (Car and Driver website, January 7,2004 ).
a. Develop an estimated regression equation using handling, dependability, and fit and finish to predict overall quality.
b. Another Sports/GT car rated by Car and Driver is the Honda Accord. The ratings for handling, dependability, and fit and finish for the Honda Accord were $8.28,9.06,$ and $8.07,$ respectively. Estimate the overall rating for this car.
c. Provide a $95 \%$ confidence interval for overall quality for all sports and GT cars with the characteristics listed in part (b).
d. Provide a $95 \%$ prediction interval for overall quality for the Honda Accord described in part (b).
e. The overall rating reported by Carand Driver for the Honda Accord was $8.65 .$ How does this rating compare to the estimates you developed in parts (b) and (d)?

Dominador Tan
Dominador Tan
Numerade Educator
01:10

Problem 32

Consider a regression study involving a dependent variable $y,$ a categorical independent variable $x_{1},$ and a categorical variable with two levels (level 1 and level 2 ).
a. Write a multiple regression equation relating $x_{1}$ and the categorical variable to $y$.
b. What is the expected value of $y$ corresponding to level 1 of the categorical variable?
c. What is the expected value of $y$ corresponding to level 2 of the categorical variable?
d. Interpret the parameters in your regression equation.

Neel Faucher
Neel Faucher
Numerade Educator
01:44

Problem 33

Consider a regression study involving a dependent variable $y,$ a quantitative independent variable $x_{1},$ and a categorical independent variable with three possible levels (level $1,$ level
$2,$ and level 3 ).
a. How many dummy variables are required to represent the categorical variable?
b. Write a multiple regression equation relating $x_{1}$ and the categorical variable to $y$
c. Interpret the parameters in your regression equation.

Shu Naito
Shu Naito
Numerade Educator
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Problem 34

Management proposed the following regression model to predict sales at a fast-food outlet.
\[
y=\beta_{0}+\beta_{1} x_{1}+\beta_{2} x_{2}+\beta_{3} x_{3}+\epsilon
\]
where
\[
\begin{aligned}
x_{1} &=\text { number of competitors within one mile } \\
x_{2} &=\text { population within one mile }(1000 \mathrm{s}) \\
x_{3} &=\left\{\begin{array}{l}
1 \text { if drive-up window present } \\
0 \text { otherwise }
\end{array}\right.\\
y &=\operatorname{sales}(\$ 1000 \mathrm{s})
\end{aligned}
\]
The following estimated regression equation was developed after 20 outlets were surveyed.
\[
\hat{y}=10.1-4.2 x_{1}+6.8 x_{2}+15.3 x_{3}
\]
a. What is the expected amount of sales attributable to the drive-up window?
b. Predict sales for a store with two competitors, a population of 8000 within one mile, and no drive-up window.
c. Predict sales for a store with one competitor, a population of 3000 within one mile, and a drive-up window.

Donna Densmore
Donna Densmore
Numerade Educator
01:39

Problem 35

Refer to the Johnson Filtration problem introduced in this section. Suppose that in addition to information on the number of months since the machine was serviced and whether a mechanical or an electrical repair was necessary, the managers obtained a list showing which repairperson performed the service. The revised data follow.
a. Ignore for now the months since the last maintenance service $\left(x_{1}\right)$ and the repairperson who performed the service. Develop the estimated simple linear regression equation to predict the repair time ( $y$ ) given the type of repair $\left(x_{2}\right) .$ Recall that $x_{2}=0$ if the type of repair is mechanical and 1 if the type of repair is electrical.
b. Does the equation that you developed in part (a) provide a good fit for the observed data? Explain.
c. Ignore for now the months since the last maintenance service and the type of repair associated with the machine. Develop the estimated simple linear regression equation to predict the repair time given the repairperson who performed the service. Let $x_{3}=0$ if Bob Jones performed the service and $x_{3}=1$ if Dave Newton performed the service.
d. Does the equation that you developed in part
(c) provide a good fit for the observed data? Explain.

Dominador Tan
Dominador Tan
Numerade Educator
04:32

Problem 36

This problem is an extension of the situation described in exercise 35
a. Develop the estimated regression equation to predict the repair time given the number of months since the last maintenance service, the type of repair, and the repairperson who performed the service.
b. At the .05 level of significance, test whether the estimated regression equation developed in part (a) represents a significant relationship between the independent variables and the dependent variable.
c. Is the addition of the independent variable $x_{3}$, the repairperson who performed the service, statistically significant? Use $\alpha=.05 .$ What explanation can you give for the results observed?

