Essentials of Modern Business Statistics

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

Chapter 15

Multiple Regression - all with Video Answers

Educators

Chapter Questions

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. Predict $y$ when $x_{1}=180$ and $x_{2}=310$.

Emily Himsel

Emily Himsel

Numerade Educator

Problem 2

Consider the following data for a dependent variable $y$ and two independent variables, $x_{1}$ and $x_{2}$
$\begin{array}{lrr}\boldsymbol{x}_{1} & \boldsymbol{x}_{2} & \boldsymbol{y} \\ & & \\ 30 & 12 & 94 \\ 47 & 10 & 108 \\ 25 & 17 & 112 \\ 51 & 16 & 178 \\ 40 & 5 & 94 \\ 51 & 19 & 175 \\ 74 & 7 & 170 \\ 36 & 12 & 117 \\ 59 & 13 & 142 \\ 76 & 16 & 211\end{array}$
a. Develop an estimated regression equation relating $y$ to $x_{1} .$ Predict $y$ if $x_{1}=45$
b. Develop an estimated regression equation relating $y$ to $x_{2}$. Predict $y$ if $x_{2}=15$.
c. Develop an estimated regression equation relating $y$ to $x_{1}$ and $x_{2}$. Predict $y$ if $x_{1}=45$ and $x_{2}=15$

Paul A.

California State Polytechnic University, Pomona

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. Predict $y$ when $x_{1}=10, x_{2}=5, x_{3}=1,$ and $x_{4}=2$

Sneha Ravi

Numerade Educator

Problem 4

Forecasting Shoe Sales. 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. Predict the sales resulting from a 15,000 investment in inventory and an advertising budget of 10,000.
b. Interpret $b_{1}$ and $b$, in this estimated regression equation.

Emily Himsel

Numerade Educator

Problem 5

The owner of Showtime Movie Theaters, Inc. would like to predict weekly gross revenue as a function of advertising expenditures. Historical data for a sample of eight weeks follow.
$\begin{array}{ccc}\begin{array}{c}\text { Weekly } \\ \text { Gross Revenue } \\ \text { (\$1000s) }\end{array} & \begin{array}{c}\text { Television } \\ \text { Advertising } \\ \text { (\$1000s) }\end{array} & \begin{array}{c}\text { Newspaper } \\ \text { Advertising } \\ \text { (\$1000s) }\end{array} \\ 96 & 5.0 & 1.5 \\ 90 & 2.0 & 2.0 \\ 95 & 4.0 & 1.5 \\ 92 & 2.5 & 2.5 \\ 95 & 3.0 & 3.3 \\ 94 & 3.5 & 2.3 \\ 94 & 2.5 & 4.2 \\ 94 & 3.0 & 2.5\end{array}$
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. Predict weekly gross revenue for a week when 3500 is spent on television advertising and 2300 is spent on newspaper advertising?

Shu Naito

Numerade Educator

Problem 6

The National Football League (NFL) records a variety of performance data for individuals and teams. To investigate the importance of passing on the percentage of games won by a team, the following data show the conference (Conf), average number of passing yards per attempt (Yds/Att), the number of interceptions thrown per attempt (Int/Att), and the percentage of games won (Win\%) for a random sample of 16 NFL teams for one full season.
a. Develop the estimated regression equation that could be used to predict the percentage of games won given the average number of passing yards per attempt.
b. Develop the estimated regression equation that could be used to predict the percentage of games won given the number of interceptions thrown per attempt.
c. Develop the estimated regression equation that could be used to predict the percentage of games won given the average number of passing yards per attempt and the number of interceptions thrown per attempt.
d. The average number of passing yards per attempt for the Kansas City Chiefs was 6.2 and the number of interceptions thrown per attempt was $.036 .$ Use the estimated regression equation developed in part (c) to predict the percentage of games won by the Kansas City Chiefs. (Note: For this season the Kansas City Chiefs' record was 7 wins and 9 losses. Compare your prediction to the actual percentage of games won by the Kansas City Chiefs.

