Essentials of Modern Business Statistics

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

Chapter 14 Simple Linear Regression - all with Video Answers

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Chapter Questions

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Problem 1

Given are five observations for two variables, $x$ and $y$.
\begin{tabular}{r|rrrrr}
$x_{i}$ & 1 & 2 & 3 & 4 & 5 \\
\hline$y_{i}$ & 3 & 7 & 5 & 11 & 14
\end{tabular}
Develop a scatter diagram for these data.
b. What does the scatter diagram developed in part (a) indicate about the relationship between the two variables?
c. Try to approximate the relationship between $x$ and $y$ by drawing a straight line through the data.
d. Develop the estimated regression equation by computing the values of $b_{0}$ and $b_{1}$ using equations (14.6) and (14.7)
e. Use the estimated regression equation to predict the value of $y$ when $x=4$

Victor Salazar

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Problem 2

Given are five observations for two variables, $x$ and $y$.
\begin{tabular}{c|rrrrr}
$\boldsymbol{x}_{i}$ & 3 & 12 & 6 & 20 & 14 \\
\hline $\boldsymbol{y}_{i}$ & 55 & 40 & 55 & 10 & 15
\end{tabular}
a. Develop a scatter diagram for these data.
b. What does the scatter diagram developed in part (a) indicate about the relationship between the two variables?
c. Try to approximate the relationship between $x$ and $y$ by drawing a straight line through the data.
d. Develop the estimated regression equation by computing the values of $b_{0}$ and $b_{1}$ using equations ( 14.6 ) and (14.7).
Use the estimated regression equation to predict the value of $y$ when $x=10$.

Victor Salazar

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Problem 3

Given are five observations collected in a regression study on two variables. \begin{tabular}{c|crrrr}
$x_{i}$ & 2 & 6 & 9 & 13 & 20 \\
\hline$y_{i}$ & 7 & 18 & 9 & 26 & 23
\end{tabular}
a. Develop a scatter diagram for these data.
b. Develop the estimated regression equation for these data.
c. Use the estimated regression equation to predict the value of $y$ when $x=6$.

Victor Salazar

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Problem 4

4. Retail and Trade: Female Managers. The following data give the percentage of women working in five companies in the retail and trade industry. The percentage of management jobs held by women in each company is also shown. \begin{tabular}{l|lllll}
\% Working & 67 & 45 & 73 & 54 & 61 \\
\hline \% Management & 49 & 21 & 65 & 47 & 33
\end{tabular}
a. Develop a scatter diagram for these data with the percentage of women working in the company as the independent variable.
b. What does the scatter diagram developed in part (a) indicate about the relationship between the two variables?
c. Try to approximate the relationship between the percentage of women working in the company and the percentage of management jobs held by women in that company.
d. Develop the estimated regression equation by computing the values of $b_{0}$ and $b_{1}$
e. Predict the percentage of management jobs held by women in a company that has $60 \%$ women employees.

Victor Salazar

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Problem 5

Production Line Speed and Quality Control. Brawdy Plastics, Inc., produces plastic seat belt retainers for General Motors at their plant in Buffalo, New York. After final assembly and painting, the parts are placed on a conveyor belt that moves the parts past a final inspection station. How fast the parts move past the final inspection station depends upon the line speed of the conveyor belt (feet per minute). Although faster line speeds are desirable, management is concerned that increasing the line speed too much may not provide enough time for inspectors to identify which parts are actually defective. To test this theory, Brawdy Plastics conducted an experiment in which the same batch of parts, with a known number of defective parts, was inspected using a variety of line speeds. The following data were collected.
\begin{tabular}{ll}
Line & Number of \\
Speed & Defective \\
20 & Parts Found \\
20 & 23 \\
30 & 21 \\
30 & 19 \\
40 & 16 \\
40 & 15 \\
50 & 17 \\
50 & 14 \\
\hline
\end{tabular}
a. Develop a scatter diagram with the line speed as the independent variable.
b. What does the scatter diagram developed in part (a) indicate about the relationship between the two variables?
c. Use the least squares method to develop the estimated regression equation.
d. Predict the number of defective parts found for a line speed of 25 feet per minute.

Victor Salazar

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Problem 6

Passing and Winning in the NFL. 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 average number of passing yards per attempt (Yds/Att) and the percentage of games won (WinPct) for a random sample of $10 \mathrm{NFL}$ teams for the 2011 season (NFL website).
$\begin{array}{lcc}\text { Team } & \text { Yds/Att } & \text { WinPct } \\ \text { Arizona Cardinals } & 6.5 & 50 \\ \text { Atlanta Falcons } & 7.1 & 63 \\ \text { Carolina Panthers } & 7.4 & 38 \\ \text { Chicago Bears } & 6.4 & 50 \\ \text { Dallas Cowboys } & 7.4 & 50 \\ \text { New England Patriots } & 8.3 & 81 \\ \text { Philadelphia Eagles } & 7.4 & 50 \\ \text { Seattle Seahawks } & 6.1 & 44 \\ \text { St. Louis Rams } & 5.2 & 13 \\ \text { Tampa Bay Buccaneers } & 6.2 & 25\end{array}$
a. Develop a scatter diagram with the number of passing yards per attempt on the horizontal axis and the percentage of games won on the vertical axis.
b. What does the scatter diagram developed in part (a) indicate about the relationship between the two variables?
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.
d. Provide an interpretation for the slope of the estimated regression equation.
e. For the 2011 season, the average number of passing yards per attempt for the Kansas City Chiefs was $6.2 .$ Use the estimated regression equation developed in part (c) to predict the percentage of games won by the Kansas City Chiefs. (Note: For the 2011 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.