Dominador Tan
Dominador Tan
Numerade Educator
01:32

Problem 37

The Consumer Reports Restaurant Customer Satisfaction Survey is based upon 148,599 visits to full-service restaurant chains (Consumer Reports website, February 11,2009 ). Assume the following data are representative of the results reported. The variable Type indicates whether the restaurant is an Italian restaurant or a seafood/steakhouse. Price indicates the average amount paid per person for dinner and drinks, minus the tip. Score reflects diners' overall satisfaction, with higher values indicating greater overall satisfaction. A score of 80 can be interpreted as very satisfied.
a. Develop the estimated regression equation to show how overall customer satisfaction is related to the independent variable average meal price.
b. At the .05 level of significance, test whether the estimated regression equation developed in part (a) indicates a significant relationship between overall customer satisfaction and average meal price.
c. Develop a dummy variable that will account for the type of restaurant (Italian or seafood/steakhouse).
d. Develop the estimated regression equation to show how overall customer satisfaction is related to the average meal price and the type of restaurant.
e. Is type of restaurant a significant factor in overall customer satisfaction?
f. Estimate the Consumer Reports customer satisfaction score for a seafood/steakhouse that has an average meal price of $\$ 20 .$ How much would the estimated score have changed for an Italian restaurant?

Sheryl Ezze
Sheryl Ezze
Numerade Educator
02:33

Problem 38

A 10 -year study conducted by the American Heart Association provided data on how age, blood pressure, and smoking relate to the risk of strokes. Assume that the following data are from a portion of this study. Risk is interpreted as the probability (times 100 ) that the patient will have a stroke over the next 10 -year period. For the smoking variable, define a dummy variable with 1 indicating a smoker and 0 indicating a nonsmoker.
a. Develop an estimated regression equation that relates risk of a stroke to the person's age, blood pressure, and whether the person is a smoker.
b. Is smoking a significant factor in the risk of a stroke? Explain. Use $a=.05$
c. What is the probability of a stroke over the next 10 years for Art Speen, a 68 -year-old smoker who has blood pressure of $175 ?$ What action might the physician recommend for this patient?

Shu Naito
Shu Naito
Numerade Educator
01:40

Problem 39

Data for two variables, $x$ and $y,$ follow.
\[
\begin{array}{c|ccccc}
\boldsymbol{x}_{i} & 1 & 2 & 3 & 4 & 5 \\
\hline \boldsymbol{y}_{i} & 3 & 7 & 5 & 11 & 14
\end{array}
\]
a. Develop the estimated regression equation for these data.
b. Plot the standardized residuals versus $\hat{y}$. Do there appear to be any outliers in these data? Explain.
c. Compute the studentized deleted residuals for these data. At the .05 level of significance, can any of these observations be classified as an outlier? Explain.

Shu Naito
Shu Naito
Numerade Educator
01:21

Problem 40

Data for two variables, $x$ and $y$, follow.
\[
\begin{array}{c|ccccc}
\boldsymbol{x}_{\boldsymbol{i}} & 22 & 24 & 26 & 28 & 40 \\
\hline \boldsymbol{y}_{\boldsymbol{i}} & 12 & 21 & 31 & 35 & 70
\end{array}
\]
a. Develop the estimated regression equation for these data.
b. Compute the studentized deleted residuals for these data. At the .05 level of significance, can any of these observations be classified as an outlier? Explain.
c. Compute the leverage values for these data. Do there appear to be any influential observations in these data? Explain.
d. Compute Cook's distance measure for these data. Are any observations influential? Explain.

Shu Naito
Shu Naito
Numerade Educator
01:34

Problem 41

Exercise 5 gave the following data on weekly gross revenue, television advertising, and newspaper advertising for Showtime Movie Theaters.
a. Find an estimated regression equation relating weekly gross revenue to television and newspaper advertising.
b. Plot the standardized residuals against $\hat{y}$. Does the residual plot support the assumptions about $\epsilon ?$ Explain.
c. Check for any outliers in these data. What are your conclusions?
d. Are there any influential observations? Explain.