Dominador Tan

Numerade Educator

Problem 7

The United States Office of Personnel Management (OPM) manages the civil service of the federal government. Results from its annual Federal Employee Viewpoint Survey (FEVS) are used to measure employee satisfaction on several work aspects, including Job Satisfaction, Pay Satisfaction, Organization Satisfaction, and an overall measure of satisfaction referred to as Global Satisfaction. In each case a 100 -point scale with higher values indicating greater satisfaction is used (OPM website). Scores for Global Satisfaction, Job Satisfaction, Pay Satisfaction, and Organization Satisfaction for a sample of
65 employees are provided in the file Satisfaction.
Develop the estimated multiple regression equation that can be used to predict the Global Satisfaction score using the Job Satisfaction, Pay Satisfaction, and Organization Satisfaction scores.
b. Predict the overall Global Satisfaction score for an employee with a Job Satisfaction score of $72,$ a Pay Satisfaction score of $54,$ and an Organization Satisfaction score of 53

Dominador Tan

Numerade Educator

Problem 8

The Condé Nast Traveler Gold List provides ratings for the top 20 small cruise ships. The following data shown are the scores each ship received based upon the results from Condé Nast Traveler's annual Readers' Choice Survey. Each score represents the percentage of respondents who rated a ship as excellent or very good on several criteria, including Shore Excursions and Food/Dining. An overall score is also reported and used to rank the ships. The highest ranked ship, the Seabourn Odyssey, has an overall score of $94.4,$ the highest component of which is 97.8 for Food/Dining.
$\begin{array}{lccc}\text { Ship } & \text { Sverall } & \text { Shore } & \\ \text { Seabourn Odyssey } & 94.4 & \text { Excursions } & \text { Food/Dining } \\ \text { Seabourn Pride } & 93.0 & 90.9 & 97.8 \\ \text { National Geographic Endeavor } & 92.9 & 84.2 & 96.7 \\ \text { Seabourn Sojourn } & 91.3 & 100.0 & 88.5 \\ \text { Paul Gauguin } & 90.5 & 94.8 & 97.1 \\ \text { Seabourn Legend } & 90.3 & 87.9 & 91.2 \\ \text { Seabourn Spirit } & 90.2 & 82.1 & 98.8 \\ \text { Silver Explorer } & 89.9 & 86.3 & 92.0 \\ \text { Silver Spirit } & 89.4 & 92.6 & 88.9 \\ \text { Seven Seas Navigator } & 89.2 & 85.9 & 90.8 \\ \text { Silver Whisperer } & 89.2 & 83.3 & 90.5 \\ \text { National Geographic Explorer } & 89.1 & 82.0 & 88.6 \\ \text { Silver Cloud } & 88.7 & 93.1 & 89.7 \\ \text { Celebrity Xpedition } & 87.2 & 78.3 & 91.3 \\ \text { Silver Shadow } & 87.2 & 91.7 & 73.6 \\ \text { Silver Wind } & 86.6 & 75.0 & 89.7 \\ \text { SeaDream II } & 86.2 & 78.1 & 91.6 \\ \text { Wind Star } & 86.1 & 77.4 & 90.9 \\ \text { Wind Surf } & 86.1 & 76.5 & 91.5 \\ \text { Wind Spirit } & 85.2 & 72.3 & 89.3 \\ \end{array}$
a. Determine an estimated regression equation that can be used to predict the overall score given the score for Shore Excursions.
b. Consider the addition of the independent variable Food/Dining. Develop the estimated regression equation that can be used to predict the overall score given the scores for Shore Excursions and Food/Dining.
c. Predict the overall score for a cruise ship with a Shore Excursions score of 80 and a Food/Dining Score of 90 .

Dominador Tan

Numerade Educator

Problem 9

Spring is a peak time for selling houses. The file SpringHouses contains the selling price, number of bathrooms, square footage, and number of bedrooms of 26 homes sold in Ft. Thomas, Kentucky, in spring 2018 (realtor.com website).
a. Develop scatter plots of selling price versus number of bathrooms, selling price versus square footage, and selling price versus number of bedrooms. Comment on the relationship between selling price and these three variables.
b. Develop an estimated regression equation that can be used to predict the selling price given the three independent variables (number of baths, square footage, and number of bedrooms).
c. It is argued that we do not need both number of baths and number of bedrooms. Develop an estimated regression equation that can be used to predict selling price given square footage and the number of bedrooms.
d. Suppose your house has four bedrooms and is 2650 square feet. What is the predicted selling price using the model developed in part (c).