Victor Salazar

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Problem 7

Sales Experience and Performance. A sales manager collected the following data on annual sales for new customer accounts and the number of years of experience for a sample of 10 salespersons.
$\begin{array}{ccc} & \text { Years of } & \text { Annual Sales } \\ \text { Salesperson } & \text { Experience } & (51000 \mathrm{~s}) \\ 1 & 1 & 80 \\ 2 & 3 & 97 \\ 3 & 4 & 92 \\ 4 & 4 & 102 \\ 5 & 6 & 103 \\ 6 & 8 & 111 \\ 7 & 10 & 119 \\ 8 & 10 & 123 \\ 9 & 11 & 117 \\ 10 & 13 & 136\end{array}$
a. Develop a scatter diagram for these data with years of experience as the independent variable.
b. Develop an estimated regression equation that can be used to predict annual sales given the years of experience.
c. Use the estimated regression equation to predict annual sales for a salesperson with 9 years of experience.

Victor Salazar

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Problem 8

Broker Satisfaction. The American Association of Individual Investors (AAII) On-Line Discount Broker Survey polls members on their experiences with discount brokers. As part of the survey, members were asked to rate the quality of the speed of execution with their broker as well as provide an overall satisfaction rating for electronic trades. Possible responses (scores) were no opinion (0), unsatisfied (I), somewhat satisfied (2), satisfied (3), and very satisfied (4). For each broker summary scores were computed by calculating a weighted average of the scores provided by each respondent. A portion of the survey results follow (AAII website, February 7,2012 ).
$\begin{array}{lcc}\text { Brokerage } & \text { Speed } & \text { Satisfaction } \\ \text { Scottrade, Inc. } & 3.4 & 3.5 \\ \text { Charles Schwab } & 3.3 & 3.4 \\ \text { Fidelity Brokerage Services } & 3.4 & 3.9 \\ \text { TD Ameritrade } & 3.6 & 3.7 \\ \text { E-Trade Financial } & 3.2 & 2.9 \\ \text { Vanguard Brokerage Services } & 3.8 & 2.8 \\ \text { USAA Brokerage Services } & 3.8 & 3.6 \\ \text { Thinkorswim } & 2.6 & 2.6 \\ \text { Wells Fargo Investments } & 2.7 & 2.3 \\ \text { Interactive Brokers } & 4.0 & 4.0 \\ \text { Zecco.com } & 2.5 & 2.5\end{array}$
a. Develop a scatter diagram for these data with the speed of execution as the independent variable.
b. What does the scatter diagram developed in part (a) indicate about the relationship between the two variables?
c. Develop the least squares estimated regression equation.
d. Provide an interpretation for the slope of the estimated regression equation.
e. Suppose Zecco.com developed new software to increase their speed of execution rating. If the new software is able to increase their speed of execution rating from the current value of 2.5 to the average speed of execution rating for the other 10 brokerage firms that were surveyed, what value would you predict for the overall satisfaction rating?

Victor Salazar

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Problem 9

Estimating Landscaping Expenditures. David's Landscaping has collected data on home values (in thousands of $\$$ ) and expenditures (in thousands of $\$$ ) on landscaping with the hope of developing a predictive model to help marketing to potential new clients. Data for 14 households may be found in the file Landscape.
a. Develop a scatter diagram with home value as the independent variable.
b. What does the scatter plot developed in part (a) indicate about the relationship between the two variables?
c. Use the least squares method to develop the estimated regression equation.
d. For every additional $\$ 1000$ in home value, estimate how much additional will be spent on landscaping.
e. Use the equation estimated in part (c) to predict the landscaping expenditures for a home valued at $\$ 575,000$.

Victor Salazar

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Problem 10

Age and the Price of Wine. For a particular red wine, the following data show the auction price for a 750 -milliliter bottle and the age of the wine as of June 2016 (WineX website).
$\begin{array}{cc}\text { Age (years) } & \text { Price (\$) } \\ 36 & 256 \\ 20 & 142 \\ 29 & 212 \\ 33 & 255 \\ 41 & 331 \\ 27 & 173 \\ 30 & 209 \\ 45 & 297 \\ 34 & 237 \\ 22 & 182\end{array}$
a. Develop a scatter diagram for these data with age as the independent variable.
b. What does the scatter diagram developed in part (a) indicate about the relationship between age and price?
c. Develop the least squares estimated regression equation.
d. Provide an interpretation for the slope of the estimated equation.

Victor Salazar

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Problem 11

Laptop Ratings. To help consumers in purchasing a laptop computer, Consumer Reports calculates an overall test score for each computer tested based upon rating factors such as ergonomics, portability, performance, display, and battery life. Higher overall scores indicate better test results. The following data show the average retail price and the overall score for ten 13 -inch models (Consumer Reports website).
$\begin{array}{lcc}\text { Brand \& Model } & \begin{array}{c}\text { Price } \\ \text { (\$) }\end{array} & \begin{array}{c}\text { Overall } \\ \text { Score }\end{array} \\ \text { Samsung Ultrabook NP900X3C-A01US } & 1250 & 83 \\ \text { Apple MacBook Air MC965LL/A } & 1300 & 83 \\ \text { Apple MacBook Air MD231LLA } & 1200 & 82 \\ \text { HP ENVY 13-2050nr Spectre XT } & 950 & 79 \\ \text { Sony VAIO SVS13112FXB } & 800 & 77 \\ \text { Acer Aspire S5-391-9880 Ultrabook } & 1200 & 74 \\ \text { Apple MacBook Pro MD101LL/A } & 1200 & 74 \\ \text { Apple MacBook Pro MD313LL/A } & 1000 & 73 \\ \text { Dell Inspiron 113Z-6591SLV } & 700 & 67 \\ \text { Samsung NP535U3C-A01US } & 600 & 63\end{array}$ a. Develop a scatter diagram with price as the independent variable.
b. What does the scatter diagram developed in part (a) indicate about the relationship between the two variables?
c. Use the least squares method to develop the estimated regression equation.
d. Provide an interpretation of the slope of the estimated regression equation.
e. Another laptop that Consumer Reports tested is the Acer Aspire $\mathrm{S} 3-951-6646 \mathrm{Ul}$ trabook; the price for this laptop was $\$ 700 .$ Predict the overall score for this laptop using the estimated regression equation developed in part (c).