Shu Naito
Shu Naito
Numerade Educator
01:30

Problem 42

The following data show the curb weight, horsepower, and $1 / 4$ -mile speed for 16 popular sports and GT cars. Suppose that the price of each sports and GT car is also available. The complete data set is as follows:
a. Find the estimated regression equation, which uses price and horsepower to predict $1 / 4-$ mile speed
b. Plot the standardized residuals against $\hat{y}$. Does the residual plot support the assumption about $\epsilon ?$ Explain.
c. Check for any outliers. What are your conclusions?
d. Are there any influential observations? Explain.

Shu Naito
Shu Naito
Numerade Educator
03:25

Problem 43

The Ladies Professional Golfers Association (LPGA) maintains statistics on performance and earnings for members of the LPGA Tour. Year-end performance statistics for the 30 players who had the highest total earnings in LPGA Tour events for 2005 appear in the file named LPGA (LPGA website, 2006 ). Earnings $(\$ 1000$ s) is the total earnings in thousands of dollars; Scoring Avg. is the average score for all events; Greens in Reg. is the percentage of time a player is able to hit the green in regulation; and Putting Avg. is the average number of putts taken on greens hit in regulation. A green is considered hit in regulation if any part of the ball is touching the putting surface and the difference between the value of par for the hole and the number of strokes taken to hit the green is at least 2
a. Develop an estimated regression equation that can be used to predict the average score for all events given the percentage of time a player is able to hit the green in regulation and the average number of putts taken on greens hit in regulation.
b. Plot the standardized residuals against $\hat{y}$. Does the residual plot support the assumption about $\epsilon ?$ Explain.
c. Check for any outliers. What are your conclusions?
d. Are there any influential observations? Explain.

Shu Naito
Shu Naito
Numerade Educator
03:26

Problem 44

Refer to the Simmons Stores example introduced in this section. The dependent variable is coded as $y=1$ if the customer used the coupon and 0 if not. Suppose that the only information available to help predict whether the customer will use the coupon is the customer's credit card status, coded as $x=1$ if the customer has a Simmons credit card and $x=0$ if not.
a. Write the logistic regression equation relating $x$ to $y$.
b. What is the interpretation of $E(y)$ when $x=0 ?$
c. For the Simmons data in Table 15.11 , use Minitab to compute the estimated logit.
d. Use the estimated logit computed in part (c) to compute an estimate of the probability of using the coupon for customers who do not have a Simmons credit card and an estimate of the probability of using the coupon for customers who have a Simmons credit card.
e. What is the estimate of the odds ratio? What is its interpretation?

Sheryl Ezze
Sheryl Ezze
Numerade Educator
01:17

Problem 45

In Table 15.12 we provided estimates of the probability using the coupon in the Simmons Stores catalog promotion. A different value is obtained for each combination of values for the independent variables.
a. Compute the odds in favor of using the coupon for a customer with annual spending of $\$ 4000$ who does not have a Simmons credit card $\left(x_{1}=4, x_{2}=0\right)$
b. Use the information in Table 15.12 and part (a) to compute the odds ratio for the Simmons credit card variable $x_{2}=0,$ holding annual spending constant at $x_{1}=4$
c. In the text, the odds ratio for the credit card variable was computed using the information in the $\$ 2000$ column of Table $15.12 .$ Did you get the same value for the odds ratio in part (b)?

Dominador Tan
Dominador Tan
Numerade Educator
02:10

Problem 46

Community Bank would like to increase the number of customers who use payroll direct deposit. Management is considering a new sales campaign that will require each branch manager to call each customer who does not currently use payroll direct deposit. As an incentive to sign up for payroll direct deposit, each customer contacted will be offered free checking for two years. Because of the time and cost associated with the new campaign, management would like to focus their efforts on customers who have the highest probability of signing up for payroll direct deposit. Management believes that the average monthly balance in a customer's checking account may be a useful predictor of whether the customer will sign up for direct payroll deposit. To investigate the relationship between these two variables, Community Bank tried the new campaign using a sample of 50 checking account customers who do not currently use payroll direct deposit. The sample data show the average monthly checking account balance (in hundreds of dollars) and whether the customer contacted signed up for payroll direct deposit (coded 1 if the customer signed up for payroll direct deposit and 0 if not.). The data are contained in the data set named Bank; a portion of the data follows.
a. Write the logistic regression equation relating $x$ to $y$
b. For the Community Bank data, use Minitab to compute the estimated logistic regression equation.
c. Conduct a test of significance using the $G$ test statistic. Use $\alpha=.05$
d. Estimate the probability that customers with an average monthly balance of $\$ 1000$ will sign up for direct payroll deposit.
e. Suppose Community Bank only wants to contact customers who have a .50 or higher probability of signing up for direct payroll deposit. What is the average monthly balance required to achieve this level of probability?
f. What is the estimate of the odds ratio? What is its interpretation?