Shu Naito

Numerade Educator

Problem 10

Major League Baseball (MLB) consists of teams that play in the American League and the National League. MLB collects a wide variety of team and player statistics. Some of the statistics often used to evaluate pitching performance are as follows:

ERA: The average number of earned runs given up by the pitcher per nine innings. An earned run is any run that the opponent scores off a particular pitcher except for runs scored as a result of errors.
SO/IP: The average number of strikeouts per inning pitched.
HR/IP: The average number of home runs per inning pitched.
R/IP: The number of runs given up per inning pitched.
The following data show values for these statistics for a random sample of 20 pitchers from the American League for a full season. Develop an estimated regression equation that can be used to predict the average number of runs given up per inning given the average number of strikeouts per inning pitched.
b. Develop an estimated regression equation that can be used to predict the average number of runs given up per inning given the average number of home runs per inning pitched.
c. Develop an estimated regression equation that can be used to predict the average number of runs given up per inning given the average number of strikeouts per inning pitched and the average number of home runs per inning pitched.
A. A. J. Burnett, a pitcher for the New York Yankees, had an average number of strikeouts per inning pitched of .91 and an average number of home runs per inning of $16 .$ Use the estimated regression equation developed in part (c) to predict the average number of runs given up per inning for A. J. Burnett. (Note: The actual value for R/IP was .6.)
e. Suppose a suggestion was made to also use the earned run average as another independent variable in part (c). What do you think of this suggestion?

Dominador Tan

Numerade Educator

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 $\mathrm{SST}$ and $\mathrm{SSR}$ are 6724.125 and $6216.375,$ respectively.
a. Find SSE.
b. Compute $R^{2}$
c. Compute $R_{\mathrm{a}}^{2}$
d. Comment on the goodness of fit.

Dominador Tan

Numerade Educator

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. 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

Numerade Educator

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 $\mathrm{SST}$ and $\mathrm{SSR}$ are 1805 and $1760,$ respectively.
a. Compute $R^{2}$
b. Compute $R_{\mathrm{a}}^{2}$
c. Comment on the goodness of fit.

Rashmi Sinha

Numerade Educator

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, $\mathrm{SST}=16,000$ 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

Numerade Educator

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 $\left(x_{1}\right)$ and newspaper advertising $\left(x_{2}\right) .$ The estimated regression equation was
$$\hat{y}=83.2+2.29 x_{1}+1.30 x_{2}$$
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

Numerade Educator

Problem 16

In exercise $6,$ data were given on the average number of passing yards per attempt (Yds/Att), the number of interceptions thrown per attempt (Int/Att), and the percentage of games won (Win\%) for a random sample of 16 National Football League (NFL) teams for one full season.
a. Did the estimated regression equation that uses only the average number of passing yards per attempt as the independent variable to predict the percentage of games won provide a good fit?
b. Discuss the benefit of using both the average number of passing yards per attempt and the number of interceptions thrown per attempt to predict the percentage of games won.

Dominador Tan

Numerade Educator

Problem 17

Revisit exercise $9,$ where we develop an estimated regression equation that can be used to predict the selling price given the number of bathrooms, square footage, and number of bedrooms in the house.
a. Does the estimated regression equation provide a good fit to the data? Explain.
b. In part (c) of exercise 9 you developed an estimated regression equation that predicts selling price given the square footage and number of bedrooms. Compare the fit for this simpler model to that of the model that also includes number of bathrooms as an independent variable.

Shu Naito

Numerade Educator

Problem 18

Refer to exercise $10,$ where Major League Baseball (MLB) pitching statistics were reported for a random sample of 20 pitchers from the American League for one full season.
a. In part (c) of exercise $10,$ an estimated regression equation was developed relating the average number of runs given up per inning pitched given the average number of strikeouts per inning pitched and the average number of home runs per inning pitched. 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.
c. Suppose the earned run average (ERA) is used as the dependent variable in part (c) instead of the average number of runs given up per inning pitched. Does the estimated regression equation that uses the ERA provide a good fit to the data? Explain.