Victor Salazar

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Problem 12

Stock Beta. In June 2016 , Yahoo Finance reported the beta value for Coca-Cola was .82 (Yahoo Finance website). Betas for individual stocks are determined by simple linear regression. The dependent variable is the total return for the stock, and the independent variable is the total return for the stock market, such as the return of the $\mathrm{S} \& \mathrm{P} 500$. The slope of this regression equation is referred to as the stock's beta. Many financial analysts prefer to measure the risk of a stock by computing the stock's beta value. The data contained in the file CocaCola show the monthly percentage returns for the S\&P 500 and the Coca-Cola Company for August 2015 to May 2016 .
$\begin{array}{lll} & \text { S\&P } 500 & \text { Coca-Cola } \\ \text { Month } & \text { \% Return } & \text { \% Return } \\ \text { August } & -3 & 3 \\ \text { September } & 8 & 6 \\ \text { October } & 0 & 1 \\ \text { November } & -2 & 1 \\ \text { December } & -5 & 0 \\ \text { January } & 0 & 0 \\ \text { February } & 7 & 8 \\ \text { March } & 0 & -3 \\ \text { April } & 2 & 0 \\ \text { May } & -5 & -1\end{array}$
a. Develop a scatter diagram with the $\mathrm{S} \& \mathrm{P} \%$ Return as the independent variable.
b. What does the scatter diagram developed in part (a) indicate about the relationship between the returns of the $\mathrm{S} \& \mathrm{P} 500$ and those of the Coca-Cola Company?
c. Develop the least squares estimated regression equation.
d. Provide an interpretation for the slope of the estimated equation (i.e., the beta).
e. Is your beta estimate close to $.82 ?$ If not, why might your estimate be different?

Victor Salazar

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Problem 13

Auditing Itemized Tax Deductions. A large city hospital conducted a study to investigate the relationship between the number of unauthorized days that employees are absent per year and the distance (miles) between home and work for the employees. A sample of 10 employees was selected and the following data were collected.
a. Develop a scatter diagram for these data. Does a linear relationship appear reasonable? Explain.
b. Develop the least squares estimated regression equation that relates the distance to work to the number of days absent.
c. Predict the number of days absent for an employee that lives 5 miles from the hospital.

Victor Salazar

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Problem 14

GPS Navigators. Consumer Reports conducted extensive tests of GPS-based navigators and developed an overall rating based on factors such as ease of use, driver information, display, and battery life. The following data show the price and rating for a sample of $20 \mathrm{GPS}$ units with a 4.3 -inch screen that Consumer Reports tested (Consumer Reports website).
a. Develop a scatter diagram with price as the independent variable.
b. What does the scatter diagram developed in part (a) indicate about the relationship between the two variables?
c. Use the least squares method to develop the estimated regression equation.
d. Predict the rating for a GPS system with a 4.3 -inch screen that has a price of $\$ 200$.

Victor Salazar

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Problem 15

The data from exercise 1 follow.
$$
\begin{array}{l|lllrr}
\boldsymbol{x}_{i} & 1 & 2 & 3 & 4 & 5 \\
\hline \boldsymbol{y}_{i} & 3 & 7 & 5 & 11 & 14
\end{array}
$$
The estimated regression equation for these data is $\hat{y}=.20+2.60 x$.
a. Compute SSE, SST, and SSR using equations (14.8), (14.9), and (14.10).
b. Compute the coefficient of determination $r^{2}$. Comment on the goodness of fit.
c. Compute the sample correlation coefficient.

Victor Salazar

Numerade Educator

05:43

Problem 16

The data from exercise 2 follow.
$$
\begin{array}{r|rrrrr}
\boldsymbol{x}_{i} & 3 & 12 & 6 & 20 & 14 \\
\hline \boldsymbol{y}_{i} & 55 & 40 & 55 & 10 & 15
\end{array}
$$
The estimated regression equation for these data is $\hat{y}=68-3 x$
a. Compute $\mathrm{SSE}, \mathrm{SST},$ and $\mathrm{SSR} .$
b. Compute the coefficient of determination $r^{2}$. Comment on the goodness of fit.
c. Compute the sample correlation coefficient.

Srikar Katta

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Problem 17

The data from exercise 3 follow.
$$
\begin{array}{l|rrrrr}
x_{i} & 2 & 6 & 9 & 13 & 20 \\
\hline y_{i} & 7 & 18 & 9 & 26 & 23
\end{array}
$$
The estimated regression equation for these data is $\hat{y}=7.6+.9 x .$ What percentage of the total sum of squares can be accounted for by the estimated regression equation? What is the value of the sample correlation coefficient?