Dominador Tan
Dominador Tan
Numerade Educator
05:07

Problem 47

Over the past few years the percentage of students who leave Lakeland College at the end of the first year has increased. Last year Lakeland started a voluntary one-week orientation program to help first-year students adjust to campus life. If Lakeland is able to show that the orientation program has a positive effect on retention, they will consider making the program a requirement for all first-year students. Lakeland's administration also suspects that students with lower GPAs have a higher probability of leaving Lakeland at the end of the first year. In order to investigate the relation of these variables to retention, Lakeland selected a random sample of 100 students from last year's entering class. The data are contained in the data set named Lakeland; a portion of the data follows.
The dependent variable was coded as $y=1$ if the student returned to Lakeland for the sophomore year and $y=0$ if not. The two independent variables are:
$x_{1}=$ GPA at the end of the first semester $x_{2}=\left\{\begin{array}{l}0 \text { if the student did not attend the orientation program } \\ 1 \text { if the student attended the orientation program }\end{array}\right.$a. Write the logistic regression equation relating $x_{1}$ and $x_{2}$ to $y$.
b. What is the interpretation of $E(y)$ when $x_{2}=0 ?$
c. Use both independent variables and Minitab to compute the estimated logit.
d. Conduct a test for overall significance using $a=.05$
e. Use $\alpha=.05$ to determine whether each of the independent variables is significant.
f. Use the estimated logit computed in part (c) to compute an estimate of the probability that students with a 2.5 grade point average who did not attend the orientation program will return to Lakeland for their sophomore year. What is the estimated probability for students with a 2.5 grade point average who attended the orientation program?
g. What is the estimate of the odds ratio for the orientation program? Interpret it.
h. Would you recommend making the orientation program a required activity? Why or why not?

Sheryl Ezze
Sheryl Ezze
Numerade Educator
02:32

Problem 48

Consumer Reports conducted a taste test on 19 brands of boxed chocolates. The following data show the price per serving, based on the FDA serving size of 1.4 ounces, and the quality rating for the 19 chocolates tested (Consumer Reports, February 2002 ).
Suppose that you would like to determine whether products that cost more rate higher in quality. For the purpose of this exercise, use the following binary dependent variable:
$y=1$ if the quality rating is very good or excellent and 0 if good or fair
a. Write the logistic regression equation relating $x=$ price per serving to $y$
b. Use Minitab to compute the estimated logit.
c. Use the estimated logit computed in part (b) to compute an estimate of the probability a chocolate that has a price per serving of $\$ 4.00$ will have a quality rating of very good or excellent.
d. What is the estimate of the odds ratio? What is its interpretation?

Breanna Ollech
Breanna Ollech
Numerade Educator
01:37

Problem 49

The admissions officer for Clearwater College developed the following estimated regression equation relating the final college GPA to the student's SAT mathematics score and high-school GPA.
\[
\hat{y}=-1.41+.0235 x_{1}+.00486 x_{2}
\]
where
$\begin{aligned} x_{1} &=\text { high-school grade point average } \\ x_{2} &=\text { SAT mathematics score } \\ y &=\text { final college grade point average } \end{aligned}$
a. Interpret the coefficients in this estimated regression equation.
b. Estimate the final college GPA for a student who has a high-school average of 84 and a score of 540 on the SAT mathematics test.

Dominador Tan
Dominador Tan
Numerade Educator
View

Problem 50

The personnel director for Electronics Associates developed the following estimated regression equation relating an employee's score on a job satisfaction test to his or her length of service and wage rate.
\[
\hat{y}=14.4-8.69 x_{1}+13.5 x_{2}
\]
where
\[
\begin{array}{l}
x_{1}=\text { length of service (years) } \\
x_{2}=\text { wage rate (dollars) }
\end{array}
\]
$y=$ job satisfaction test score (higher scores indicate greater job satisfaction)
a. Interpret the coefficients in this estimated regression equation.
b. Develop an estimate of the job satisfaction test score for an employee who has four years of service and makes $\$ 6.50$ per hour.

