Solved: Accuracy of software effort estimates. Refer to

Problem 170SE IQs and The Bell Curve . The Bell Curve (Free Press,1994), written by Richard Herrnstein and Charles Murray(H&M), is a controversial book about race, genes, IQ,and economic mobility. The book heavily employs statistics and statistical methodology in an attempt to supportthe authors’ positions on the relationships among thesevariables and their social consequences. The main themeof The Bell Curve can be summarized as follows: (1) Measured intelligence (IQ) is largely geneticallyinherited. (2) IQ is correlated positively with a variety of socioeconomic status success measures, such as prestigious job, high annual income, and high educationalattainment. (3) From 1 and 2, it follows that socioeconomic successes are largely genetically caused and thereforeresistant to educational and environmental interventions (such as affirmative action). The statistical methodology (regression) employedby the authors and the inferences derived from thestatistics were critiqued in Chance (Summer 1995) andThe Journal of the American Statistical Association (Dec.1995). The following are just a few of the problems withH&M’s use of regression that are identified: Problem 1 H&M consistently use a trio of independent variables—IQ, socioeconomic status, and age—in a series of first-order models designed to predictdependent social outcome variables such as income and unemployment. (Only on a single occasion are interaction terms incorporated.) Consider, for example, themodel E(y) = ?0 + ?1x1 + ?2x2 + ?3x3 where y = income, x1 = IQ, x2 = socioeconomic status,and x3 = age. H&M employ t-tests on the individual bparameters to assess the importance of the independentvariables. As with most of the models considered in TheBell Curve, the estimate of ?1 in the income model ispositive and statistically significant at ? = .05, and theassociated t-value is larger (in absolute value) than thet-values associated with the other independent variables.Consequently, H&M claim that IQ is a better predictorof income than the other two independent variables. Noattempt was made to determine whether the model wasproperly specified or whether the model provides an adequate fit to the data. Problem 2 In an appendix, the authors describemultiple regression as a “mathematical procedure thatyields coefficients for each of [the independent variables],indicating how much of a change in [the dependentvariable] can be anticipated for a given change in anyparticular [independent] variable, with all the others heldconstant.” Armed with this information and the fact thatthe estimate of ?1 in the model above is positive, H&Minfer that a high IQ necessarily implies (or causes) a highincome, and a low IQ inevitably leads to a low income.(Cause-and-effect inferences like this are made repeatedly throughout the book.) Problem 3 The title of the book refers to the normal distribution and its well-known “bell-shaped” curve.There is a misconception among the general public thatscores on intelligence tests (IQ) are normally distributed. In fact, most IQ scores have distributions thatare decidedly skewed. Traditionally, psychologists andpsychometricians have transformed these scores so thatthe resulting numbers have a precise normal distribution.H&M make a special point to do this. Consequently, themeasure of IQ used in all the regression models is normalized (i.e., transformed so that the resulting distributionis normal), despite the fact that regression methodologydoes not require predictor (independent) variables to benormally distributed. Problem 4 A variable that is not used as a predictor of social outcome in any of the models in The BellCurve is level of education. H&M purposely omit education from the models, arguing that IQ causes education,not the other way around. Other researchers who haveexamined H&M’s data report that when education isincluded as an independent variable in the model, theeffect of IQ on the dependent variable (say, income) isdiminished. a. Co/mment on each of the problems identified. Whydo each of these problems cast a shadow on the inferences made by the authors? b. Using the variables specified in the model above,describe how you would conduct the multiple regression analysis. (Propose a more complex modeland describe the appropriate model tests, including aresidual analysis.)

Problem 1E Write a first-order model relating E(y) to a. two quantitative independent variables. b. four quantitative independent variables. c. five quantitative independent variables

Problem 2E Minitab was used to fit the model E(y) = ?0 + ?1x1 + ?2x2 to n = 20 data points, and the printout shown on the next page was obtained. a. What are the sample estimates of ? 0, ? 1, and ? 2? b. What is the least squares prediction equation? c. Find SSE, MSE, and s. Interpret the standard deviation in the context of the problem. d. Test H0: ?1 = 0 against H0: ?1 ? 0. Use ? = .05. e. Use a 95% confidence interval to estimate ?2. f. Find R2 and R2a and interpret these values. g. Find the test statistic for testing H0: ?1 = ?2 = 0. h. Find the observed significance level of the test, part g. Interpret the result.

Problem 3E Suppose you fit the multiple regression model y = ?0 + ?1x1 + ?2x2 + ?3x3 + ? to n = 30 data points and obtain the following result: The estimated standard errors of are 1.86 and .29, respectively. a. Test the null hypothesis H0: ?2 = 0 against the alternative hypothesis Ha: ?2 ? 0. Use a = .05. b. Test the null hypothesis H0: ?3 = 0 against the alternative hypothesis Ha: ?3 ? 0. Use a = .05. c. The null hypothesis H0: ?2 = 0 is not rejected. In contrast, the null hypothesis H0: ?3 = 0 is rejected. Explain how this can happen even though .

Problem 4E Suppose you fit the first-order multiple regression model y = ? 0 + ? 1 x 1 + ? 2 x 2 + ? to n = 25 data points and obtain the prediction equation The estimated standard deviations of the sampling distributions of are 2.3 and .27, respectively. a. Test H0: ?1 = 0 against Ha: ?1 7 0. Use a = .05. b. Test H0: ?2 = 0 against Ha: ?2 ? 0. Use a = .05. c. Find a 90% confidence interval for ?1. Interpret the interval. d. Find a 99% confidence interval for ?2. Interpret the interval.

Problem 6E Consider the first-order model equation in three quantitative independent variables E ( y ) = 1 + 2 x 1 + x 2 - 3 x 3 a. Graph the relationship between y and x1 for x2 = 1 and x3 = 3. b. Repeat part a for x2 = -1 and x3 = 1. c. How do the graphed lines in parts a and b relate to each other? What is the slope of each line? d. If a linear model is first-order in three independent variables, what type of geometric relationship will you obtain when E(y) is graphed as a function of one of the independent variables for various combinations of values of the other independent variables?

How is the number of degrees of freedom available for estimating \(\sigma^2\) (the variance of \(\epsilon\)) related to the number of independent variables in a regression model?

Suppose you fit the first-order model \(y=\beta_{0}+\beta_{1} x_{1}+\beta_{2} x_{2}+\beta_{3} x_{3}+\beta_{4} x_{4}+\beta_{5} x_{5}+\varepsilon\) to n = 30 data points and obtain \(SSE=.33\ \ \ \ \ R^2=.92\) a. Do the values of SSE and \(R^2\) suggest that the model provides a good fit to the data? Explain. b. Is the model of any use in predicting y? Test the null hypothesis \(H_0:\ \beta_1=\ \beta_2\ =\ \beta_3\ =\ \beta_4\ =\ \beta_5\ =\ 0\) against the alternative hypothesis \(H_a:\) At least one of the parameters \(\beta_1,\ \beta_2,\ .\ .\ .\ ,\ \beta_5\) is nonzero. Use \(\alpha\ =\ .05\).

Problem 10E Forecasting movie revenues with Twitter. Refer to the IEEE International Conference on Web Intelligence and Intelligent Agent Technology (2010) study on using the volume of chatter on Twitter.com to forecast movie box office revenue, Exercise 11.23 (p. 617). Recall that opening weekend box office revenue data (in millions of dollars) were collected for a sample of 24 recent movies. In addition to each movie’s tweet rate, i.e., the average number of tweets referring to the movie per hour 1 week prior to the movie’s release, the researchers also computed the ratio of positive to negative tweets (called the PN-ratio). a. Give the equation of a first-order model relating revenue (y) to both tweet rate (x1) and PN-ratio (x2). b. Which b in the model, part a, represents the change in revenue (y) for every 1-tweet increase in the tweet rate (x1), holding PN-ratio (x2) constant? c. Which b in the model, part a, represents the change in revenue (y) for every 1-unit increase in the PN-ratio (x2), holding tweet rate (x1) constant? d. The following coefficients were reported: R2 = .945 and R2a = .940. Give a practical interpretation for both R2 and R2a. e. Conduct a test of the null hypothesis, H0:?1 = ?2 = 0. Use ? = .05. f. The researchers reported the p-values for testing H0: ?1 = 0 and H0: ?2 = 0 as both less than .0001. Interpret these results (use ? = .01). Forecasting movie revenues with Twitter. Marketers are keenly interested in how social media (e.g., Facebook, Twitter) may influence consumers who buy their products. Researchers at HP Labs (Palo Alto, CA) investigated whether the volume of chatter on Twitter.com could be used to forecast the box office revenues of movies (IEEE International Conference on Web Intelligence and Intelligent Agent Technology, 2010). Opening weekend box office revenue data (in smillions of dollars) were collected for a sample of 23 recent movies. In addition, the researchers computed each movie’s tweet rate, i.e., the average number of tweets (at Twitter.com) referring to the movie per hour 1 week prior to the movie’s release. The data (simulated based on information provided in the study) are listed in the accompanying table. Assuming that movie revenue and tweet rate are linearly related, how much do you estimate a movie’s opening weekend revenue to change as the tweet rate for the movie increases by an average of 100 tweets per hour? Tweet Rate Revenue (millions) 1365 .8 142 1212 .8 77 581 .5 61 310 .1 32 455 31 290 30 250 21 680 .5 18 150 18 164 .5 17 113 .9 16 144 .5 15 418 14 98 14 100 .8 12 115 .4 11 74 .4 10 87 .5 9 127 .6 9 52 .2 9 144 .1 8 41 .3 2 2.75 0.3

If the analysis of variance F-test leads to the conclusion that at least one of the model parameters is nonzero, can you conclude that the model is the best predictor for the dependent variable y? Can you conclude that all of the terms in the model are important for predicting y? What is the appropriate conclusion?

Problem 11E Accounting and Machiavellianism. Refer to the Behavioral Research in Accounting (Jan. 2008) study of Machiavellian traits (e.g., manipulation, cunning, duplicity, deception, and bad faith) in accountants, Exercise 9.8 (p. 492). Recall that a Mach rating score was determined for each in a sample of accounting alumni of a large southwestern university. For one portion of the study, the researcher modeled an accountant’s Mach score (y) as a function of age, gender, education, and income. Data on n = 198 accountants yielded the results shown in the table. a. Conduct a test of overall model utility. Use a = .05. b. Interpret the coefficient of determination, R2. c. Is there sufficient evidence (at a = .05) to say that income a. is a statistically useful predictor of Mach score? Accounting and Machiavellianism. A study of Machiavellian traits in accountants was published in Behavioral Research in Accounting (January 2008). Recall (from Exercise 1.31, p. 27) that Machiavellian describes negative character traits such as manipulation, cunning, duplicity, deception, and bad faith. A Mach rating score was determined for each in a sample of accounting alumni of a large southwestern university. The accountants were then classified as having high, moderate, or low Mach rating scores. For one portion of the study, the researcher investigated the impact of both Mach score classification and gender on the average income of an accountant. For this experiment, identify each of the following: a. Experimental unit b. Response variable c. Factors d. Levels of each factor e. Treatments Accounting and Machiavellianism. Behavioral Research in Accounting (January 2008) published a study of Machiavellian traits in accountants. Machiavellian describes negative character traits that include manipulation, cunning, duplicity, deception, and bad faith. A questionnaire was administered to a random sample of 700 accounting alumni of a large southwestern university; however, due to nonresponse and incomplete answers, only 198 questionnaires could be analyzed. Several variables were measured, including age, gender, level of education, income, job satisfaction score, and Machiavellian (“Mach”) rating score. The research findings suggest that Machiavellian behavior is not required to achieve success in the accounting profession. a. What is the population of interest to the researcher? b. What type of data (quantitative or qualitative) is produced a. by each of the variables measured? b. Identify the sample. c. Identify the data-collection method used. d. What inference was made by the researcher? e. How might the nonresponses impact the inference?