Victor Salazar

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Problem 18

Consumer Reports: Headphones. The following data show the brand, price (\$), and the overall score for six stereo headphones that were tested by Consumer Reports (Consumer Reports website). The overall score is based on sound quality and effectiveness of ambient noise reduction. Scores range from 0 (lowest) to 100 (highest). The estimated regression equation for these data is $\hat{y}=23.194+.318 x,$ where $x=$ price
$(\$)$ and $y=$ overall score.
a. Compute SST, SSR, and SSE.
b. Compute the coefficient of determination $r^{2}$. Comment on the goodness of fit.
c. What is the value of the sample correlation coefficient?

Victor Salazar

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Problem 19

Sales Experience and Sales Performance. In exercise 7 a sales manager collected the following data on $x=$ annual sales and $y=$ years of experience. The estimated regression equation for these data is $\hat{y}=80+4 x$
a. Compute $\mathrm{SST}, \mathrm{SSR},$ and $\mathrm{SSE}$.
b. Compute the coefficient of determination $r^{2}$. Comment on the goodness of fit.
c. What is the value of the sample correlation coefficient?

Victor Salazar

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Problem 20

The cost of Lighter Racing Bikes. Bicycling, the world's leading cycling magazine, reviews hundreds of bicycles throughout the year. Their "Road-Race" category contains reviews of bikes used by riders primarily interested in racing. One of the most important factors in selecting a bike for racing is the weight of the bike. The following data show the weight (pounds) and price (\$) for 10 racing bikes reviewed by the magazine (Bicycling website).
a. Use the data to develop an estimated regression equation that could be used to estimate the price for a bike given the weight.
b. Compute $r^{2}$. Did the estimated regression equation provide a good fit?
c. Predict the price for a bike that weighs 15 pounds.

Victor Salazar

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Problem 21

cost Estimation. An important application of regression analysis in accounting is in the estimation of cost. By collecting data on volume and cost and using the least squares method to develop an estimated regression equation relating volume and cost, an accountant can estimate the cost associated with a particular manufacturing volume. Consider the following sample of production volumes and total cost data for a manufacturing operation.
a. Use these data to develop an estimated regression equation that could be used to predict the total cost for a given production volume.
b. What is the variable cost per unit produced?
c. Compute the coefficient of determination. What percentage of the variation in total cost can be explained by production volume?
d. The company's production schedule shows 500 units must be produced next month. Predict the total cost for this operation.

Victor Salazar

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Problem 22

Rental Car Revenue. The following data were used to investigate the relationship between the number of cars in service ( 1000 s) and the annual revenue (\$ millions) for six smaller car rental companies (Auto Rental News website).
With $x=$ cars in service $(1000 \mathrm{~s})$ and $y=$ annual revenue $(\$$ millions $),$ the estimated regression equation is $\hat{y}=-17.005+12.966 x .$ For these data $\mathrm{SSE}=1043.03$
a. Compute the coefficient of determination $r^{2}$.
b. Did the estimated regression equation provide a good fit? Explain.
c. What is the value of the sample correlation coefficient? Does it reflect a strong or weak relationship between the number of cars in service and the annual revenue?

Victor Salazar

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Problem 23

The data from exercise 1 follow.
$$
\begin{array}{r|rrrrr}
x_{i} & 1 & 2 & 3 & 4 & 5 \\
\hline y_{i} & 3 & 7 & 5 & 11 & 14
\end{array}
$$
a. Compute the mean square error using equation (14.15)
b. Compute the standard error of the estimate using equation (14.16)
c. Compute the estimated standard deviation of $b_{1}$ using equation (14.18).
d. Use the $t$ test to test the following hypotheses $(\alpha=.05)$
$$
\begin{array}{l}
H_{0}: \beta_{1}=0 \\
H_{\mathrm{a}}: \beta_{1} \neq 0
\end{array}
$$
e. Use the $F$ test to test the hypotheses in part (d) at a .05 level of significance. Present the results in the analysis of variance table format.

Victor Salazar

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Problem 24

$$
\begin{aligned}
&\text { The data from exercise } 2 \text { follow. }\\
&\begin{array}{r|rrrrr}
\boldsymbol{x}_{i} & 3 & 12 & 6 & 20 & 14 \\
\hline \boldsymbol{y}_{i} & 55 & 40 & 55 & 10 & 15
\end{array}
\end{aligned}
$$
a. Compute the mean square error using equation (14.15).
b. Compute the standard error of the estimate using equation (14.16).
c. Compute the estimated standard deviation of $b_{1}$ using equation (14.18).
d. Use the $t$ test to test the following hypotheses $(\alpha=.05)$
$$
\begin{array}{l}
H_{0}: \beta_{1}=0 \\
H_{\mathrm{a}}: \beta_{1} \neq 0
\end{array}
$$
e. Use the $F$ test to test the hypotheses in part (d) at a .05 level of significance. Present the results in the analysis of variance table format.

Victor Salazar

Numerade Educator

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Problem 25

$$
\begin{aligned}
&\text { The data from exercise } 3 \text { follow. }\\
&\begin{array}{r|rrrrr}
\boldsymbol{x}_{i} & 2 & 6 & 9 & 13 & 20 \\
\hline \boldsymbol{y}_{\boldsymbol{i}} & 7 & 18 & 9 & 26 & 23
\end{array}
\end{aligned}
$$
a. What is the value of the standard error of the estimate?
b. Test for a significant relationship by using the $t$ test. Use $\alpha=.05$.
c. Use the $F$ test to test for a significant relationship. Use $\alpha=.05 .$ What is your conclusion?