Donna Densmore
Donna Densmore
Numerade Educator
01:55

Problem 51

A partial computer output from a regression analysis follows.
a. Compute the missing entries in this output.
b. Use the $F$ test and $\alpha=.05$ to see whether a significant relationship is present.
c. Use the $t$ test and $\alpha=.05$ to test $H_{0}: \beta_{1}=0$ and $H_{0}: \beta_{2}=0$
d. Compute $R_{\mathrm{a}^{\prime}}^{2}$.

Dominador Tan
Dominador Tan
Numerade Educator
01:17

Problem 52

Recall that in exercise $49,$ the admissions officer for Clearwater College developed the following estimated regression equation relating final college GPA to the student's SAT mathematics score and high-school GPA.
\[
\hat{y}=-1.41+.0235 x_{1}+.00486 x_{2}
\]
where
$\begin{aligned} x_{1} &=\text { high-school grade point average } \\ x_{2} &=\text { SAT mathematics score } \\ y &=\text { final college grade point average } \end{aligned}$
A portion of the Minitab computer output follows.
a. Complete the missing entries in this output.
b. Use the $F$ test and a .05 level of significance to see whether a significant relationship is present.
c. Use the $t$ test and $\alpha=.05$ to test $H_{0}: \beta_{1}=0$ and $H_{0}: \beta_{2}=0$
d. Did the estimated regression equation provide a good fit to the data? Explain.

Dominador Tan
Dominador Tan
Numerade Educator
00:52

Problem 53

Recall that in exercise 50 the personnel director for Electronics Associates developed the following estimated regression equation relating an employee's score on a job satisfaction test to length of service and wage rate.
\[
\hat{y}=14.4-8.69 x_{1}+13.5 x_{2}
\]
where
$\begin{aligned} x_{1} &=\text { length of service (years) } \\ x_{2} &=\text { wage rate (dollars) } \\ y &=\text { job satisfaction test score (higher scores } \end{aligned}$ indicate greater job satisfaction)
A portion of the Minitab computer output follows.
a. Complete the missing entries in this output.
b. Compute $F$ and test using $\alpha=.05$ to see whether a significant relationship is present.
c. $\quad$ Did the estimated regression equation provide a good fit to the data? Explain.
d. Use the $t$ test and $\alpha=.05$ to test $H_{0}: \beta_{1}=0$ and $H_{0}: \beta_{2}=0$.

Dominador Tan
Dominador Tan
Numerade Educator
02:58

Problem 54

The Tire Rack, America's leading online distributor of tires and wheels, conducts extensive testing to provide customers with products that are right for their vehicle, driving style, and driving conditions. In addition, the Tire Rack maintains an independent consumer survey to help drivers help each other by sharing their long-term tire experiences. The following data show survey ratings (1 to 10 scale with 10 the highest rating) for 18 maximum performance summer tires (Tire Rack website, February 3,2009 ). The variable Steering rates the tire's steering responsiveness, Tread Wear rates quickness of wear based on the driver's expectations, and Buy Again rates the driver's overall tire satisfaction and desire to purchase the same tire again.
a. Develop an estimated regression equation that can be used to predict the Buy Again rating given based on the Steering rating. At the .05 level of significance, test for a significant relationship.
b. Did the estimated regression equation developed in part (a) provide a good fit to the data? Explain.
c. Develop an estimated regression equation that can be used to predict the Buy Again rating given the Steering rating and the Tread Wear rating.
d. Is the addition of the Tread Wear independent variable significant? Use $a=.05$.

Shu Naito
Shu Naito
Numerade Educator
View

Problem 55

Consumer Reports provided extensive testing and ratings for 24 treadmills. An overall score, based primarily on ease of use, ergonomics, exercise range, and quality, was developed for each treadmill tested. In general, a higher overall score indicates better performance. The following data show the price, the quality rating, and overall score for the 24 treadmills (Consumer Reports, February 2006).
a. Use these data to develop an estimated regression equation that could be used to estimate the overall score given the price.
b. Use $\alpha=.05$ to test for overall significance.
c. To incorporate the effect of quality, a categorical variable with three levels, we used two dummy variables: Quality-E and Quality-VG. Each variable was coded 0 or 1 as follows.
\[
\begin{aligned}
\text { Quality-E } &=\left\{\begin{array}{l}
1 \text { if quality rating is excellent } \\
0 \text { otherwise }
\end{array}\right.\\
\text { Quality-VG } &=\left\{\begin{array}{l}
1 \text { if quality rating is very good } \\
0 \text { otherwise }
\end{array}\right.
\end{aligned}
\]
Develop an estimated regression equation that could be used to estimate the overall score given the price and the quality rating.
d. For the estimated regression equation developed in part (c), test for overall significance using $\alpha=.10$
e. For the estimated regression equation developed in part (c), use the $t$ test to determine the significance of each independent variable. Use $\alpha=.10$
f. Develop a standardized residual plot. Does the pattern of the residual plot appear to be reasonable?
g. Do the data contain any outliers or influential observations?
h. Estimate the overall score for a treadmill with a price of $\$ 2000$ and a good quality rating. How much would the estimate change if the quality rating were very good? Explain.