Usability professionals salary survey. The Usability Professionals’ Association (UPA) supports people who research, design, and evaluate the user experience of products and services. Recently, the UPA conducted a salary survey of its members (UPA Salary Survey, August 18, 2009). One of the report’s authors, Jeff Sauro, investigated how much having a PhD affects salaries in this profession and discussed his analysis on the blog, www.measuringusability. com. Sauro fit a first-order multiple regression model for salary (y, in dollars) as a function of years of experience (\(x_1\)), PhD status (\(x_2 = 1\) if PhD, 0 if not), and manager status (\(x_3 = 1\) if manager, 0 if not). The following prediction equation was obtained: \(\hat{y}=52,484+2,941 x_{1}+16,880 x_{2}+11,108 x_{3}\) a. Predict the salary of a UPA member with 10 years of experience who does not have a PhD, but is a manager. b. Predict the salary of a UPA member with 10 years of experience who does have a PhD, but is not a manager. c. The following coefficient was reported: \(R_a^2 = .32\). Give a practical interpretation of this value. d. A 95% confidence interval for \(\beta_1\) was reported as (2700, 3200). Give a practical interpretation of this result. e. A 95% confidence interval for \(\beta_2\) was reported as (11,500, 22,300). Give a practical interpretation of this result. f. A 95% confidence interval for \(\beta_3\) was reported as (7,600, 14,600). Give a practical interpretation of this result.

Problem 13E Highway crash data analysis. Researchers at Montana State University have written a tutorial on an empirical method for analyzing before and after highway crash data (Montana Department of Transportation, Research Report, May 2004). The initial step in the methodology is to develop a Safety Performance Function (SPF)—a mathematical model that estimates crash occurrence for a given roadway segment. Using data collected for over 100 roadway segments, the researchers fit the model, E(y) = ?0 + ?1x1 + ?2x2, where y = number of crashes per 3 years, x1 = roadway length (miles), and x2 = AADT = average annual daily traffic (number of vehicles). The results are shown in the following tables. Interstate Highways Variable Parameter Estimate Standard Error t-value Intercept 1.81231 .50568 3.58 Length (x1) .10875 .03166 3.44 AADT (x2) .00017 .00003 5.19 Noninterstate Highways Variable Parameter Estimate Standard Error t-value Intercept 1.20785 .28075 4.30 Length (x1) .06343 .01809 3.51 AADT (x2) .00056 .00012 4.86 a. Give the least squares prediction equation for the interstate highway model. b. Give practical interpretations of the b estimates, part a. c. Refer to part a. Find a 99% confidence interval for ?1 and interpret the result. d. Refer to part a. Find a 99% confidence interval for ?2 and interpret the result. e. Repeat parts a–d for the noninterstate highway model.

Problem 12E Characteristics of lead users. During new product development, companies often involve “lead users,” i.e., creative individuals who are on the leading edge of an important market trend. Creativity and Innovation Management (Feb. 2008) published an article on identifying the social network characteristics of lead users of children’s computer games. Data were collected for n = 326 children, and the following variables were measured: lead-user rating (y, measured on a 5-point scale), gender (x1 = 1 if female, 0 if male), age (x2, years), degree of centrality (x3, measured as the number of direct ties to other peers in the network), and betweenness centrality (x4, measured as the number of shortest paths between peers). A first-order model for y was fit to the data, yielding the following least squares prediction equation: a. Give two properties of the errors of prediction that result from using the method of least squares to obtain the parameter estimates. b. Give a practical interpretation of the estimate of ?4 in the model. c. A test of H0: ?4 = 0 resulted in a p-value of .002. Make the appropriate conclusion at a = .05.

Novelty of a vacation destination. Many tourists choose a NW vacation destination based on the newness or uniqueness (i.e., the novelty) of the itinerary. Texas A\&M University professor J. Petrick investigated the relationship between novelty and vacationing golfers' demographics (Annals of Tourism Research, Vol. 29, 2002). Data were obtained from a mail survey of 393 golf vacationers to a large coastal resort in the southeastern United States. Several measures of novelty level (on a numerical scale) were obtained for each vacationer, including "change from routine," "thrill," "boredom-alleviation," and "surprise." The researcher employed four independent variables in a regression model to predict each of the novelty measures. The independent variables were \(x_{1}=\) number of rounds of golf per year, \(x_{2}=\) total number of golf vacations taken, \(x_{3}=\) number of years played golf, and \(x_{4}=\) average golf score. a. Give the hypothesized equation of a first-order model for y = change from routine. b. A test of \(H_{0}: \beta_{3}=0\) versus \(H_{\mathrm{a}}: \beta_{3}<0\) yielded a p-value of .005. Interpret this result if \(\alpha=.01\). c. The estimate of \(\beta_{3}\) was found to be negative. Based on this result (and the result of part b), the researcher concluded that "those who have played golf for more years are less apt to seek change from their normal routine in their golf vacations." Do you agree with this statement? Explain. d. The regression results for three dependent novelty measures, based on data collected for n = 393 golf vacationers, are summarized in the table below. Give the null hypothesis for testing the overall adequacy of the first-order regression model. e. Give the rejection region for the test, part d, for \(\alpha=.01\). f. Use the test statistics reported in the table and the rejection region from part e to conduct the test for each of the dependent measures of novelty. g. Verify that the p-values reported in the table support your conclusions in part f. h. Interpret the values of \(R^{2}\) reported in the table. Text Transcription: x_1 = x_2 = x_3 = x_4 = H_{0}: beta_3 = 0 H_a: beta_3 < 0 alpha = .01 beta_3 R^2

Predicting runs scored in baseball. In Chance (Fall 2000), statistician Scott Berry built a multiple regression model for predicting total number of runs scored by a Major League Baseball team during a season. Using data on all teams over a 9-year period (a sample of n = 234), the results in the following table were obtained. a. Write the least squares prediction equation for y = total number of runs scored by a team in a season. b. Interpret, practically, the \(\beta\) estimates in the model. c. Conduct a test of \(H_0: \beta_7=0\) against \(H_{\mathrm{a}}: \beta_7<0\) at \(\alpha=.05\). Interpret the results. d. Form a 95% confidence interval for \(\beta_5\). Interpret the results.

Problem 16E Arsenic in groundwater. Environmental Science & Technology (Jan. 2005) reported on a study of the reliability of a commercial kit to test for arsenic in groundwater. The field kit was used to test a sample of 328 groundwater wells in Bangladesh. In addition to the arsenic level (micrograms per liter), the latitude (degrees), longitude (degrees), and depth (feet) of each well were measured. The data are saved in the file. (The first and last 5 observations are listed in the table below.) a. Write a first-order model for arsenic level (y) as a function of latitude, longitude, and depth. b. Fit the model to the data using the method of least squares. c. Give practical interpretations of the ? estimates. d. Find the model standard deviation, s, and interpret its value. e. Find and interpret the values of R2 and R2a. f. Conduct a test of overall model utility at ? = .05. g. Based on the results, parts d–f, would you recommend using the model to predict arsenic level (y)? Explain.

Problem 20E Extracting water from oil. In the oil industry, water that mixes with crude oil during production and transportation must be removed. Chemists have found that the oil can be extracted from the water/oil mix electrically. Researchers at the University of Bergen (Norway) conducted a series of experiments to study the factors that influence the voltage (y) required to separate the water from the oil (Journal of Colloid and Interface Science, Aug. 1995). The seven independent variables investigated in the study are listed in the table below. (Each variable was measured at two levels—a “low” level and a “high” level.) Sixteen water/oil mixtures were prepared using different combinations of the independent variables; then each emulsion was exposed to a high electric field. In addition, three mixtures were tested when all independent variables were set to 0. The data for all 19 experiments are saved in the file (selected observations are shown in the table at the bottom of the page). a. Propose a first-order model for y as a function of all seven independent variables. b. Use a statistical software package to fit the model to the data in the table. c. Fully interpret the ? estimates. d. Evaluate the overall utility of the model at ? = .10.

Problem 18E Contamination from a plant’s discharge. Refer to the U.S. Army Corps of Engineers data (Example 1.5, p. 13) on fish contaminated from the toxic discharges of a chemical plant located on the banks of the Tennessee River in Alabama. Recall that the engineers measured the length (in centimeters), weight (in grams), and DDT level (in parts per million) for 144 captured fish. In addition, the number of miles upstream from the river was recorded. The data are saved in the file. (The first and last five observations are shown in the table in the next column.) a. Fit the first-order model, E(y) = ?0 + ?1x1 + ?2x2 + ?3x3, to the data, where y = DDT level, x1 = mile, x2 = length, and x3 = weight. Report the least squares prediction equation. b. Find the estimate of the standard deviation of e for the model and give a practical interpretation of its value. c. Conduct a test of the global utility of the model. Use ? = .05. d. Do the data provide sufficient evidence to conclude that DDT level increases as length increases? Report the observed significance level of the test and reach a conclusion using ? = .05. e. Find and interpret a 95% confidence interval for ?3.

Problem 17E Reality TV and cosmetic surgery. How much influence does the media, especially reality television programs, have on one’s decision to undergo cosmetic surgery? This was the question of interest to psychologists who published an article in ?ody Image: An International Journal of Research (March 2010). In the study, 170 college students answered questions about their impressions of reality TV shows featuring cosmetic surgery, level of self-esteem, satisfaction with one’s own body, and desire to have cosmetic surgery to alter one’s body. The variables analyzed in the study were measured as follows: DESIRE—scale ranging from 5 to 25, where the higher the value, the greater the interest in having cosmetic surgery; GENDER—1 if male, 0 if female; SELFESTM—scale ranging from 4 to 40, where the higher the value, the greater the level of self-esteem; ?ODYSAT—scale ranging from 1 to 9, where the higher the value, the greater the satisfaction with one’s own body; and IMPREAL—scale ranging from 1 to 7, where the higher the value, the more one believes reality television shows featuring cosmetic surgery are realistic. The data for the study (simulated based on statistics reported in the journal article) are saved in the file. Selected observations are listed in the next table on page 683. The psychologists used multiple regression to model desire to have cosmetic surgery (y) as a function of gender (x1), self-esteem (x2),body satisfaction (x3), and impression of reality TV (x4). a. Fit the first-order model, E(y) = ?0 + ?1x1 + ?2x2 + ?3x3 + ?4x4, to the data in the file. Give the least squares prediction equation. b. Interpret the ? estimates in the words of the problem. Data for Exercise 12.17 (first and last five observations) c. Is the overall model statistically useful for predicting desire to have cosmetic surgery? Test using ? = .01. d. Which statistic, R2 or R2a, is the preferred measure of model fit? Practically interpret the value of this statistic. a. Conduct a test to determine whether desire to have cosmetic surgery decreases linearly as level of body satisfaction increases. Use ? = .05. e. Find a 95% confidence interval for ?4. Practically interpret the result.

Occupational safety study. An important goal in occupational safety is “active caring.” Employees demonstrate active caring (AC) about the safety of their coworkers when they identify environmental hazards and unsafe work practices and then implement appropriate corrective actions for these unsafe conditions or behaviors. Three factors hypothesized to increase the propensity for an employee to actively care for safety are (1) high self-esteem, (2) optimism, and (3) group cohesiveness. Applied & Preventive Psychology (Winter 1995) attempted to establish empirical support for the AC hypothesis by fitting the model \(E(y)=\beta_0+\beta_1x_1+\beta_2x_2+\beta_3x_3\), where y = AC score (active caring score, 15-point scale) \(x_1\) = self-esteem score \(x_2\) = Optimism score \(x_3\) = Group cohesion score The regression analysis, based on data collected for n = 31 hourly workers at a large fiber-manufacturing plant, yielded a multiple coefficient of determination of \(R^2 = .362\). a. Interpret the value of \(R^2\). b. Use the \(R^2\) value to test the global utility of the model. Use \(\alpha\ =\ .05\).

\(R^2\) and model fit. Because the coefficient of determination always increases when a new independent variable is added to the model, it is tempting to include many variables in a model to force \(R^2\) to be near 1 . However, doing so reduces the degrees of freedom available for estimating \(\sigma^2\), which adversely affects our ability to make reliable inferences. Suppose you want to use 18 economic indicators to predict next year's gross domestic product (GDP). You fit the model \(y=\beta_{0}+\beta_{1} x_{1}+\beta_{2} x_{2}+\cdots+\beta_{17} x_{17}+\beta_{18} x_{18}+\varepsilon\) where y = GDP and \(x_1,\ x_2,\ .\ .\ .\ ,\ x_{18}\) are the economic indicators. Only 20 years of data (n = 20) are used to fit the model, and you obtain \(R^2\ =\ .95\). Test to see whether this impressive-looking \(R^2\) is large enough for you to infer that the model is useful-that is, that at least one term in the model is important for predicting GDP. Use \(\alpha\ =\ .05\).