Victor Salazar

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Problem 26

Headphones. In exercise 18 the data on price $(\$)$ and the overall score for six stereo headphones tested by Consumer Reports were as follows (Consumer Reports website).
a. Does the $t$ test indicate a significant relationship between price and the overall score? What is your conclusion? Use $\alpha=.05$.
b. Test for a significant relationship using the $F$ test. What is your conclusion? Use $\alpha=.05$
c. Show the ANOVA table for these data.

Victor Salazar

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Problem 27

College GPA and Salary. Do students with higher college grade point averages (GPAs) earn more than those graduates with lower GPAs (Civic Science)? Consider the college GPA and salary data ( 10 years after graduation) provided in the file GPASalary.
a. Develop a scatter diagram for these data with college GPA as the independent variable. What does the scatter diagram indicate about the relationship between the two variables?
b. Use these data to develop an estimated regression equation that can be used to predict annual salary 10 years after graduation given college GPA.
c. At the .05 level of significance, does there appear to be a significant statistical relationship between the two variables?

Victor Salazar

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Problem 28

Broker Satisfaction Conclusion. In exercise 8 , ratings data on $x=$ the quality of the speed of execution and $y=$ overall satisfaction with electronic trades provided the estimated regression equation $\hat{y}=.2046+.9077 x$ (AAII website). At the .05 level of significance, test whether speed of execution and overall satisfaction are related. Show the ANOVA table. What is your conclusion?

Victor Salazar

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Problem 29

cost Estimation Conclusion. Refer to exercise $21,$ where data on production volume and cost were used to develop an estimated regression equation relating production volume and cost for a particular manufacturing operation. Use $\alpha=.05$ to test whether the production volume is significantly related to the total cost. Show the ANOVA table. What is your conclusion?

Victor Salazar

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Problem 30

Rental Car Revenue and Fleet Size. Refer to exercise $9,$ where the following data were used to investigate the relationship between the number of cars in service ( $1000 \mathrm{~s}$ ) and the annual revenue (\$ millions) for six smaller car rental companies (Auto Rental News website).
With $x=$ cars in service $(1000 \mathrm{~s})$ and $y=$ annual revenue $(\$$ millions $),$ the estimated regression equation is $\hat{y}=-17.005+12.966 x .$ For these data $\mathrm{SSE}=1043.03$ and $\mathrm{SST}=10,568 .$ Do these results indicate a significant relationship between the number of cars in service and the annual revenue?

Victor Salazar

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Problem 31

Racing Bike Significance of Weight on cost. In exercise $20,$ data on $x=$ weight (pounds) and $y=$ price (\$) for 10 road-racing bikes provided the estimated regression equation $\hat{y}=28,574-1439 x$ ( Bicycling website). For these data $\mathrm{SSE}=7,102,922.54$ and $\mathrm{SST}=52,120,800 .$ Use the $F$ test to determine whether the weight for a bike and the price are related at the .05 level of significance.

Victor Salazar

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Problem 32

$$
\begin{aligned}
&\text { The data from exercise } 1 \text { follow. }\\
&\begin{array}{c|ccccc}
\boldsymbol{x}_{i} & 1 & 2 & 3 & 4 & 5 \\
\hline \boldsymbol{y}_{i} & 3 & 7 & 5 & 11 & 14
\end{array}
\end{aligned}
$$
a. Use equation (14.23) to estimate the standard deviation of $\hat{y}^{\text {? }}$ when $x=4$.
b. Use expression (14.24) to develop a $95 \%$ confidence interval for the expected value of $y$ when $x=4$
c. Use equation ( 14.26 ) to estimate the standard deviation of an individual value of $y$ when $x=4$
d. Use expression (14.27) to develop a $95 \%$ prediction interval for $y$ when $x=4$

Victor Salazar

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Problem 33

$$
\begin{aligned}
&\text { The data from exercise } 2 \text { follow. }\\
&\begin{array}{r|rrrrr}
\boldsymbol{x}_{i} & 3 & 12 & 6 & 20 & 14 \\
\hline \boldsymbol{y}_{i} & 55 & 40 & 55 & 10 & 15
\end{array}
\end{aligned}
$$
a. Estimate the standard deviation of $\hat{y}$ when $x=8$.
b. Develop a $95 \%$ confidence interval for the expected value of $y$ when $x=8$.
c. Estimate the standard deviation of an individual value of $y$ when $x=8$.
d. Develop a $95 \%$ prediction interval for $y$ when $x=8$.

Victor Salazar

Numerade Educator

11:23

Problem 34

$$
\begin{aligned}
&\text { The data from exercise } 3 \text { follow. }\\
&\begin{array}{c|crrrr}
\boldsymbol{x}_{i} & 2 & 6 & 9 & 13 & 20 \\
\hline \boldsymbol{y}_{i} & 7 & 18 & 9 & 26 & 23
\end{array}
\end{aligned}
$$
Develop the $95 \%$ confidence and prediction intervals when $x=12 .$ Explain why these two intervals are different.

Srikar Katta

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Problem 35

Restaurant Lines. Many small restaurants in Portland, Oregon, and other cities across the United States do not take reservations. Owners say that with smaller capacity, noshows are costly, and they would rather have their staff focused on customer service rather than maintaining a reservation system (pressherald.com). However, it is important to be able to give reasonable estimates of waiting time when customers arrive and put their name on the waiting list. The file Restaurantline contains 40 observations of number of people in line ahead of a customer (independent variable $x$ ) and actual waiting time (dependent variable $y$ ). The estimated regression equation is:
$\hat{y}=4.35+8.81 x$ and $\mathrm{MSE}=94.42$
a. Develop a point estimate for a customer who arrive with three people on the wait-list.
b. Develop a $95 \%$ confidence interval for the mean waiting time for a customer who arrives with three customers already in line.
c. Develop a $95 \%$ prediction interval for Roger and Sherry Davy's waiting time if there are three customers in line when they arrive.
d. Discuss the difference between parts (b) and (c).