Shu Naito

Numerade Educator

Problem 19

In exercise $1,$ the following estimated regression equation based on 10 observations was presented.
$$\hat{y}=29.1270+.5906 x_{1}+.4980 x_{2}$$
Here $\mathrm{SST}=6724.125, \mathrm{SSR}=6216.375, s_{b_{1}}=.0813,$ and $s_{b_{2}}=.0567$
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

Numerade Educator

Problem 20

Refer to the data presented in exercise $2 .$ The estimated regression equation for these data is
$$\hat{y}=-18.4+2.01 x_{1}+4.74 x_{2}$$
Here $\mathrm{SST}=15,182.9, \mathrm{SSR}=14,052.2, s_{b_{1}}=.2471,$ and $s_{b_{2}}=.9484$
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. Is $\beta_{2}$ significant? Use $\alpha=.05$

Dominador Tan

Numerade Educator

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

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 $\mathrm{SST}=16,000$ 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

Numerade Educator

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 $/ \mathrm{or} \beta_{2}$ 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

Numerade Educator

Problem 24

The National Football League (NFL) records a variety of performance data for individuals and teams. A portion of the data showing the average number of passing yards obtained per game on offense (OffPass $Y \mathrm{ds} / \mathrm{G}$ ), the average number of yards given up per game on defense (DefYds/G), and the percentage of games won (Win\%) for one full season follows.
$\begin{array}{lccc}\text { Team } & \text { OffPassYds/G } & \text { DefYds/G } & \text { Win\% } \\ \text { Arizona } & 222.9 & 355.1 & 50.0 \\ \text { Atlanta } & 262.0 & 333.6 & 62.5 \\ \text { Baltimore } & 213.9 & 288.9 & 75.0 \\ \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot \\ \text { St. Louis } & 179.4 & 358.4 & 12.5 \\ \text { Tampa Bay } & 228.1 & 394.4 & 25.0 \\ \text { Tennessee } & 245.2 & 355.1 & 56.3 \\ \text { Washington } & 235.8 & 339.8 & 31.3\end{array}$
a. Develop an estimated regression equation that can be used to predict the percentage of games won given the average number of passing yards obtained per game on offense and the average number of yards given up per game on defense.
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?

Shu Naito

Numerade Educator

Problem 25

The Honda Accord was named the best midsized car for resale value for 2018 by the Kelley Blue Book (Kelley Blue Book website). The file AutoResale contains mileage, age, and selling price for a sample of 33 Honda Accords.
a. Develop an estimated regression equation that predicts the selling price of a used Honda Accord given the mileage and age of the car.
b. Is multicollinearity an issue for this model? Find the correlation between the independent variables to answer this question.
c. Use the $F$ test to determine the overall significance of the relationship. What is your conclusion at the .05 level of significance?
d. Use the $t$ test to determine the significance of each independent variable. What is your conclusion at the .05 level of significance?

Shu Naito

Numerade Educator

Problem 26

In exercise $10,$ data showing the values of several pitching statistics for a random sample of 20 pitchers from the American League of Major League Baseball were provided. In part (c) of this exercise an estimated regression equation was developed to predict the average number of runs given up per inning pitched (R/IP) given the average number of strikeouts per inning pitched (SO/IP) and the average number of home runs per inning pitched (HR/IP).
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?

Dominador Tan

Numerade Educator

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. Predict an individual value of $y$ when $x_{1}=180$ and $x_{2}=310$.

ME

Mina Eskandar

Numerade Educator

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 point estimate of 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 $x_{2}=15$

Shu Naito

Numerade Educator

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
$$\hat{y}=83.23+2.29 x_{1}+1.30 x_{2}$$
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 $95 \%$ prediction interval for next week's revenue, assuming that the advertising expenditures will be allocated as in part (a).