Problem 19E Cooling method for gas turbines. Refer to the Journal of Engineering for Gas Turbines and Power (Jan. 2005) study of a high-pressure inlet fogging method for a gas turbine engine, Exercise 7.40 (p. 378). Recall that the heat rate (kilo - joules per kilowatt per hour) was measured for each in a sample of 67 gas turbines augmented with high-pressure inlet fogging. In addition, several other variables were measured, including cycle speed (revolutions per minute), inlet temperature (°C), exhaust gas temperature (°C), cycle pressure ratio, and air mass flow rate (kilograms per second). The data are saved in the file. (The first and last five observations are listed in the table at the bottom of the page.) a. Write a first-order model for heat rate (y) as a function of speed, inlet temperature, exhaust temperature, cycle pressure ratio, and airflow rate. b. Fit the model to the data using the method of least squares. c. Give practical interpretations of the ? estimates. d. Find the model standard deviation, s, and interpret its value. e. Conduct a test for overall model utility using ? = .01. f. Find and interpret R2a. g. Is there sufficient evidence (at ? = .01) to indicate that heat rate (y) is linearly related to inlet temperature?

Problem 23E Bordeaux wine sold at auction. The vineyards in the Bordeaux region of France are known for producing excellent red wines. However, the uncertainty of the weather during the growing season, the phenomenon that wine tastes better with age, and the fact that some Bordeaux vineyards produce better wines than others encourage speculation concerning the value of a case of wine produced by a certain vineyard during a certain year (or vintage). As a result, many wine experts attempt to predict the auction price of a case of Bordeaux wine. The publishers of a newsletter titled Liquid Assets: The International Guide to Fine Wine discussed a multiple regression approach to predicting the London auction price of red Bordeaux wine in Chance (Fall 1995). The natural logarithm of the price y (in dollars) of a case containing a dozen bottles of red wine was modeled as a function of weather during growing season and age of vintage using data collected for the vintages of 1952–1980. Three models were fit to the data. The results of the regressions are summarized in the table on the next page. a. For each model, conduct a t-test (at ? = .05) for each of the ? parameters in the model. Interpret the results. b. When the natural log of y is used as a dependent variable, the antilogarithm of ? ? coefficient minus 1—that is e?i - 1 - represents the percentage change in y for every 1-unit increase in the associated x value. Use this information to interpret the b estimates of each model. c. Based on the values of R2 and s, which of the three models would you recommend for predicting Bordeaux wine prices? Explain.

Problem 26E Predicting runs scored in baseball. Refer to the Chance (Fall 2000) study of runs scored in Major League Baseball games, Exercise 12.14 (p. 681). Multiple regression was used to model total number of runs scored (y) of a team during the season as a function of number of walks (x1), number of singles (x2), number of doubles (x3), number of triples (x4), number of home runs (x5), number of stolen bases (x6), number of times caught stealing (x7), number of strikeouts (x8), and total number of outs (x9). Using the b estimates given in Exercise 12.14, predict the number of runs scored by your favorite Major League Baseball team last year. How close is the predicted value to the actual number of runs scored by your team? [Note: You can find data on your favorite team on the Internet at www.mlb.com.]

Cost analysis for a shipping department. Multiple regression is used by accountants in cost analysis to shed light on the factors that cause costs to be incurred and the magnitudes of their effects. Sometimes, it is desirable to use physical units instead of cost as the dependent variable in a cost analysis (e.g., if the cost associated with the activity of interest is a function of some physical unit, such as hours of labor). The advantage of this approach is that the regression model will provide estimates of the number of labor hours required under different circumstances, and these hours can then be costed at the current labor rate (Horngren, Foster, and Datar, Cost Accounting, 2006). The sample data shown in the table below have been collected from a firm’s accounting and production records to provide cost information about the firm’s shipping department. These data are saved in the file. Consider the model y = ?0 + ?1x1 + ?2x2 + ?3x3 + ? a. Find the least squares prediction equation. b. Use an F-test to investigate the usefulness of the model specified in part a. Use ? = .01 and state your conclusion in the context of the problem. c. Test H0 : ?2 = 0 versus H? : ?2 ? 0 using ? = .05. What do the results of your test suggest about the magnitude of the effects of x2 on labor costs? d. Find R2 and interpret its value in the context of the problem. e. If shipping department employees are paid $7.50 per hour, how much less, on average, will it cost the company per week if the average number of pounds per shipment increases from a level of 20 to 21? Assume that x1 and x2 remain unchanged. Your answer is an estimate of what is known in economics as the expected marginal cost associated with a 1-pound increase in x3. f. With what approximate precision can this model be used to predict the hours of labor? [Note: The precision of multiple regression predictions is discussed in Section 12.4.] g. Can regression analysis alone indicate what factors cause costs to increase? Explain.

Reality TV and cosmetic surgery. Refer to the Body Image: An International Journal of Research (March 2010) study of the impact of reality TV shows on one's desire to undergo cosmetic surgery, Exercise 12.17 (p. 682). Recall that psychologists used multiple regression to model desire to have cosmetic surgery (y) as a function of gender \(\left(x_1\right)\), self esteem \(\left(x_2\right)\), body satisfaction \(\left(x_3\right)\), and impression of reality TV \(\left(x_4\right)\). The SPSS printout below shows a confidence interval for E(y) for each of the first five students in the study. a. Interpret the confidence interval for E(y) for student 1 . b. Interpret the confidence interval for E(y) for student 4 .

Arsenic in groundwater. Refer to the EnvironmentalScience & Technology (Jan. 2005) study of the reliability of a commercial kit to test for arsenic in groundwater,Exercise 12.16 (p. 682). You fit a first-order model for arsenic level (y) as a function of latitude, longitude, anddepth. Based on the model statistics, the researchers concluded that the arsenic level is highest at a low latitude, high longitude, and low depth. Do you agree? If so,find a 95% prediction interval for arsenic level for the lowest latitude, highest longitude, and lowest depth that are within the range of the sample data. Interpret the result.

Problem 29E Cooling method for gas turbines. Refer to the Journal ofEngineering for Gas Turbines and Power (Jan. 2005) studyof a high-pressure inlet fogging method for a gas turbineengine, Exercise 12.19 (p. 683). Recall that you fit a firstorder model for heat rate (y) as a function of speed (x1),inlet temperature (x2), exhaust temperature (x3), cycle pressure ratio (x4), and airflow rate (x5). A Minitab printout withboth a 95% confidence interval for E(y) and predictioninterval for y for selected values of the x’s is shown below. a. Interpret the 95% prediction interval for y in the wordsof the problem. b. Interpret the 95% confidence interval for E(y) in thewords of the problem. c. Will the confidence interval for E(y) always be narrower than the prediction interval for y? Explain.

Pay-for-performance efficiency of CEOs. In Chapter 1(p. 4), we presented Forbes magazine’s “ExecutiveCompensation Scoreboard.” Forbes (April 13, 2011) determined which CEOs are worth their pay based on a comparison of the firm’s stock performance and the CEO’s annual salary. Data for 175 of the top performers are saved in the file. Some of the variables measured for each CEO include pay-for-performance efficiency rating (\(x_1\)), value of shares owned (\(x_2\)), age (\(x_3\)), and the CEO’s average salary over the past 5 years (y). (Recall that the lower the efficiency rating, the more the CEO is worth his/her pay.) Consider The first-order model \(E(y)=\beta_0+\beta_1x_1+\beta_2x_2+\beta_3x_3\). a. Fit the model to the data and give the least squares prediction equation. b. Conduct a test of overall model adequacy using \(\alpha\ =\ .05\). c. Predict, with 95% confidence, the 5-year pay of a CEOwith \(x_1 = 173\), \(x_2 = +102.9\) million, and \(x_3 = 59\) years. (Note: These values represent the data for Sam Palmisano,CEO of IBM.) Interpret the result.

Problem 25E Characteristics of lead users. Refer to the Creativity and Innovation Management (Feb. 2008) study of lead users of children’s computer games, Exercise 12.12 (p. 681). Recall that the researchers modeled lead-user rating (y, measured on a 5-point scale) as a function of gender (x1 = 1 if female, 0 if male), age (x2, years), degree of centrality (x3, measured as the number of direct ties to other peers in the network), and betweenness centrality (x4, measured as the number of shortest paths between peers). The least squares prediction equation was a. Compute the predicted lead-user rating of a 10-year-old female child with 5 direct ties to other peers in her social network and with 2 shortest paths between peers. b. Compute an estimate for the mean lead-user rating of all 8-year-old male children with 10 direct ties to other peers and with 4 shortest paths between peers.

Extracting water from oil. Refer to Exercise 12.20 (p. 684)and the study on separating water from oil. The researchers concluded that “in order to break a water-oil mixture with the lowest possible voltage, the volume fraction of the dispersed phase (\(x_1\)) should be high, while the salinity (\(x_2\))and the amount of surfactant (\(x_5\)) should be low.” Use this information and the first-order model of Exercise 12.20 to find a 95% prediction interval for this “low” voltage (y).Interpret the interval.

Chemical plant contamination. Refer to Exercise 12.18 (p. 683) and the U.S. Army Corps of Engineers study. You fit the first-order model, \(E(y)=\beta_0+\beta_1 x_1+\beta_2 x_2+\beta_3 x_3\), to the data, where y = DDT level (parts per million), \(x_1=\) number of miles upstream, \(x_2=\) length (centimeters), and \(x_3=\) weight (grams). Use the Excel/XLSTAT printout above to predict, with 90% confidence, the DDT level of a fish caught 300 miles upstream with a length of 40 centimeters and a weight of 1,000 grams. Interpret the result.

Boiler drum production. In a production facility, an accurate estimate of man-hours needed to complete a task is crucial to management in making such decisions as the proper number of workers to hire, an accurate deadline to quote a client, or cost-analysis decisions regarding budgets. A manufacturer of boiler drums wants to use regression to predict the number of man-hours needed to erect the drums in future projects. To accomplish this, data for 35 boilers were collected. In addition to man-hours (y), the variables measured were boiler capacity \(\left(x_1=\mathrm{lb} / \mathrm{hr}\right)\), boiler design pressure \(\left(x_2=\right.\) pounds per square inch or psi), boiler type \(\left(x_3=1\right.\) if industry field erected, 0 if utility field erected), and drum type \(\left(x_4=1\right.\) if steam, 0 if mud). The data are saved in the file. (The first five and last five observations are listed in the accompanying table.) a. Fit the model \(E(y)=\beta_0+\beta_1 x_1+\beta_2 x_2+\beta_3 x_3+\beta_4 x_4\) to the data. Give the estimates of the \(\beta\)'s. b. Conduct a test for the global utility of the model. Use \(\alpha=.01\) c. Find a 95% confidence interval for E(y) when \(x_1=\) \(150,000, x_2=500, x_3=1\), and \(x_4=0\). Interpret the result.

Write an interaction model relating the mean value of y, E(y) to a. two quantitative independent variables b. three quantitative independent variables [Hint: Include all possible two-way cross-product terms.]

Problem 35E Suppose the true relationship between E(y) and the quantitative independent variables x1 and x2 is E(y) = 3 + x1 + 2x2 - x1x2 a. Describe the corresponding three-dimensional responsesurface. b. Plot the linear relationship between y and x2 forx1 = 0, 1, 2, where 0 ? x2 ? 5. c. Explain why the lines you plotted in part b are notparallel. d. Use the lines you plotted in part b to explain howchanges in the settings of x1 and x2 affect E(y). e. Use your graph from part b to determine how muchE(y) changes when x1 is changed from 2 to 0 and x2 issimultaneously changed from 4 to 5.

Suppose you fit the interaction model \(y=\beta_{0}+\beta_{1} x_{1}+\beta_{2} x_{2}+\beta_{3} x_{1} x_{2}+\varepsilon\) to n = 32 data points and obtain the following results: \(\begin{array}{llll} \mathrm{SS}_{y y}=479 & \mathrm{SSE}=21 & \hat{\beta}_{3}=10 & s_{\hat{\beta}_{3}}=4 \end{array}\) a. Find \(R^2\) and interpret its value. b. Is the model adequate for predicting y? Test at \(\alpha\ =\ .05\). c. Use a graph to explain the contribution of the \(x_1x_2\) term to the model. d. Is there evidence that \(x_1\) and \(x_2\) interact? Test at \(\alpha\ =\ .05\).