Victor Salazar

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Problem 36

Sales Performance. In exercise $7,$ the data on $y=$ annual sales (\$ 1000 s) for new customer accounts and $x=$ number of years of experience for a sample of 10 salespersons provided the estimated regression equation $\hat{y}=80+4 x .$ For these data $\bar{x}=7$ $\Sigma\left(x_{i}-\bar{x}\right)^{2}=142,$ and $s=4.6098$
a. Develop a $95 \%$ confidence interval for the mean annual sales for all salespersons with nine years of experience.
b. The company is considering hiring Tom Smart, a salesperson with nine years of experience. Develop a $95 \%$ prediction interval of annual sales for Tom Smart.
c. Discuss the differences in your answers to parts (a) and (b).

Victor Salazar

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Problem 37

Auditing Itemized Deductions. In exercise $13,$ data were given on the adjusted gross income $x$ and the amount of itemized deductions taken by taxpayers. Data were reported in thousands of dollars. With the estimated regression equation $\hat{y}=4.68+.16 x$, the point estimate of a reasonable level of total itemized deductions for a taxpayer with an adjusted gross income of $\$ 52,500$ is $\$ 13,080$.
a. Develop a $95 \%$ confidence interval for the mean amount of total itemized deductions for all taxpayers with an adjusted gross income of $\$ 52,500 .$
b. Develop a $95 \%$ prediction interval estimate for the amount of total itemized deductions for a particular taxpayer with an adjusted gross income of $\$ 52,500$.
c. If the particular taxpayer referred to in part (b) claimed total itemized deductions of $\$ 20,400,$ would the IRS agent's request for an audit appear to be justified?
d. Use your answer to part (b) to give the IRS agent a guideline as to the amount of total itemized deductions a taxpayer with an adjusted gross income of $\$ 52,500$ should claim before an audit is recommended.

Victor Salazar

Numerade Educator

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Problem 38

Prediction Intervals for cost Estimation. Refer to exercise $21,$ where data on the production volume $x$ and total cost $y$ for a particular manufacturing operation were used to develop the estimated regression equation $\hat{y}=1246.67+7.6 x$.
a. The company's production schedule shows that 500 units must be produced next month. Predict the total cost for next month.
b. Develop a $99 \%$ prediction interval for the total cost for next month.
c. If an accounting cost report at the end of next month shows that the actual production cost during the month was $\$ 6000$, should managers be concerned about incurring such a high total cost for the month? Discuss.

Victor Salazar

Numerade Educator

02:16

Problem 39

Entertainment Spend. The Wall Street Journal asked Concur Technologies, Inc., an expense management company, to examine data from 8.3 million expense reports to provide insights regarding business travel expenses. Their analysis of the data showed that New York was the most expensive city. The following table shows the average daily hotel room rate $(x)$ and the average amount spent on entertainment (y) for a random sample of 9 of the 25 most-visited U.S. cities. These data lead to the estimated regression equation $\hat{y}=17.49+1.0334 x .$ For these data $\mathrm{SSE}=1541.4$
a. Predict the amount spent on entertainment for a particular city that has a daily room rate of $\$ 89$.
b. Develop a $95 \%$ confidence interval for the mean amount spent on entertainment for all cities that have a daily room rate of $\$ 89$.
c. The average room rate in Chicago is $\$ 128 .$ Develop a $95 \%$ prediction interval for the amount spent on entertainment in Chicago.

Neel Faucher

Numerade Educator

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Problem 40

Apartment Selling Price. The commercial division of a real estate firm conducted a study to determine the extent of the relationship between annual gross rents (\$1000s) and the selling price (\$ 1000 s) for apartment buildings. Data were collected on several properties sold, and Excel's Regression tool was used to develop an estimated regression equation. A portion of the regression output follows.
a. How many apartment buildings were in the sample?
b. Write the estimated regression equation.
c. Use the $t$ test to determine whether the selling price is related to annual gross rents. Use $\alpha=.05$.
d. Use the $F$ test to determine whether the selling price is related to annual gross rents. Use $\alpha=.05$
e. Predict the selling price of an apartment building with gross annual rents of $\$ 50,000$

Victor Salazar

Numerade Educator

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Problem 41

Computer Terminal Maintenance. Following is a portion of the regression output for an application relating maintenance expense (dollars per month) to usage (hours per week) for a particular brand of computer terminal.a. Write the estimated regression equation.
b. Use a $t$ test to determine whether monthly maintenance expense is related to usage at the .05 level of significance.
c. Did the estimated regression equation provide a good fit? Explain.

Victor Salazar

Numerade Educator

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Problem 42

Annual Sales and Salespeople. A regression model relating the number of salespersons at a branch office to annual sales at the office (in thousands of dollars) provided the following regression output.
a. Write the estimated regression equation.
b. Compute the $F$ statistic and test the significance of the relationship at the .05 level of significance.
c. Compute the $t$ statistic and test the significance of the relationship at the .05 level of significance.
d. Predict the annual sales at the Memphis branch office. This branch employs 12 salespersons.