Shu Naito

Numerade Educator

Problem 30

In exercise $24,$ an estimated regression equation was developed relating the percentage of games won by a team in the National Football League during a complete season to the average number of passing yards obtained per game on offense and the average number of yards given up per game on defense during the season. a. Predict the percentage of games won for a particular team that averages 225 passing yards per game on offense and gives up an average of 300 yards per game on defense.
b. Develop a $95 \%$ prediction interval for the percentage of games won for a particular team that averages 225 passing yards per game on offense and gives up an average of 300 yards per game on defense.

Shu Naito

Numerade Educator

Problem 31

Refer to exercise $25 .$ Use the estimated regression equation from part (a) to answer the following questions.
a. Estimate the selling price of a four-year-old Honda Accord with mileage of 40,000 miles.
a. Develop a $95 \%$ confidence interval for the selling price of a car with the data in part (a).
b. Develop a $95 \%$ prediction interval for the selling price of a particular car having the data in part (a).

Shu Naito

Numerade Educator

Problem 32

Consider a regression study involving a dependent variable $y,$ a quantitative independent variable $x_{1}$, and a categorical independent 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.

VK

Victor Kilel

Numerade Educator

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

Numerade Educator

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
$$\begin{aligned}x_{1} &=\text { number of competitors within one mile } \\x_{2} &=\text { population within one mile (1000s) } \\x_{3} &=\left\{\begin{array}{l}1 \text { if drive-up window present } \\0 \text { otherwise }\end{array}\right.\\y=\operatorname{sales}(\$ 1000 \mathrm{~s})\end{aligned}$$ 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 1 mile, and a drive-up window.

Donna Densmore

Numerade Educator

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. $\begin{array}{llll}\begin{array}{l}\text { Repair Time } \\ \text { in Hours }\end{array} & \begin{array}{l}\text { Months since } \\ \text { Last Service }\end{array} & \text { Type of Repair } & \text { Repairperson } \\ 2.9 & 2 & \text { Electrical } & \text { Dave Newton } \\ 3.0 & 6 & \text { Mechanical } & \text { Dave Newton } \\ 4.8 & 8 & \text { Electrical } & \text { Bob Jones } \\ 1.8 & 3 & \text { Mechanical } & \text { Dave Newton } \\ 2.9 & 2 & \text { Electrical } & \text { Dave Newton } \\ 4.9 & 7 & \text { Electrical } & \text { Bob Jones } \\ 4.2 & 9 & \text { Mechanical } & \text { Bob Jones } \\ 4.8 & 8 & \text { Mechanical } & \text { Bob Jones } \\ 4.4 & 4 & \text { Electrical } & \text { Bob Jones } \\ 4.5 & 6 & \text { Electrical } & \text { Dave Newton }\end{array}$
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.
boes 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.
Does the equation that you developed in part (c) provide a good fit for the observed data? Explain.

Dominador Tan

Numerade Educator

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

Numerade Educator

Problem 37

Best Buy, a nationwide retailer of electronics, computers, and appliances, sells several brands of refrigerators. A random sample of models of full size refrigerators prices sold by Best Buy and the corresponding cubic feet (cu. ft.) and list price follow (Best Buy website). a. Develop the estimated simple linear regression equation to show how list price is related to the independent variable cubic feet.
b. At the .05 level of significance, test whether the estimated regression equation developed in part (a) indicates a significant relationship between list price and cubic feet.
c. Develop a dummy variable that will account for whether the refrigerator has the thru-the-door ice and water feature. Code the dummy variable with a value of 1 if the refrigerator has the thru-the-door ice and water feature and with 0 otherwise. Use this dummy variable to develop the estimated multiple regression equation to show how list price is related to cubic feet and the thru-the-door ice and water feature.
d. At $\alpha=.05$, is the thru-the-door ice and water feature a significant factor in the list price of a refrigerator?

Dominador Tan

Numerade Educator

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 smoker variable, 1 indicates a smoker and 0 indicates 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 $\alpha=.05$.
c. What is the probability of a stroke over the next 10 years for Art Speen, a 68 -yearold smoker who has blood pressure of $175 ?$ What action might the physician recommend for this patient?