Consumer behavior while waiting in line. While waiting in a long line for service (e.g., to use an ATM or at the post office), at some point you may decide to leave the queue. The Journal of Consumer Research (Nov. 2003) published a study of consumer behavior while waiting in a queue. A sample of n = 148 college students was asked to imagine that they were waiting in line at a post office to mail a package and that the estimated waiting time is 10 minutes or less. After a 10-minute wait, students were asked about their level of negative feelings (annoyed, anxious) on a scale of 1 (strongly disagree) to 9 (strongly agree). Before answering, however, the students were informed about how many people were ahead of them and behind them in the line. The researchers used regression to relate negative feelings score (y) to number ahead in line \(\left(x_1\right)\) and number behind in line \(\left(x_2\right)\). a. The researchers fit an interaction model to the data. Write the hypothesized equation of this model. b. In the words of the problem, explain what it means to say that " \(x_1\) and \(x_2\) interact to affect y." c. A t-test for the interaction \(\beta\) in the model resulted in a p-value greater than .25. Interpret this result. d. From their analysis, the researchers concluded that "the greater the number of people ahead, the higher the negative feeling score" and "the greater the number of people behind, the lower the negative feeling score." Use this information to determine the signs of \(\beta_1\) and \(\beta_2\) in the model.

Defects in nuclear missile housing parts. The technique of multivariable testing (MVT) was discussed in the Journal Of the Reliability Analysis Center (First Quarter, 2004).MVT was shown to improve the quality of carbon-foam rings used in nuclear missile housings. The rings are produced via a casting process that involves mixing ingredients, oven curing, and carving the finished part. One type of defect analyzed was the number y of black streaks in the manufactured ring. Two variables found to impact the number of defects were turntable speed (revolutions per minute), \(x_1\), and cutting-blade position (inches from center), \(x_2\). a. The researchers discovered “an interaction between blade position and turntable speed.” Hypothesize a regression model for E(y) that incorporates this interaction. b. The researchers reported a positive linear relationship between number of defects (y) and turntable speed (\(x_1\)) but found that the slope of the relationship was much steeper for lower values of cutting-blade position (\(x_2\)).What does this imply about the interaction term in the model, part a? Explain.

Problem 40E Role of retailer interest on shopping behavior. Retail interest is defined by marketers as the level of interest a consumer has in a given retail store. Marketing professors at theUniversity of Tennessee at Chattanooga and the Universityof Alabama investigated the role of retailer interest in consumers’ shopping behavior (Journal of Retailing, Summer2006). Using survey data collected for n = 375 consumers, the professors developed an interaction model fory = willingness of the consumer to shop at a retailer’s storein the future (called repatronage intentions) as a function ofx1 = consumer satisfaction and x2 = retailer interest. Theregression results are shown below. a. Is the overall model statistically useful for predicting y?Test using ? = .05. b. Conduct a test for interaction at ? = .05. c. Use the ? estimates to sketch the estimated relationship between repatronage intentions (y) and satisfaction (x1) when retailer interest is x2 = 1 (a low value). d. Repeat part c when retailer interest is x2 = 7 (a highvalue). e. Sketch the two lines, parts c and d, on the same graph toillustrate the nature of the interaction.

Problem 37E The Minitab printout below was obtained from fitting themodel y = ?0 + ?1x1 + ?2x2 + ?3x1x2 + ? to n = 15 data points. a. What is the prediction equation for the responsesurface? b. Describe the geometric form of the response surface ofpart a.

Forecasting movie revenues with Twitter. Refer to the IEEE International Conference on Web Intelligence and Intelligent Agent Technology (2010) study on using the volume of chatter on Twitter.com to forecast movie box office revenue, Exercise 12.10(p. 680). The researchers modeled a movie's opening weekend box office revenue (y) as a function of tweet rate \(\left(x_1\right)\) and ratio of positive to negative tweets \(\left(x_2\right)\) using a first-order model a. Write the equation of an interaction model for E(y) as a function of \(x_1\) and \(x_2\). b. In terms of the \(\beta\)'s in the model, part a, what is the change in revenue \((\mathbf{y})\) for every 1-tweet increase in the tweet rate \(\left(x_1\right)\), holding PN-ratio \(\left(x_2\right)\) constant at a value of 2.5? c. In terms of the \(\beta\)'s in the model, part a, what is the change in revenue (y) for every 1-tweet increase in the tweet rate \(\left(x_1\right)\), holding PN-ratio \(\left(x_2\right)\) constant at a value of 5.0? d. In terms of the \(\beta\)'s in the model, part a, what is the change in revenue (y) for every 1-unit increase in the PN-ratio \(\left(x_2\right)\), holding tweet rate \(\left(x_1\right)\) constant at a value of 100? e. Give the null hypothesis for testing whether tweet rate \(\left(x_1\right)\) and PN-ratio \(\left(x_2\right)\) interact to affect revenue (y).

Factors that impact an auditor’s judgment. A study was conducted to determine the effects of linguistic deliverystyle and client credibility on auditors’ judgments(Advances in Accounting and Behavioral Research, 2004).Two hundred auditors from Big 5 accounting firms were each asked to assume that he or she was an audit team supervisor of a new manufacturing client and was performing an analytical review of the client’s financial statement.The researchers gave the auditors different information on the client’s credibility and linguistic delivery style of the client's explanation. Each auditor then provided an assessment of the likelihood that the client-provided explanation accounted for the fluctuation in the financial statement.The three variables of interest—credibility (\(x_1\)), linguistic delivery style (\(x_2\)), and likelihood (y)—were all measured on a numerical scale. Regression analysis was used to fit the interaction model, \(y=\beta_0+\beta_1x_1+\beta_2x_2+\beta_3x_1x_2+\epsilon\).The results are summarized in the table on the next page. a. Interpret the phrase client credibility and linguistic delivery style interact in the words of the problem. b. Give the null and alternative hypotheses for testing the overall adequacy of the model. c. Conduct the test, part b, using the information in the table. d. Give the null and alternative hypotheses for testing whether client credibility and linguistic delivery style interact. e. Conduct the test, part d, using the information in the table. f. The researchers estimated the slope of the likelihood–linguistic delivery style line at a low level of client credibility (\(x_1 = 22\)). Obtain this estimate and interpret it in the words of the problem. g. The researchers also estimated the slope of the likelihood–linguistic delivery style line at a high level of client credibility (\(x_1 = 46\)). Obtain this estimate and interpret it in the words of the problem.

Problem 43E Reality TV and cosmetic surgery. Refer to the Body Image:An International Journal of Research (March 2010) study ofthe impact of reality TV shows on a college student’s decisionto undergo cosmetic surgery, Exercise 12.17 (p. 682). Recall that the data for the study (simulated based on statisticsreported in the journal article) are saved in the file. Considerthe interaction model, E(y) = ?0 + ?1x1 + ?2x4 + ?3x1x4,where y = desire to have cosmetic surgery (25-point scale),x1 = {1 if male, 0 if female}, and x4 = impression of realityTV (7-point scale). The model was fit to the data and theresulting SPSS printout appears at the bottom of the page. a. Give the least squares prediction equation. b. Find the predicted level of desire (y) for a male college student with an impression-of-reality-TV-scale score of 5. c. Conduct a test of overall model adequacy. Use? = .10. d. Give a practical interpretation of e. Give a practical interpretation of s. f. Conduct a test (at ? = .10) to determine if gender (x1)and impression of reality TV show (x4) interact inthe prediction of level of desire for cosmetic surgery(y).

Problem 45E Arsenic in groundwater. Refer to the Environmental Science& Technology (Jan. 2005) study of the reliability of a commercial kit to test for arsenic in groundwater, Exercise 12.16(p. 682). Recall that you fit a first-order model for arsenic level(y) as a function of latitude (x1), longitude (x2), and depth (x3). a. Write a model for arsenic level (y) that includes firstorder terms for latitude, longitude, and depth, as wellas terms for interaction between latitude and depth andinteraction between longitude and depth. b. Use statistical software to fit the interaction model, part a,to the data. Give the least squares prediction equation. c. Conduct a test (at ? = .05) to determine whether latitude and depth interact to affect arsenic level. d. Conduct a test (at ? = .05) to determine whether longitude and depth interact to affect arsenic level. e. Practically interpret the results of the tests, parts c and d.

Cooling method for gas turbines. Refer to the Journal ofEngineering for Gas Turbines and Power (Jan. 2005) study of a high-pressure inlet fogging method for a gas turbine engine, Exercise 12.19 (p. 683). Recall that you fit a first order model for heat rate (y) as a function of speed (x1),inlet temperature (x2), exhaust temperature (x3), cycle pressure ratio (x4), and airflow rate (x5). a. Researchers hypothesize that the linear relationship between heat rate (y) and temperature (both inlet and exhaust) depends on airflow rate. Write a model for heat rate that incorporates the researchers’ theories. b. Use statistical software to fit the interaction model, part a,to the data. Give the least squares prediction equation. c. Conduct a test (at ? = .05) to determine whether inlet temperature and airflow rate interact to affect heat rate. d. Conduct a test (at ? = .05) to determine whether exhaust temperature and airflow rate interact to affect heat rate. e. Practically interpret the results of the tests, parts c and d.

Therapists' reactions to child-abuse reports. Licensed therapists are mandated by law to report child abuse by their clients. This requires the therapist to breach confidentiality and possibly lose the client's trust. A national survey of licensed psychotherapists was conducted to investigate clients' reactions to legally mandated child-abuse reports (American Journal of Orthopsychiatry, Jan. 1997). The sample consisted of 303 therapists who had filed a child-abuse report against one of their clients. The researchers were interested in finding the best predictors of a client's reaction (y) to the report, where y is measured on a 30-point scale. (The higher the value, the more favorable the client's response to the report.) The independent variables found to have the most predictive power are listed here. \(x_1\) = Therapist’s age (years) \(x_2\) = Therapist’s gender (1 if male, 0 if female) \(x_3\) = Degree of therapist’s role strain (25-point scale) \(x_4\) = Strength of client-therapist relationship (40-point scale) \(x_5\) = Type of case (1 if family, 0 if not) \(x_1 x_2\) = Age * gender interaction a. Hypothesize a first-order model relating y to each of the five independent variables. b. Give the null hypothesis for testing the contribution of \(x_4\), strength of client-therapist relationship, to the model. c. The test statistic for the test, part b, was t = 4.408 with an associated p-value of .001. Interpret this result. d. The estimated \(\beta\) coefficient for the \(x_1 x_2\) interaction term was positive and highly significant (p < .001). According to the researchers, "This interaction suggests that.. as the age of the therapist increased,... male therapists were less likely to get negative client reactions than were female therapists." Do you agree? e. For this model, \(R^2=.2946\). Interpret this value.

Consider the following quadratic models: (1) y = 1 - 2x + \(x^2\) (2) y = 1 + 2x + \(x^2\) (3) y = 1 + \(x^2\) (4) y = 1 - \(x^2\) (5) y = 1 + \(3x^2\) a. Graph each of the quadratic models, side by side, on the same sheet of graph paper. b. What effect does the first-order term (2x) have on the graph of the curve? c. What effect does the second-order term (\(x^2\)) have on the graph of the curve?

Write a second-order model relating the mean of y, E(y), to a. one quantitative independent variable b. two quantitative independent variables c. three quantitative independent variables [Hint: Include all possible two-way cross-product terms and squared terms.]

Suppose you fit the quadratic model \(E(y)=\beta 0+\beta 1 x+\beta 2 \times 2\) to a set of n = 20 data points and found \(R^2\) =.91, \(SS_yy\) =29.94, and SSE = 2.63. a. Is there sufficient evidence to indicate that the model contributes information for predicting y? Test using \(\alpha\) =.05. b. What null and alternative hypotheses would you test to determine whether upward curvature exists? c. What null and alternative hypotheses would you test to determine whether downward curvature exists?