Victor Salazar
Victor Salazar
Numerade Educator
03:16

Problem 56

A portion of a data set containing information for 45 mutual funds that are part of the Morningstar Funds 500 for 2008 follows. The complete data set is available in the file named MutualFunds. The data set includes the following five variables:

Type: The type of fund, labeled DE (Domestic Equity), IE (International Equity), and FI (Fixed Income).
Net Asset Value (\$): The closing price per share on December 31, 2007.
5-Year Average Return (\%): The average annual return for the fund over the past 5 years.
Expense Ratio (\%): The percentage of assets deducted each fiscal year for fund expenses.
Morningstar Rank: The risk adjusted star rating for each fund; Morningstar ranks go from a low of $1-$ Star to a high of 5 -Stars.
a. Develop an estimated regression equation that can be used to predict the 5 -year average return given fund type. At the .05 level of significance, test for a significant relationship.
b. Did the estimated regression equation developed in part (a) provide a good fit to the data? Explain.
c. Develop the estimated regression equation that can be used to predict the 5 -year average return given the type of fund, the net asset value, and the expense ratio. At the .05 level of significance, test for a significant relationship. Do you think any variables should be deleted from the estimated regression equation? Explain.
d. Morningstar Rank is a categorical variable. Because the data set contains only funds with four ranks (2-Star through 5-Star), use the following dummy variables: 3StarRank $=1$ for a 3 -Star fund, 0 otherwise; 4 StarRank $=1$ for a 4 -Star fund, 0 otherwise; and 5 StarRank $=1$ for a 5 -Star fund, 0 otherwise. Develop an estimated regression equation that can be used to predict the 5 -year average return given the type of fund, the expense ratio, and the Morningstar Rank. Using $\alpha=.05,$ remove any independent variables that are not significant.
e. Use the estimated regression equation developed in part
(d) to estimate the 5 -year average return for a domestic equity fund with an expense ratio of $1.05 \%$ and a $3-$ Star Morningstar Rank.

Dominador Tan
Dominador Tan
Numerade Educator
09:38

Problem 57

The U.S. Department of Energy's Fuel Economy Guide provides fuel efficiency data for cars and trucks (U.S. Department of Energy website, February 22,2008 ). A portion of the data for 311 compact, midsize, and large cars follows. The column labeled Class identifies the size of the car; Compact, Midsize, or Large. The column labeled Displacement shows the engine's displacement in liters. The column labeled Fuel Type shows whether the car uses premium (P) or regular (R) fuel, and the column labeled Hwy MPG shows the fuel efficiency rating for highway driving in terms of miles per gallon. The complete data set is contained in the file named FuelData.
a. Develop an estimated regression equation that can be used to predict the fuel efficiency for highway driving given the engine's displacement. Test for significance using $\alpha=.05$
b. Consider the addition of the dummy variables ClassMidsize and ClassLarge. The value of ClassMidsize is 1 if the car is a midsize car and 0 otherwise; the value of ClassLarge is 1 if the car is a large car and 0 otherwise. Thus, for a compact car, the value of ClassMidsize and the value of ClassLarge is $0 .$ Develop the estimated regression equation that can be used to predict the fuel efficiency for highway driving given the engine's displacement and the dummy variables ClassMidsize and ClassLarge.
c. Use $\alpha=.05$ to determine whether the dummy variables added in part (b) are significant.
d. Consider the addition of the dummy variable FuelPremium, where the value of FuelPremium is 1 if the car uses premium fuel and 0 if the car uses regular fuel. Develop the estimated regression equation that can be used to predict the fuel efficiency for highway driving given the engine's displacement, the dummy variables ClassMidsize and ClassLarge, and the dummy variable FuelPremium.
e. For the estimated regression equation developed in part (d), test for overall significance and individual significance using $\alpha=.05$.

Carson Merrill
Carson Merrill
Numerade Educator