Shu Naito

Numerade Educator

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 residuals against $\hat{y}$. Does the residual plot support the assumptions about $\epsilon$ ? Explain.
c. Plot the standardized residuals against $\hat{y}$. Do any outliers appear in these data? Explain.

Shu Naito

Numerade Educator

Problem 40

Data for two variables, $x$ and $y,$ follow.
$$\begin{array}{c|ccccc}\boldsymbol{x}_{i} & 22 & 24 & 26 & 28 & 40 \\\hline \boldsymbol{y}_{i} & 12 & 21 & 31 & 35 & 70\end{array}$$
a. Develop the estimated regression equation for these data.
b. Compute the standardized residuals for these data. Can any of these observations be classified as an outlier? Explain.
c. Develop a standardized residual plot against $\hat{y}$. Does the residual plot support the assumptions about $\epsilon$ ? Explain.

Shu Naito

Numerade Educator

Problem 41

Exercise 5 gave the following data on weekly gross revenue 1000, television advertising expenditures (\$1000s), and newspaper advertising expenditures (\$1000s) for Showtime Movie Theaters.
a. Find an estimated regression equation relating weekly gross revenue to television advertising expenditures and newspaper advertising expenditures.
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?

Shu Naito

Numerade Educator

Problem 42

The following table reports the price, horsepower, and $1 / 4$ -mile speed for 16 popular sports and GT cars.
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?

Shu Naito

Numerade Educator

Problem 43

The Ladies Professional Golfers Association (LPGA) maintains statistics on performance and earnings for members of the LPGA Tour. Yearend performance statistics for 134 golfers for 2014 appear in the file 2014 LPGAStats (LPGA website). Earnings 1000 is the total earnings in thousands of dollars; Scoring Avg. is the scoring average for all events; Greens in Reg. is the percentage of time a player is able to hit the greens 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 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 scoring average given the percentage of time a player is able to hit the greens 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

Numerade Educator

Problem 44

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. Predict 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.

Donna Densmore

Numerade Educator

Problem 45

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 $\begin{aligned} x_{1} &=\text { length of service (years) } \\ x_{2} &=\text { wage rate (dollars) } \\ y &=\text { job satisfaction test score (higher scores } \end{aligned}$ indicate greater job satisfaction)
a. Interpret the coefficients in this estimated regression equation.
b. Predict the job satisfaction test score for an employee who has four years of service and makes $\$ 6.50$ per hour.

Donna Densmore

Numerade Educator

Problem 46

A partial computer output from a regression analysis using Excel's Regression tool follows.
a. Compute the missing entries in this output.
b. Using $\alpha=.05,$ test for overall significance.
c. Use the $t$ test and $\alpha=.05$ to test $H_{0}: \beta_{1}=0$ and $H_{0}: \beta_{2}=0$

Shu Naito

Numerade Educator

Problem 47

Analyzing College Grad Point Average. Recall that in exercise $44,$ 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 Excel Regression tool output follows.
a. Complete the missing entries in this output.
b. Using $\alpha=.05,$ test for overall significance.
c. 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

Numerade Educator

Problem 48

Analyzing Job Satisfaction. Recall that in exercise 45 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 $\begin{aligned} x_{1}=& \text { length of service (years) } \\ x_{2}=& \text { wage rate (dollars) } \\ y=& \text { job satisfaction test score (higher scores } \\ & \text { indicate greater job satisfaction) } \end{aligned}$
a. Complete the missing entries in this output.
b. Using $\alpha=.05,$ test for overall significance.
c. 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 $H_{0}: \beta_{2}=0$

Shu Naito

Numerade Educator

Problem 49

For the holiday season of 2017 , nearly 59 percent of consumers planned to buy gift cards. According to the National Retail Federation, millennials like to purchase gift cards (Dayton Daily News website). Consider the sample data in the file Gift Cards. The following data are given for a sample of 600 millennials: the amount they reported spending on gift cards over the last year, annual income, marital status $(1=$ yes $, 0=$ no $),$ and whether they are male $(1=$ yes, $0=$ no $)$
a. Develop an estimated regression equation that predicts annual spend on gift cards given annual income, marital status, and gender.
b. Test the overall significance at the .05 level.
c. Test the significance of cach individual variable using a .05 level of significance.