Suppose you fit the second-order model \(y=\beta_0+\beta_1 x+\beta_2 x^2+\varepsilon\) to n = 25 data points. Your estimate of \(\beta_2\) is \(\hat{\beta}_2=.47\), and the estimated standard error of the estimate is .5. a. Test \(H_0: \beta_2=0\) against \(H_{\mathrm{a}}: \beta_2 \neq 0\). Use \(\alpha=.05\). b. Suppose you want to determine only whether the quadratic curve opens upward; that is, as x increases, the slope of the curve increases. Give the test statistic and the rejection region for the test for \(\alpha=.05\). Do the data support the theory that the slope of the curve increases as x increases? Explain.

Extracting water from oil. Refer to the Journal of Colloidand Interface Science study of water/oil mixtures, Exercise12.20 (p. 684). Recall that three of the seven variables used to predict voltage (y) were volume (\(x_1\)), salinity (\(x_2\)), and surfactant concentration (\(x_5\)). The model the researchers fit is \(E(y)=\beta_{0}+\beta_{1} x_{1}+\beta_{2} x_{2}+\beta_{3} x_{5}+\beta_{4} x_{1} x_{2}+\beta_{5} x_{1} x_{5}\) a. Note that the model includes interaction between disperse phase volume (\(x_1\)) and salinity (\(x_2\)) as well as interaction between disperse phase volume (\(x_1\)) and surfactant concentration (\(x_5\)). Discuss how these interaction terms affect the hypothetical relationship between y and \(x_1\). Draw a sketch to support your answer. b. Fit the interaction model to the data. Does this model appear to fit the data better than the first-order modeling Exercise 12.20? Explain. c. Interpret the \(\beta\) estimates of the interaction model.

Minitab was used to fit the complete second-order model \(E(y)=\beta_{0}+\beta_{1} x_{1}+\beta_{2} x_{2}+\beta_{3} x_{1} x_{2}+\beta_{4} x_{1}^{2}+\beta_{5} x_{2}^{2}\) to n = 39 data points. The printout is shown below. a. Is there sufficient evidence to indicate that at least one of the parameters \(-\beta_1, \beta_2, \beta_3, \beta_4\), and \(\beta_5\)-is nonzero? Test using \(\alpha=.05\). b. Test \(H_0: \beta_4=0\) against \(H_{\mathrm{a}}: \beta_4 \neq 0\). Use \(\alpha=.01\). c. Test \(H_0: \beta_5=0\) against \(H_{\mathrm{a}}: \beta_5 \neq 0\). Use \(\alpha=.01\). d. Use graphs to explain the consequences of the tests in parts b and c.

Testing tires for wear. Underinflated or overinflated tires canincrease tire wear. A new tire was tested for wear at different pressures, with the results shown in the following table. a. Plot the data on a scatterplot. b. If you were given only the information for x = 30, 31,32, 33, what kind of model would you suggest? Forx = 33, 34, 35, 36? For all the data?

Assertiveness and leadership. Management professors at Columbia University examined the relationship between assertiveness and leadership (Journal of Personality and Social Psychology, Feb. 2007). The sample represented 388people enrolled in a full-time MBA program. Based on answers to a questionnaire, the researchers measured two variables for each subject: assertiveness score (x) and leadership ability score (y). A quadratic regression model was fit to the data, with the following results: a. Conduct a test of overall model utility. Use \(\alpha = .05\). b. The researchers hypothesized that leadership ability increases at a decreasing rate with assertiveness. Set up the null and alternative hypotheses to test this theory. c. Use the reported results to conduct the test, part b. Give your conclusion (at \(\alpha = .05)\) in the words of the problem. Text Transcription: alpha = .05

Going for it on fourth-down in the NFL. Refer to the Chance(Winter 2009) study of fourth-down decisions by coaches in the National Football League (NFL), Exercise 11.63(p. 638). Recall that statisticians at California State University,Northridge, fit a straight-line model for predicting the number of points scored (y) by a team that has a first-down with a given number of yards (x) from the opposing goalline. A second model fit to data collected on five NFL teams from a recent season was the quadratic regression model \(E(y)=B_{0}+B_{1} x+B_{1} x^{2}\). The regression yielded the following results: \(\hat{y}=6.13+.141 x-.0009 x^{2}, R^{2}=.226\). a. If possible, give a practical interpretation of each of the \(\beta\) estimates in the model. b. Give a practical interpretation of the coefficient of determination, \(R^2\). c. In Exercise 11.63, the coefficient of correlation for the straight-line model was reported as \(R^2\) = .18.Does this statistic alone indicate that the quadratic model is a better fit than the straight-line model? Explain. d. What test of hypothesis would you conduct to determine if the quadratic model is a better fit than the straight-line model?

Personality traits and job performance. When attempting to predict job performance using personality traits, researchers typically assume that the relationship is linear. A study published in the Journal of Applied Psychology (Jan.2011) investigated a curvilinear relationship between job task performance and a specific personality trait—conscientiousness. Using data collected for 602 employees of a large public organization, task performance was measured on a 30-point scale (where higher scores indicate better performance) and conscientiousness was measured on a scale of -3 to +3 (where higher scores indicate a higher level of conscientiousness). a. The coefficient of correlation relating task performance score to conscientiousness score was reported as r = .18. Explain why the researchers should not use this statistic to investigate the curvilinear relationship between task performance and conscientiousness. b. Give the equation of a curvilinear (quadratic) model relating task performance score (y) to conscientiousness score (x). c. The researchers theorized that task performance increases as level of conscientiousness increases, but at a decreasing rate. Draw a sketch of this relationship. d. If the theory in part c is supported, what is the expected sign of \(\beta_{2}\) in the model, part b? e. The researchers reported \(\hat{\beta}_{2}=-.32\) with an associated p-value of less than .05. Use this information to test the researchers' theory at \(\alpha\) = .05.

Catalytic converters in cars. A quadratic model was applied to motor vehicle toxic emissions data collected in Mexico City (Environmental Science & Engineering, Sept. 1, 2000). The following equation was used to predict the percentage (y) of motor vehicles without catalytic converters in the Mexico City fleet for a given year (x): \(\hat{y}=325,790-321.67 x+.0794 x^{2}\) a. Explain why the value \(\hat{\beta}_{0}=325,790\) has no practical interpretation. b. Explain why the value \(\hat{\beta}_{1}=-321.67\) should not be interpreted as a slope. c. Examine the value of \(\hat{\beta}_{2}\) to determine the nature of the curvature (upward or downward) in the sample data. d. The researchers used the model to estimate "that just after the year 2021 the fleet of cars with catalytic converters will completely disappear." Comment on the danger of using the model to predict y in the year 2021. (Note: The model was fit to data collected between 1984 and 1999.) Text Transcription: hat{y} = 325,790 - 321.67x + .0794 x^2 hat{beta}_0 = 325,790 hat{beta}_1 = -321.67

Goal congruence in top management teams. Do chief executive officers (CEOs) and their top managers always agree on the goals of the company? Goal importance congruence between CEOs and vice presidents (VPs)was studied in the Academy of Management Journal (Feb.2008). The researchers used regression to model a VPs attitude toward the goal of improving efficiency (y) as a function of the two quantitative independent variables level of CEO leadership (\(x_1\)) and level of congruence between the CEO and the VP (\(x_2\)). A complete second-order model in \(x_1\) and \(x_2\) was fit to data collected for n = 517 top management team members at U.S. credit unions. a. Write the complete second-order model for E(y). b. The coefficient of determination for the model, part a,was reported as \(R^2 = .14\). Interpret this value. c. The estimate of the ?-value for the \((x_2)^2\) term in the model was found to be negative. Interpret this result, practically. d. A t-test on the \(\beta\)-value for the interaction term in the model, \(x_1x_2\), resulted in a p-value of .02. Practically interpret this result, using \(\alpha\ =\ .05\).

Failure times of silicon wafer microchips. Researchers atNational Semiconductor experimented with tin-lead solder bumps used to manufacture silicon wafer integrated circuit chips (International Wafer Level Packaging Conference,Nov. 3–4, 2005). The failure times of the microchips(in hours) was determined at different solder temperatures(degrees Celsius). The data for one experiment are given in the table. The researchers want to predict failure time (y)based on solder temperature (x). a. Construct a scatterplot for the data. What type of relationship, linear or curvilinear, appears to exist between failure time and solder temperature? b. Fit the model, \(E(y)=\beta_0+\beta_1x+\beta_2x^2\), to the data.Give the least squares prediction equation. c. Conduct a test to determine if there is upward curvature in the relationship between failure time and solder temperature. (Use \(\alpha\ =\ .05\).)

Problem 62E Estimating change-point dosage. A standard method forstudying toxic substances and their effects on humans is toobserve the responses of rodents exposed to various dosesof the substance over time. In the Journal of Agricultural,Biological, and Environmental Statistics (June 2005), researchers used least squares regression to estimate thechange-point dosage—defined as the largest dose levelthat has no adverse effects. Data were obtained from adose-response study of rats exposed to the toxic substanceaconiazide. A sample of 50 rats was evenly divided intofive dosage groups: 0, 100, 200, 500, and 750 milligramsper kilogram of body weight. The dependent variabley measured was the weight change (in grams) after a 2-week exposure. The researchers fit the quadratic modelE(y) = ?0 + ?1x + ?2x2, where x = dosage level, with thefollowing results: = 10.25 + .0053x - .0000266x2. a. Construct a rough sketch of the least squares predictionequation. Describe the nature of the curvature in theestimated model. b. Estimate the weight change (y) for a rat given a dosageof 500 mg/kg of aconiazide. c. Estimate the weight change (y) for a rat given a dosage of0 mg/kg of aconiazide. (This dosage is called the controldosage level.) d. Of the five dosage groups in the study, find the largest dosage level x that yields an estimated weightchange that is closest to but below the estimatedweight change for the control group. This value is thechange-point dosage.

Shopping on Black Friday. Refer to the International Journal of Retail and Distribution Management (Vol. 39, 2011) study of shopping on Black Friday (the day after Thanksgiving), Exercise 6.16 (p. 309). Recall that researchers conducted interviews with a sample of 38 women shopping on BlackFriday to gauge their shopping habits. Two of the variables measured for each shopper were age (x) and number of years shopping on Black Friday (y). Data on these two variables for the 38 shoppers are listed in the accompanying table. a. Fit the quadratic model, E(y) = ?0 + ?1x + ?2x2, to the data using statistical software. Give the prediction equation. b. Conduct a test of the overall adequacy of the model. Use ? = .01. c. Conduct a test to determine if the relationship betweenage (x) and number of years shopping on Black Friday (y) is best represented by a linear or quadratic function. Use ? = .01. .

Problem 65E Orange juice demand study. A chilled orange juice warehousing operation in New York City was experiencing toomany out-of-stock situations with its 96-ounce containers.To better understand current and future demand for thisproduct, the company examined the last 40 days of sales,which are shown in the table on the next page. One of thecompany’s objectives is to model demand, y, as a functionof sale day, x (where x = 1, 2, 3,...., 40). a. Construct a scatterplot for these data. b. Does it appear that a second-order model might betterexplain the variation in demand than a first-ordermodel? Explain. c. Fit a first-order model to these data. d. Fit a second-order model to these data. e. Compare the results in parts c and d and decide whichmodel better explains variation in demand. Justify yourchoice.

Problem 61E Revenues of popular movies. The Internet Movie Database(www.imdb.com) monitors the gross revenues for all majormotion pictures. The accompanying table gives both thedomestic (United States and Canada) and internationalgross revenues for a sample of 19 popular movies. a. Write a first-order model for foreign gross revenues (y)as a function of domestic gross revenues (x). b. Write a second-order model for international gross revenues y as a function of domestic gross revenues x. c. Construct a scatterplot for these data. Which of themodels from parts a and b appears to be the betterchoice for explaining the variation in foreign grossrevenues? d. Fit the model of part b to the data and investigate itsusefulness. Is there evidence of a curvilinear relationship between international and domestic gross revenues? Try using ? = .05. e. Based on your analysis in part d, which of the modelsfrom parts a and b better explains the variation in international gross revenues? Compare your answer to yourpreliminary conclusion from part c.