Victor Salazar

Numerade Educator

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Problem 43

Estimating Setup Time. Sherry is a production manager for a small manufacturing shop and is interested in developing a predictive model to estimate the time to produce an order of a given size - that is, the total time to produce a certain quantity of the product. She has collected data on the total time to produce 30 different orders of various quantities in the file Setup.
a. Develop a scatter diagram with quantity as the independent variable.
b. What does the scatter diagram developed in part (a) indicate about the relationship between the two variables?
c. Develop the estimated regression equation. Interpret the intercept and slope.
d. Test for a significant relationship. Use .05 .
e. Did the estimated regression equation provide a good fit?

Victor Salazar

Numerade Educator

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Problem 44

Auto Racing Helmet. Automobile racing, high-performance driving schools, and driver education programs run by automobile clubs continue to grow in popularity. All these activities require the participant to wear a helmet that is certified by the Snell Memorial Foundation, a not-for-profit organization dedicated to research, education, testing, and development of helmet safety standards. Snell "SA" (Sports Application) rated professional helmets are designed for auto racing and provide extreme impact resistance and high fire protection. One of the key factors in selecting a helmet is weight, since lower weight helmets tend to place less stress on the neck. The following data show the weight and price for 18 SA helmets (SoloRacer website). a. Develop a scatter diagram with weight as the independent variable.
boes there appear to be any relationship between these two variables?
c. Develop the estimated regression equation that could be used to predict the price given the weight.
d. Test for the significance of the relationship at the .05 level of significance.
e. Did the estimated regression equation provide a good fit? Explain.

Victor Salazar

Numerade Educator

01:27

Problem 45

$$
\begin{aligned}
&\text { Given are data for two variables, } x \text { and } y \text { . }\\
&\begin{array}{c|ccccc}
\boldsymbol{x}_{i} & 6 & 11 & 15 & 18 & 20 \\
\hline \boldsymbol{y}_{i} & 6 & 8 & 12 & 20 & 30
\end{array}
\end{aligned}
$$
a. Develop an estimated regression equation for these data.
b. Compute the residuals.
c. Develop a plot of the residuals against the independent variable $x$. Do the assumptions about the error terms seem to be satisfied?
d. Compute the standardized residuals.
e. Develop a plot of the standardized residuals against $\hat{y}$. What conclusions can you draw from this plot?

Adriano Chikande

Numerade Educator

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Problem 46

The following data were used in a regression study.
a. Develop an estimated regression equation for these data.
b. Construct a plot of the residuals. Do the assumptions about the error term seem to be satisfied?

Victor Salazar

Numerade Educator

01:31

Problem 47

Restaurant Advertising and Revenue. Data on advertising expenditures and revenue (in thousands of dollars) for the Four Seasons Restaurant follow.
a. Let $x$ equal advertising expenditures and $y$ equal revenue. Use the method of least squares to develop a straight line approximation of the relationship between the two variables.
b. Test whether revenue and advertising expenditures are related at a .05 level of significance.
c. Prepare a residual plot of $y-\hat{y}$ versus $\hat{y}$. Use the result from part (a) to obtain the values of $\hat{y}$
d. What conclusions can you draw from residual analysis? Should this model be used, or should we look for a better one?

Adriano Chikande

Numerade Educator

02:27

Problem 48

Experience and Sales. Refer to exercise $7,$ where an estimated regression equation relating years of experience and annual sales was developed.
a. Compute the residuals and construct a residual plot for this problem.
b. Do the assumptions about the error terms seem reasonable in light of the residual plot?

Jameson Kuper

Numerade Educator

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Problem 49

cost of Living Index. The file cost Living contains the cost of living indexes and the population densities (number of people per square mile) for 61 cities in the United States. The cost of living index measures the cost of living in a particular city relative to the cost of living in New York City. San Francisco has an index of 112.15 , meaning that it is $12.15 \%$ more expensive to live in San Francisco than New York City. Washington, $\mathrm{DC},$ has an index of $84.64,$ which means that the cost to live in Washington, $\mathrm{DC},$ is only $84.64 \%$ of what it costs to live in New York City (Numbeo website).
a. Develop the estimated regression equation that can be used to predict the cost of living index for a U.S. city, given the city's population density.
b. Construct a residual plot against the independent variable.
c. Review the residual plot constructed in part (b). Do the assumptions of the error term and model form seem reasonable?

Victor Salazar

Numerade Educator

01:39

Problem 50

$$
\begin{aligned}
&\text { Consider the following data for two variables, } x \text { and } y\\
&\begin{array}{r|lllllll}
\boldsymbol{x}_{i} & 135 & 110 & 130 & 145 & 175 & 160 & 120 \\
\hline \boldsymbol{y}_{i} & 145 & 100 & 120 & 120 & 130 & 130 & 110
\end{array}
\end{aligned}
$$
a. Develop a scatter diagram for these data. Does the scatter diagram indicate any outliers in the data? In general, what implications does this finding have for simple linear regression?
b. Compute the standardized residuals for these data. Do the data include any outliers? Explain.

Adriano Chikande

Numerade Educator

01:59

Problem 51

$$
\begin{aligned}
&\text { Consider the following data for two variables, } x \text { and } y \text { . }\\
&\begin{array}{r|rrrrrrrr}
\boldsymbol{x}_{i} & 4 & 5 & 7 & 8 & 10 & 12 & 12 & 22 \\
\hline \boldsymbol{y}_{i} & 12 & 14 & 16 & 15 & 18 & 20 & 24 & 19
\end{array}
\end{aligned}
$$
a. Develop a scatter diagram for these data. Does the scatter diagram indicate any influential observations? Explain.
b. Compute the standardized residuals for these data. Do the data include any outliers? Explain.
c. Compute the leverage values for these data. Does there appear to be any influential observations in these data? Explain.