Shu Naito

Numerade Educator

Problem 50

The Cincinnati Zoo and Botanical Gardens had a record attendance of 1.87 million visitors in 2017 (Cincinnati Business Courier website). Nonprofit organizations such as zoos and museums are becoming more sophisticated in their use of data to improve the customer experience. Being able to better estimate expected revenue is one use of analytics that allows nonprofits to better manage their operations. The file ZooSpend contains sample data on zoo attendance. The file contains the following data on 125 visits by families to a zoo: amount spent, size of the family, the distance the family lives from the zoo (the gate attendee asks for the zip code of each family entering the $\mathrm{z} 00$ ), and whether or not the family has a zoo membership $(1=$ yes $, 0=\mathrm{no})$
a. Develop an estimated regression equation that predicts the amount of money spent by a family given family size, whether or not it has a zoo membership, and the distance the family lives from the zoo.
b. Test the significance of the zoo membership independent variable at the .05 level.
c. Give an explanation for the sign of the estimate you tested in part (b).
d. Test the overall significance of the model at the .05 level.
e. Estimate the amount of money spent in a visit by a family of five that lives 125 miles from the $z 00$ and does not have a zoo membership.

Shu Naito

Numerade Educator

Problem 51

a. Develop an estimated simple linear regression equation that can be used to predict the Buy Again rating given the Tread Wear rating. At the .01 level of significance, test for a significant relationship. Does this estimated regression equation provide a good fit to the data? Explain.
b. Develop an estimated multiple regression equation that can be used to predict the Buy Again rating given the Tread Wear rating and the Dry Traction rating. Is the addition of the Dry Traction independent variable significant at $\alpha=.01 ?$ Explain.
c. Develop an estimated multiple regression equation that can be used to predict the Buy Again rating given the Tread Wear rating, the Dry Traction rating, and the Steering rating. Is the addition of the Steering independent variable significant at $\alpha=.01 ?$ Explain. The Tire Rack, an 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 Tire Rack website). The following data show survey ratings ( 1 to 10 scale with 10 the highest rating) for 18 high-performance all-season tires. The variable Tread Wear rates quickness of wear based on the driver's expectations, the variable Dry Traction rates the grip of a tire on a dry road, the variable Steering rates the tire's steering responsiveness, and the variable Buy Again rates the driver's desire to purchase the same tire again.

Shu Naito

Numerade Educator

Problem 52

Percentage in the NBA. The National Basketball Association (NBA) records a variety of statistics for each team. Five of these statistics are the percentage of games won (Win\%), the percentage of field goals made (FG\%), the percentage of three-point shots made $(3 \mathrm{P} \%),$ the percentage of free throws made $(\mathrm{FT} \%)$ the average number of offensive rebounds per game (RBOff), and the average number of defensive rebounds per game (RBDef). The data contained in the file $N B A$ Stats show the values of these statistics for the 30 teams in the NBA for one full season. A portion of the data follows. a. Develop an estimated regression equation that can be used to predict the percentage of games won given the percentage of field goals made. At the .05 level of significance, test for a significant relationship.
b. Provide an interpretation for the slope of the estimated regression equation developed in part (a).
c. Develop an estimated regression equation that can be used to predict the percentage of games won given the percentage of field goals made, the percentage of three-point shots made, the percentage of free throws made, the average number of offensive rebounds per game, and the average number of defensive rebounds per game (RBDef).
d. For the estimated regression equation developed in part (c), remove any independent variables that are not significant at the .05 level of significance and develop a new estimated regression equation using the remaining independent variables.
e. Assuming the estimated regression equation developed in part (d) can be used for the $2012-2013$ season, predict the percentage of games won for a team with the following values for the four independent variables: $\mathrm{FG} \%=45,3 \mathrm{P} \%=35$, $\mathrm{RBOff}=12,$ and $\mathrm{RBDef}=30$

Shu Naito

Numerade Educator