Public perceptions of health risks. In the Journal of Experimental Psychology: Learning, Memory, and Cognition(July 2005), University of Basel (Switzerland) psychologists tested the ability of people to judge risk of an infectious disease. The researchers asked German college students to estimate the number of people who are infected with a certain disease in a typical year. The median estimates as well as the actual incidence rate for each in a sample of 24 infections are provided in the table. Consider the quadratic model \(E(y)=\beta_0+\beta_1 x+\beta_2 x^2\), where y = actual incidence rate and x = estimated rate. a. Fit the quadratic model to the data and then conducta test to determine if incidence rate is curvilinearly related to estimated rate. (Use \(\alpha\) = .05.) b. Construct a scatterplot for the data. Locate the data point for botulism on the graph. What do you observe? c. Repeat part a but omit the data point for botulism from the analysis. Has the fit of the model improved?Explain.

Problem 67E Write a regression model relating E(y) to a qualitativeindependent variable that can assume three levels.Interpret all the terms in the model.

Write a regression model relating the mean value of y to a qualitative independent variable that can assume two levels.Interpret all the terms in the model.

The Minitab printout below resulted from fitting the following model to n = 15 data points: \(y=\beta_{0}+\beta_{1} x_{1}+\beta_{2} x_{2}+\varepsilon\) where \(x_{1}=\left\{\begin{array}{ll} 1 & \text { if level } 2 \\ 0 & \text { if not } \end{array} \quad x_{2}=\left\{\begin{array}{l} 1 \text { if level } 3 \\ 0 \text { if not } \end{array}\right.\right.\) a. Report the least squares prediction equation. b. Interpret the values of \(\beta_1\) and \(\beta_2\). c. Interpret the following hypotheses in terms of \(\mu_1\), \(\mu_2\),and \(\mu_3\): \(H_0:\ \beta_1=\beta_2=0\) \(H_a\): At least one of the parameters \(\beta_1\) and \(\beta_2\) differs from 0 d. Conduct the hypothesis test of part c.

The following model was used to relate E(y) to a single quantitative variable with four levels: \(E(y)=\beta_{0}+\beta_{1} x_{1}+\beta_{2} x_{2}+\beta_{3} x_{3}\) where \(x_{1}=\left\{\begin{array}{ll} 1 & \text { if level } 2 \\ 0 & \text { if not } \end{array} x_{2}=\left\{\begin{array}{l} 1 \text { if level } 3 \\ 0 \text { if not } \end{array} \quad x_{3}=\left\{\begin{array}{l} 1 \text { if level } 4 \\ 0 \text { if not } \end{array}\right.\right.\right.\) This model was fit to n = 30 data points, and the following result was obtained: \(\hat{y}=10.2-4 x_{1}+12 x_{2}+2 x_{3}\) a. Use the least squares prediction equation to find the estimate of E(y) for each level of the qualitative independent variable. b. Specify the null and alternative hypotheses you would use to test whether E(y) is the same for all four levels of the independent variable.

Accuracy of software effort estimates. Periodically, software engineers must provide estimates of their effort in developing new software. In the Journal of Empirical SoftwareEngineering (Vol. 9, 2004), multiple regression was used to predict the accuracy of these effort estimates. The dependent variable, defined as the relative error in estimating effort, y = (actual effort - estimated effort)/(actual effort) was determined for each in a sample of n = 49 software development tasks. Several qualitative independent variables were evaluated as potential predictors of relative error. Some of these variables are described in the table. a. Write a model for E(y) as a function of estimator role. Interpret the \(\beta\)'s. b. Write a model for E(y) as a function of task complexity. Interpret the \(\beta\)'s. c. Write a model for E(y) as a function of contract type. Interpret the \(\beta\)'s. d. Write a model for E(y) as a function of customer priority. Interpret the \(\beta\)'s.

Impact of race on football card values. University ofColorado sociologists investigated the impact of race on the value of professional football players’ “rookie” cards(Electronic Journal of Sociology, 2007). The sample consisted of 148 rookie cards of National Football League(NFL) players who were inducted into the Football Hall Of Fame. The price of the card (in dollars) was modeled as a function of several qualitative independent variables:race of player (black or white), card availability (high orlow), and player position (quarterback, running back, wide receiver, tight end, defensive lineman, linebacker, defensive back, or offensive lineman). a. Create the appropriate dummy variables for each of the qualitative independent variables. b. Write a model for price (y) as a function of race.Interpret the \(\beta\)’s in the model. c. Write a model for price (y) as a function of card availability. Interpret the \(\beta\)’s in the model. d. Write a model for price (y) as a function of position.Interpret the \(\beta\)’s in the model.

Can money spent on gifts buy love? Refer to the Journal Of Experimental Social Psychology (Vol. 45, 2009) study of whether buying gifts truly buys love, Exercise 9.9 (p. 492).Recall that study participants were randomly assigned to play the role of gift-giver or gift-receiver. Gift-receiverswere asked to provide the level of appreciation (measured on a 7-point scale where 1 = “not at all” and 7 = “to a great extent”) they had for the last birthday gift they received from a loved one. Gift-givers were asked to recall the last birthday gift they gave to a loved one and to provide the level of appreciation the loved one had for the gift. a. Write a dummy variable regression model that will allow the researchers to compare the average level of appreciation for birthday gift-givers \(\left(\mu_{\mathrm{G}}\right)\) to the average for birthday gift-receivers \(\left(\mu_{\mathrm{R}}\right)\). b. Express each of the model's \(\beta\) parameters in terms of \(\mu_{\mathrm{G}}\) and \(\mu_{\mathrm{R}}\) c. The researchers hypothesize that the average level of appreciation is higher for birthday gift-givers than for birthday gift-receivers. Explain how to test this hypothesis using the regression model.

Production technologies, terroir, and quality of Bordeaux wine. In addition to state-of-the-art technologies, the production of quality wine is strongly influenced by the natural endowments of the grape-growing region—called the “terroir.” The Economic Journal (May 2008) published an empirical study of the factors that yield a qualityBordeaux wine. A quantitative measure of wine quality (y)was modeled as a function of several qualitative independent variables, including grape-picking method (manual or automated), soil type (clay, gravel, or sand), and slope orientation (east, south, west, southeast, or southwest). a. Create the appropriate dummy variables for each of the qualitative independent variables. b. Write a model for wine quality (y) as a function of grape-picking method. Interpret the \(\beta\)’s in the model. c. Write a model for wine quality (y) as a function of soil type. Interpret the \(\beta\)’s in the model. d. Write a model for wine quality (y) as a function of slope orientation. Interpret the \(\beta\)’s in the model.

Buy-side vs. sell-side analysts’ earnings forecasts. Refer to the Financial Analysts Journal (Jul./Aug. 2008) comparison of earnings forecasts of buy-side and sell-side analysts,Exercise 2.86 (p. 86). The Harvard Business School professors used regression to model the relative optimism (y) ofthe analysts’ 3-month horizon forecasts. One of the independent variables used to model forecast optimism was the dummy variable x = {1 if the analyst worked for a buy-side firm, 0 if the analyst worked for a sell-side firm}. a. Write the equation of the model for E(y) as a function of type of firm. b. Interpret the value of \(\beta_0\) in the model, part a. c. The professors write that the value of \(\beta_1\) in the model,part a, “represents the mean difference in relative forecast optimism between buy-side and sell-side analysts.”Do you agree? d. The professors also argue that “if buy-side analysts make less optimistic forecasts than their sell-side counterparts, the [estimated value of \(\beta_1\)] will be negative.”Do you agree?

Do blondes raise more funds? During fundraising, does the physical appearance of the solicitor impact the level of capital raised? An economist at the University of Nevada-Reno designed an experiment to answer this question and published the results in Economic Letters (Vol. 100, 2008).Each in a sample of 955 households was contacted by a female solicitor and asked to contribute to the Center forNatural Hazards Mitigation Research. The level of contribution (in dollars) was recorded as well as the hair color of the solicitor (blond Caucasian, brunette Caucasian, or minority female). a. Consider a model for the mean level of contribution,E(y), that allows for different means depending on the hair color of the solicitor. Create the appropriate number of dummy variables for hair color. (Use minority female as the base level.) b. Write the equation of the model, part a, incorporating the dummy variables. c. In terms of the \(\beta\)’s in the model, what is the mean level of contribution for households contacted by a blondCaucasian solicitor? d. In terms of the \(\beta\)’s in the model, what is the difference between the mean level of contribution for households contacted by a blond solicitor and those contacted by a minority female? e. One theory posits that blond solicitors will achieve the highest mean contribution level, but that there will be no difference between the mean contribution levels attained by brunette Caucasian and minority females. Ifthis theory is true, give the expected signs of the \(\beta\)’s inthe model. f. The researcher found the \(\beta\)-estimate for the dummy variable for blond Caucasian to be positive and significantly different from 0 (p-value < .01). The \(\beta\)-estimate for the dummy variable for brunette Caucasian was also positive, but not significantly different from 0 (p-value > .10).Do these results support the theory, part e?

Deferred tax allowance study. A study was conducted to identify accounting choice variables that influence a manager's decision to change the level of the deferred tax asset allowance at the firm (The Engineering Economist, Jan./ Feb. 2004). Data were collected for a sample of 329 firms that reported deferred tax assets in 2000. The dependent variable of interest (DTVA) is measured as the change in the deferred tax asset valuation allowance divided by the deferred tax asset. The independent variables used as predictors of DTVA are listed as follows: LEVERAGE: \(x_1\) = ratio of debt book value to shareholder’s equity BONUS: \(x_2\) = 1 if firm maintains a management bonus plan, 0 if not MVALUE: \(x_3\) = market value of common stock BBATH: \(x_4\) = 1 if operating earnings negative and lower than last year, 0 if not EARN: \(x_5\) = change in operating earnings divided by total assets A first-order model was fit to the data with the following results ( p-values in parentheses): \(R_{\mathrm{a}}^{2}=.280\) \(\hat{y}=.044+.006 x_{1}-.035 x_{2}-.001 x_{3}+.296 x_{4}+.010 x_{5}\) (.070) (.228) (.157) (.678) (.001) (.869) a. Interpret the estimate of the \(\beta\) coefficient for \(x_4\). b. The “Big Bath” theory proposed by the researchers stated that the mean DTVA for firms with negative earnings and earnings lower than last year will exceed the mean DTVA of other firms. Is there evidence to support this theory? Test using \(\alpha\) = .05. c. Interpret the value of \(R_{\mathrm{a}}^{2}\).

Manipulating rates of return with stock splits. Some firms have been accused of using stock splits to manipulate their stock prices before being acquired by another firm. An article in Financial Management (Winter 2008) investigated the impact of stock splits on long-run stock performance of acquiring firms. A simplified version of the model fit by the researchers follows: \(E(y)=\beta_{0}+\beta_{1} x_{1}+\beta_{2} x_{2}+\beta_{3} x_{1} x_{2} \text {, }\) where y = Firm’s 3-year buy-and-hold return rate (%) \(x_1\) = {1 if stock split prior to acquisition, 0 if not} \(x_2\) = {1 if firm’s discretionary accrual is high, 0 if discretionary accrual is low} a. In terms of the \(\beta\)’s in the model, what is the mean buy-and-hold return rate (BAR) for a firm with no stock split and a high discretionary accrual (DA)? b. In terms of the \(\beta\)’s in the model, what is the mean BAR for a firm with no stock split and a low DA? c. For firms with no stock split, find the difference between the mean BAR for firms with high and low DA. (Hint:Use your answers to parts a and b.) d. Repeat part c for firms with a stock split. e. Note that the differences, parts c and d, are not the same. Explain why this illustrates the notion of interaction between \(x_1\) and \(x_2\). f. A test for \(H_0:\ \beta_3\ =\ 0\) yielded a p-value of .027. Using \(\alpha\ =\ .05\), interpret this result. g. The researchers reported that the estimated values of both \(\beta_2\) and \(\beta_3\) are negative. Consequently, they conclude that “high-DA acquirers perform worse compared with low-DA acquirers. Moreover, the underperformance is even greater if high-DA acquirers have a stock split before acquisition.” Do you agree?