Adriano Chikande

Numerade Educator

00:01

Problem 52

Predicting Charity Expenses. Charity Navigator is America's leading independent charity evaluator. The following data show the total expenses (\$), the percentage of the total budget spent on administrative expenses, the percentage spent on fundraising, and the percentage spent on program expenses for 10 supersized charities (Charity Navigator website). Administrative expenses include overhead, administrative staff and associated costs, and organizational meetings. Fundraising expenses are what a charity spends to raise money, and program expenses are what the charity spends on the programs and services it exists to deliver. The sum of the three percentages does not add to $100 \%$ because of rounding.
a. Develop a scatter diagram with fundraising expenses (\%) on the horizontal axis and program expenses (\%) on the vertical axis. Looking at the data, do there appear to be any outliers and/or influential observations?
b. Develop an estimated regression equation that could be used to predict program expenses (\%) given fundraising expenses (\%).
c. Does the value for the slope of the estimated regression equation make sense in the context of this problem situation?
d. Use residual analysis to determine whether any outliers and/or influential observations are present. Briefly summarize your findings and conclusions.

Oluwadamilola Ameobi

Numerade Educator

03:54

Problem 53

Supermarket Checkout Lines. Retail chain Kroger has more than 2700 locations and is the largest supermarket in the United States based on revenue. Kroger has invested heavily in data, technology, and analytics. Feeding predictive models with data from an infrared sensor system called QueVision to anticipate when shoppers will reach the checkout counters, Kroger is able to alert workers to open more checkout lines as needed. This has allowed Kroger to lower its average checkout time from four minutes to less than 30 seconds (Retail Touchpoints). Consider the data in the file Checkout. The file contains 32 observations. Each observation gives the arrival time (measured in minutes before 6 P.M.) and the shopping time (measured in minutes).
a. Develop a scatter diagram for arrival time as the independent variable.
b. What does the scatter diagram developed in part (a) indicate about the relationship between the two variables? Do there appear to be any outliers or influential observations? Explain.
c. Using the entire data set, develop the estimated regression equation that can be used to predict the shopping time given the arrival time.
d. Use residual analysis to determine whether any outliers or influential observations are present.
e. After looking at the scatter diagram in part (a), suppose you were able to visually identify what appears to be an influential observation. Drop this observation from the data set and fit an estimated regression equation to the remaining data. Compare the estimated slope for the new estimated regression equation to the estimated slope obtained in part (c). Does this approach confirm the conclusion you reached in part (d)? Explain.

Stanley Enemuo

Numerade Educator

02:06

Problem 54

The Value of a Major League Baseball Team. The following data show the annual revenue (\$ millions) and the estimated team value (\$ millions) for the 30 Major League Baseball teams (Forbes website).
a. Develop a scatter diagram with Revenue on the horizontal axis and Value on the vertical axis. Looking at the scatter diagram, does it appear that there are any outliers and/or influential observations in the data?
b. Develop the estimated regression equation that can be used to predict team value given the value of annual revenue.
c. Use residual analysis to determine whether any outliers and/or influential observations are present. Briefly summarize your findings and conclusions.

Adriano Chikande

Numerade Educator

02:33

Problem 55

Stock Market Performance. The Dow Jones Industrial Average (DJIA) and the Standard \& Poor's 500 (S\&P 500) indexes are used as measures of overall movement in the stock market. The DJIA is based on the price movements of 30 large companies; the $\mathrm{S} \& \mathrm{P} 500$ is an index composed of 500 stocks. Some say the $\mathrm{S} \& \mathrm{P} 500$ is a better measure of stock market performance because it is broader based. The closing price for the DJIA and the S\&P 500 for 15 weeks, beginning with January $6,2012,$ follow (Barron's website).
a. Develop a scatter diagram with DJIA as the independent variable.
b. Develop the estimated regression equation.
c. Test for a significant relationship. Use $\alpha=.05$.
d. Did the estimated regression equation provide a good fit? Explain.
e. Suppose that the closing price for the DJIA is $13,500 .$ Predict the closing price for the $\mathrm{S\&P} 500$.
f. Should we be concerned that the DJIA value of 13,500 used to predict the $\mathrm{S} \& \mathrm{P}$ 500 value in part (e) is beyond the range of the data used to develop the estimated regression equation?

Dominador Tan

Numerade Educator

01:42

Problem 56

Home Size and Price. Is the number of square feet of living space a good predictor of a house's selling price? The following data show the square footage and selling price for 15 houses in Winston Salem, North Carolina, in April 2015 (Zillow.com).
a. Develop a scatter diagram with square feet of living space as the independent variable and selling price as the dependent variable. What does the scatter diagram indicate about the relationship between the size of a house and the selling price?
b. Develop the estimated regression equation that could be used to predict the selling price given the number of square feet of living space.
c. At the .05 level, is there a significant relationship between the two variables?
d. Use the estimated regression equation to predict the selling price of a 2000 square foot house in Winston Salem, North Carolina.
e. Do you believe the estimated regression equation developed in part (b) will provide a good prediction of selling price of a particular house in Winston Salem, North Carolina? Explain.
f. Would you be comfortable using the estimated regression equation developed in part (b) to predict the selling price of a particular house in Seattle, Washington? Why or why not?

Sheryl Ezze

Numerade Educator