Problem 77E Corporate sustainability and firm characteristics. Refer tothe Business and Society (March 2011) study on how firmsize and firm type impact corporate sustainability behaviors,Exercise 9.7 (p. 492). Recall that Certified Public Accountants(CPAs) were surveyed on their firms’ likelihood of reportingsustainability policies (measured as a probability between 0and 1). The CPAs were divided into four groups depending on firm size (large or small) and firm type (public orprivate): large/public, large/private, small/public, and small/private. One goal of the analysis is to determine whether themean likelihood of reporting sustainability policies differsdepending on firm size and firm type. a. Consider a single qualitative variable representingthe four size/type categories. Create the appropriatedummy variables for representing this qualitative variable as an independent variable in a regression modelfor predicting likelihood of reporting sustainabilitypolicies (y). b. Give the equation of the model, part a, and interpreteach of the model parameters. c. The global F-test for the model resulted in p-value < .001.Give a practical interpretation of this result. d. Now consider treating firm size and firm type as twodifferent qualitative independent variables in a modelfor likelihood of reporting sustainability policies (y).Create the appropriate dummy variables for representing these qualitative variables in the model. e. Refer to part d. Write a model for E(y) as a function of firm size and firm type, but do not includeinteraction. (This model is called the main effectsmodel.) f. Refer to the model, part e. For each combination offirm size and firm type (e.g., large/public), write E(y) asa function of the model parameters. g. Use the results, part f, to show that for the main effectsmodel, the difference between the mean likelihoodsfor large and small firms does not depend on firmtype. h. Write a model for E(y) as a function of firm size, firmtype, and size X type interaction. i. Refer to the model, part h. For each combination offirm size and firm type (e.g., large/public), write E(y) asa function of the model parameters. j. Use the results, part i, to show that for the interactionmodel, the difference between the mean likelihoods forlarge and small firms does depend on firm type.

Problem 79E Comparing mosquito repellents. Which insect repellentsprotect best against mosquitoes? Consumer Reports (June2000) tested 14 products that all claim to be an effectivemosquito repellent. Each product was classified as eitherlotion/cream or aerosol/spray. The cost of the product (indollars) was divided by the amount of the repellent neededto cover exposed areas of the skin (about 1/3 ounce) toobtain a cost-per-use value. Effectiveness was measured asthe maximum number of hours of protection (in half-hour increments) provided when human testers exposed theirarms to 200 mosquitoes. The data from the report are listedin the table. a. Suppose you want to use repellent type to modelthe cost per use (y). Create the appropriate numberof dummy variables for repellent type and write themodel. b. Fit the model, part a, to the data. c. Give the null hypothesis for testing whether repellenttype is a useful predictor of cost per use (y). d. Conduct the test, part c, and give the appropriate conclusion. Use ? = .10. e. Repeat parts a–d if the dependent variable is the maximum number of hours of protection (y).

Consider a multiple regression model for a response y,with one quantitative independent variable \(x_1\) and one qualitative variable at three levels. a. Write a first-order model that relates the mean responseE(y) to the quantitative independent variable. b. Add the main effect terms for the qualitative independent variable to the model of part a. Specify the coding scheme you use. c. Add terms to the model of part b to allow for interaction between the quantitative and qualitative independent variables. d. Under what circumstances will the response lines of the model in part c be parallel? e. Under what circumstances will the model in part c have only one response line?

Refer to Exercise 12.82. a. Write a complete second-order model that relates E(y)to the quantitative variable. b. Add the main effect terms for the qualitative variable(at three levels) to the model of part a. c. Add terms to the model of part b to allow for interaction between the quantitative and qualitative independent variables. d. Under what circumstances will the response curves of the model have the same shape but different-intercepts? e. Under what circumstances will the response curves of the model be parallel lines? f. Under what circumstances will the response curves of the model be identical?

Study of recall of TV commercials. Refer to the Journal ofApplied Psychology (June 2002) study of recall of television commercials, Exercise 9.28 (p. 509). Participants were assigned to watch one of three types of TV programs,with nine commercials embedded in each show. GroupV watched a TV program with a violent content coderating (e.g., Tour of Duty); group S viewed a show witha sex content code rating (e.g., Strip Mall); and group Nwatched a neutral TV program with neither a V nor anS rating (e.g., Candid Camera). The dependent variable measured for each participant was a score (y) on his/herrecall of the brand names in the commercial messages,with scores ranging from 0 (no brands recalled) to 9 (allbrands recalled). a. Write a model for E(y) as a function of the viewer group. b. Fit the model, part a, to the data saved in the file. Give The least squares prediction equation. c. Conduct a test of overall model utility at \(\alpha\ =\ .01\).Interpret the results. d. The sample mean recall scores for the three groups were \(\bar{y}_{\mathrm{V}}=2.08,\ \bar{y}_{\mathrm{S}}=1.71,\ \text { and }\ \bar{y}_{\mathrm{N}}=3.17\). Show how to find these sample means using only the \(\beta\) estimates obtained in part b.

Consider the model: \(y=\beta_0+\beta_1 x_1+\beta_2 x_2+\beta_3 x_3+\varepsilon\) where \(x_1\) is a quantitative variable and \(x_2\) and \(x_3\) are dummy variables describing a qualitative variable at three levels using the coding scheme \(x_2= \begin{cases}1 & \text { if level } 2 \\ 0 & \text { otherwise }\end{cases}\) \(x_3= \begin{cases}1 & \text { if level } 3 \\ 0 & \text { otherwise }\end{cases}\) The resulting least squares prediction equation is \(\hat{y}=44.8+2.2 x_1+9.4 x_2+15.6 x_3\) a. What is the response line (equation) for E(y) when \(x_2=x_3=0\)? When \(x_2=1\) and \(x_3=0\)? When \(x_2=0\) and \(x_3=1?\) b. What is the least squares prediction equation associated with level 1? Level 2? Level 3? Plot these on the same graph.

Write a model that relates E(y) to two independent variables—one quantitative and one qualitative at four levels. Construct a model that allows the associated response curves to be second-order but does not allow for interaction between the two independent variables.

Consider the model: \(y= \beta_0+\beta_1 x_1+\beta_2 x_1^2+\beta_3 x_2+\beta_4 x_3+\) \(\beta_5 x_1 x_2+\beta_6 x_1 x_3+\beta_7 x_1^2 x_2+\beta_8 x_1^2 x_3+\varepsilon\) where \(x_1\) is a quantitative variable and \(x_2= \begin{cases}1 & \text { if level } 2 \\ 0 & \text { otherwise }\end{cases}\) \(x_3= \begin{cases}1 & \text { if level } 3 \\ 0 & \text { otherwise }\end{cases}\) The resulting least squares prediction equation is \(\hat{y}= 48.8-3.4 x_1+.07 x_1^2-2.4 x_2-7.5 x_3+\) \(3.7 x_1 x_2+2.7 x_1 x_3-.02 x_1^2 x_2-.04 x_1^2 x_3\) a. What is the equation of the response curve for E(y) when \(x_2=0\) and \(x_3=0\)? When \(x_2=1\) and \(x_3=0\)? When \(x_2=0\) and \(x_3=1\)? b. On the same graph, plot the least squares prediction equation associated with level 1, with level 2 , and with level 3.

Homework assistance for accounting students. Refer to theJournal of Accounting Education (Vol. 25, 2007) study of assisting accounting students with their homework, Exercise9.30 (p. 510). Recall that 175 accounting students took a pretest on a topic not covered in class and then each was given homework problem to solve on the same topic. The students were assigned to one of three homework assistance groups.Some students received the completed solution, some were given check figures at various steps of the solution, andsome received no help at all. After finishing the homework,the students were all given a posttest on the subject. Thedependent variable of interest was the knowledge gain(or test score improvement). These data are saved in the file. a. Propose a model for the knowledge gain (y) as a function of the qualitative variable, homework assistance group. b. In terms of the ?’s in the model, give an expression for the difference between the mean knowledge gains of students in the “completed solution” and “no help” groups. c. Fit the model to the data and give the least squares prediction equation. d. Conduct the global F-test for model utility using? = .05. Interpret the results, practically. e. Show that the results, part d, agree with the conclusions reached in Exercise 9.30.

Reality TV and cosmetic surgery. Refer to the Body Image: An International Journal of Research (March 2010) study of the impact of reality TV shows on a college student's decision to undergo cosmetic surgery, Exercise 12.43 (p. 696). The data saved in the file were used to fit the interaction model, \(E(y)=\beta_0+\beta_1 x_1+\beta_2 x_4+\beta_3 x_1 x_4\), where y = desire to have cosmetic surgery (25-point scale), \(x_1\) = {1 if male, 0 if female }, and \(x_4=\) impression of reality TV (7-point scale). From the SPSS printout (p. 696), the estimated equation is: \(\hat{y}=11.78-1.97 x_1+.58 x_4-.55 x_1 x_4\) a. Give an estimate of the change in desire (y) for every 1-point increase in impression of reality TV show \(\left(x_4\right)\) for female students. b. Repeat part a for male students.

Impact of race on football card values. Refer to theElectronic Journal of Sociology (2007) study of the impact of race on the value of professional football players’ “rookie” cards, Exercise 12.72 (p. 712). Recall that the sample consisted of 148 rookie cards of NFL players who were inducted into the Football Hall of Fame(HOF). The researchers modeled the natural logarithm of card price (y) as a function of the following independent variables: a. The model \(E(y)=\beta_{0}+\beta_{1} x_{1}+\beta_{2} x_{2}+\beta_{3} x_{3}+\beta_{4} x_{4}+\beta_{5} x_{5}+\beta_{6} x_{6}+\beta_{7} x_{7}+\beta_{8} x_{8}+\beta_{9} x_{9}+\beta_{10} x_{10}+\beta_{11} x_{11}+\beta_{12} x_{12}\) was fit to the data with the following results: \(R^2=.705,\ R_a^2\ =\ .681,\ F=26.9\) Interpret the results, practically. Make an inference about the overall adequacy of the model. b. Refer to part a. Statistics for the race variable were reported as follows: \(\hat{\beta}_{1}=-.147,\ s_{\hat{\beta}_{1}}=.145,\ t=-1.014\), p-value = .312. Use this information to make an inference about the impact of race on the value of professional football players' rookie cards. c. Refer to part a. Statistics for the card vintage variable were reported as follows: \(\hat{\beta}_3=-.074,\ s_{\hat{\beta}_3}=.007,\ t=-10.92\), p-value = .000. Use this information to make an inference about the impact of card vintage on the value of professional football players' rookie cards. d. Write a first-order model for E(y) as a function of card vintage (\(x_3\)) and position (\(x_5-x_{12}\)) that allows for the relationship between price and vintage to vary depending on position.

Workplace bullying and intention to leave. Workplace Bullying (e.g., work-related harassment, persistent criticism, withholding key information, spreading rumors,intimidation) has been shown to have a negative psychological effect on victims, often leading the victim to quit or resign. In Human Resource Management Journal(Oct. 2008), researchers employed multiple regression to examine whether perceived organizational support(POS) would moderate the relationship between workplace bullying and victims’ intention to leave the firm.The dependent variable in the analysis, intention to leave (y), was measured on a quantitative scale. The twokey independent variables in the study were bullying(x1, measured on a quantitative scale) and perceived organizational support (measured qualitatively as “low,”“neutral,” or “high”). a. Set up the dummy variables required to represent POSin the regression model. b. Write a model for E(y) as a function of bullying andPOS that hypothesizes three parallel straight lines, one for each level of POS. c. Write a model for E(y) as a function of bullying andPOS that hypothesizes three nonparallel straight lines,one for each level of POS. d. The researchers discovered that the effect of bullying on intention to leave was greater at the low level ofPOS than at the high level of POS. Which of the two models, parts b and c, support these findings?

Do blondes raise more funds? Refer to the EconomicLetters (Vol. 100, 2008) study of whether the color of a female solicitor’s hair impacts the level of capital raised, Exercise 12.75 (p. 712). Recall that 955 households were contacted by a female solicitor to raise funds for hazard mitigation research. In addition to the household’s level of contribution (in dollars) and the hair color of the solicitor (blond Caucasian, brunette Caucasian, or minority female), the researcher also recorded the beauty rating of the solicitor (measured quantitatively, on a 10-point scale). a. Write a first-order model (with no interaction) for mean contribution level, E(y), as a function of a solicitor’s hair color and her beauty rating. b. Refer to the model, part a. For each hair color, express the change in contribution level for each 1-point increase in a solicitor’s beauty rating in terms of the model parameters. c. Write an interaction model for mean contribution level, E(y), as a function of a solicitor’s hair color and her beauty rating. d. Refer to the model, part c. For each hair color, express the change in contribution level for each 1-point increase in a solicitor’s beauty rating in terms of the model parameters. e. Refer to the model, part c. Illustrate the interaction with a graph.

Chemical plant contamination. Refer to Exercise12.18 (p. 683) and the model relating the mean DDTlevel E(y) of contaminated fish to \(x_1\) = miles captured upstream, \(x_2\) = length, and \(x_3\) = weight. Now consider a model for E(y) as a function of both weight and species (channel catfish, largemouth bass, and smallmouth buffalo). a. Set up the appropriate dummy variables for species. b. Write the equation of a model that proposes parallel straight-line relationships between mean DDT levelE(y) and weight, one line for each species. c. Write the equation of a model that proposes nonparallel straight-line relationships between mean DDT levelE(y) and weight, one line for each species. d. Fit the model, part b, to the data saved in the file. Give The least squares prediction equation. e. Refer to part d. Interpret the value of the least squares estimate of the beta coefficient multiplied by weight. f. Fit the model, part c, to the data saved in the file. Give The least squares prediction equation. g. Refer to part f. Find the estimated slope of the line relating DDT level (y) to weight for the channel catfish species.

Problem 90E Buy-side vs. sell-side analysts’ earnings forecasts. Refer tothe Financial Analysts Journal (Jul./Aug. 2008) comparisonof earnings forecasts of buy-side and sell-side analysts,Exercise 12.74 (p. 712). Recall that the Harvard BusinessSchool professors used regression to model the relativeoptimism (y) of the analysts’ 3-month horizon forecasts.The following model was fit to data collected on 11,121forecasts: E(y) = ?0 + ?1x1 + ?2x2 + ?3x3, where x 1 = {1 if the analyst worked for a buy-side firm, 0 if theanalyst worked for a sell-side firm} x 2 = number of days between forecast and fiscal year-end(i.e., forecast horizon) x 3 = natural logarithm of the number of quarters the analyst had worked with the firm a. The coefficient of determination for the model wasreported as R2 = .069. Interpret this value. b. Use the value of R2 in part a to conduct a test of theglobal utility of the model. Use ? = .01. c. The value of ?1 was estimated as 1.07, with an associatedt-value of 4.3. Use this information to test (? = .01.)whether x1 contributes significantly to the prediction of y. d. The professors concluded that “after controlling forforecast horizon and analyst experience, earnings forecasts by the analysts at buy-side firms are more optimistic than forecasts made by analysts at sell-side firms.”Do you agree?

Agreeableness, gender, and wages. Do agreeable individuals get paid less, on average, than those who are less agreeable on the job? And is this gap greater for males than for females? These questions were addressed in the Journal Of Personality and Social Psychology (Feb. 2012). Several Variables were measured for each in a sample of individuals enrolled in the National Survey of Midlife Development in the U.S. Three of these variables are: (1) level of agreeableness score (where higher scores indicate a greater level of agreeableness), (2) gender (male or female), and (3)annual income (dollars). The researchers modeled mean income, E(y), as a function of both agreeableness score (\(x_1\)) and a dummy variable for gender (\(x_2\) = 1 if male, 0 female). Data for a sample of 100 individuals (simulated,based on information provided in the study) are saved in the file. The first 10 observations are listed in the accompanying table. a. Consider the model, \(E(y)=\beta_0+\beta_1 x_1+\beta_2 x_2\). The researchers theorized that for either gender, income would decrease as agreeableness score increases. If this theory is true, what is the expected sign of \(\beta_1\) in the model? b. The researchers also theorized that the rate of decrease of income with agreeableness score would be steeper for males than for females (i.e., the income gap between males and females would be greater the less agreeable the individuals are). Can this theory be tested using the model, part a? Explain. c. Consider the interaction model, \(E(y)=\beta_0+\beta_1 x_1+\beta_2 x_2\) \(+\beta_3 x_1 x_2\). If the theory, part b, is true, give the expected sign of \(\beta_1\). The expected sign of \(\beta_3\). d. Fit the model, part c, to the sample data. Check the signs of the estimated \(\beta\) coefficients. How do they compare to the expected values, part c? e. Refer to the interaction model, part c. Give the null and alternative hypotheses for testing whether the rate of decrease of income with agreeableness score is steeper for males than for females. f. Conduct the test, part e. Use \(\alpha=.05\). Is the researchers' theory supported?

Recently sold, single-family homes. The NationalAssociation of Realtors maintains a database consisting of sales information on homes sold in the United States. The Next table lists the sale prices for a sample of 28 recently sold, single-family homes. The table also identifies the region of the country in which the home is located and the total number of homes sold in the region during the month the home sold. a. Propose a complete second-order model for the sale price of a single-family home as a function of region and sales volume. b. Give the equation of the curve relating sale price to sales volume for homes sold in the West. c. Repeat part b for homes sold in the Northwest. d. Which \(\beta\)’s in the model, part a, allow for differences among the mean sale prices for homes in the four regions? e. Fit the model, part a, to the data using an available statistical software package. Is the model statistically useful for predicting sale price? Test using \(\alpha\ =\ .01\).

Problem 97E Determine which pairs of the following models are“nested” models. For each pair of nested models, identifythe complete and reduced model.

Volatility of foreign stocks. The relationship between country credit ratings and the volatility of the countries stock markets was examined in the Journal of PortfolioManagement (Spring 1996). The researchers point out that this volatility can be explained by two factors: the countries’ credit ratings and whether the countries in question have developed or emerging markets. Data on the volatility (measured as the standard deviation of stock returns),credit rating (measured as a percentage), and market type(developed or emerging) for a sample of 30 fictitious countries are saved in the file. (Selected observations are shown in the table below.) a. Write a model that describes the relationship between volatility (y) and credit rating (\(x_1\)) as two nonparallel lines, one for each type of market. Specify the dummy variable coding scheme you use. b. Plot volatility y against credit rating \(x_1\) for all the developed markets in the sample. On the same graph, plot y against \(x_1\) for all emerging markets in the sample. Doesit appear that the model specified in part a is appropriate? Explain. c. Fit the model, part a, to the data using a statistical software package. Report the least squares prediction equation for each of the two types of markets. d. Plot the two prediction equations of part c on a scatterplot of the data. e. Is there evidence to conclude that the slope of the linear relationship between volatility y and credit rating \(x_1\) depends on market type? Test using \(\alpha\ =\ .01\).

Personality traits and job performance. Refer to the Journal of Applied Psychology (Jan. 2011) study of the relationship between task performance and conscientiousness, Exercise 12.54 (p. 704). Recall that the researchers used a quadratic model to relate y = task performance score (measured on a 30 -point scale) to \(x_1=\) conscientiousness score (measured on a scale of -3 to +3). In addition, the researchers included job complexity in the model, where \(x_2\) = {1 if highly complex job, 0 if not }. The complete model took the form \(E(y)=\beta_0+\beta_1 x_1+\beta_2\left(x_1\right)^2+\beta_3 x_2+\beta_4 x_1 x_2+\beta_5\left(x_1\right)^2 x_2\) a. For jobs that are not highly complex, write the equation of the model for \(\mathrm{E}(\mathrm{y})\) as a function of \(x_1\). (Substitute \(x_2=0\) into the equation.) b. Refer to part a. What do each of the \(\beta\)'s represent in the model? c. For highly complex jobs, write the equation of the model for \(\mathrm{E}(\mathrm{y})\) as a function of \(x_1\). (Substitute \(x_2=1\) into the equation.) d. Refer to part c. What do each of the \(\beta\)'s represent in the model? e. Does the model support the researchers' theory that the curvilinear relationship between task performance score (y) and conscientiousness score ( \(\left.x_1\right)\) depends on job complexity \(\left(x_2\right)\)? Explain.

Suppose you fit the regression model \(y=\beta_{0}+\beta_{1} x_{1}+\beta_{2} x_{2}+\beta_{3} x_{1} x_{2}+\beta_{4} x_{1}^{2}+\beta_{5} x_{2}^{2}+\varepsilon\) to n = 30 data points and wish to test \(H_0:\ \beta_3=\beta_4=\beta_5=0\) a. State the alternative hypothesis \(H_a\). b. Give the reduced model appropriate for conducting the test. c. What are the numerator and denominator degrees of freedom associated with the F-statistic? d. Suppose the SSE's for the reduced and complete models are \(SSE_R=1,250.2\) and \(SSE_C=1,125.2\). Conduct the hypothesis test and interpret the results of your test. Test using \(\alpha\ =\ .05\).

The complete model \(y=\beta_{0}+\beta_{1} x_{1}+\beta_{2} x_{2}+\beta_{3} x_{3}+\beta_{4} x_{4}+\varepsilon\) was fit to n = 20 data points, with SSE = 152.66. The Reduced model, \(y=\beta_{0}+\beta_{1} x_{1}+\beta_{2} x_{2}+\varepsilon\), was also fit,with SSE = 160.44. a. How many \(\beta\) parameters are in the complete model?The reduced model? b. Specify the null and alternative hypotheses you would use to investigate whether the complete model contributes more information for the prediction of y than the reduced model. c. Conduct the hypothesis test of part b. Use \(\alpha\ =\ .05\).

Mental health of a community. An article in theCommunity Mental Health Journal (Aug. 2000) use multiple regression analysis to model the level of community adjustment of clients of the Department ofMental Health and Addiction Services in Connecticut.The dependent variable, community adjustment (y), was measured quantitatively based on staff ratings of the clients. (Lower scores indicate better adjustment.) The complete model was a first-order model with 21 independent variables. The independent variables were categorized as Demographic (4 variables), Diagnostic (7 variables),Treatment (4 variables), and Community (6 variables). a. Write the equation of E(y) for the complete model. b. Give the null hypothesis for testing whether the7 Diagnostic variables contribute information for the prediction of y. c. Give the equation of the reduced model appropriate for the test, part b. d. The test, part b, resulted in a test statistic of F = 59.3and p-value < .0001. Interpret this result in the words of the problem.

Problem 101E Buy-side vs. sell-side analysts’ earnings forecasts. Referto the Financial Analysts Journal (Jul./Aug. 2008) comparison of earnings forecasts of buy-side and sell-sideanalysts, Exercise 12.90 (p. 721). Recall that the HarvardBusiness School professors used regression to model therelative optimism (y) of the analysts’ 3-month horizonforecasts as a function of x1 = {1 if the analyst workedfor a buy-side firm, 0 if the analyst worked for a sell-sidefirm} and x2 = number of days between forecast andfiscal year-end (i.e., forecast horizon). Consider the complete second-order model E(y) = ?0 + ?1x1 + ?2x2 + ?3x1x2 + ?4(x2)2 + ?5x1(x2)2 a. What null hypothesis would you test to determine whether the quadratic terms in the model arestatistically useful for predicting relative optimism (y)? b. Give the complete and reduced models for conducting the test, part a. c. What null hypothesis would you test to determinewhether the interaction terms in the model are statistically useful for predicting relative optimism(y)? d. Give the complete and reduced models for conducting the test, part c. e. What null hypothesis would you test to determinewhether the dummy variable terms in the modelare statistically useful for predicting relativeoptimism (y)? f. Give the complete and reduced models for conducting the test, part e.

Workplace bullying and intention to leave. Refer to theHuman Resource Management Journal (Oct. 2008) study of workplace bullying, Exercise 12.91 (p. 722). Recall that multiple regression was used to model an employee's intention to leave (y) as a function of bullying (x1, measured on a quantitative scale) and perceived organizational support (measured qualitatively as “low POS,”“neutral POS,” or “high POS”). In Exercise 12.91b, you wrote a model for E(y) as a function of bullying and POSthat hypothesizes three parallel straight lines, one for each level of POS. In Exercise 12.91c, you wrote a model for E(y) as a function of bullying and POS that hypothesizes three nonparallel straight lines, one for each level of POS. a. Explain why the two models are nested. Which is the complete model? Which is the reduced model? b. Give the null hypothesis for comparing the two models. c. If you reject \(H_0\) in part b, which model do you prefer?Why? d. If you fail to reject \(H_0\) in part b, which model do you prefer? Why?

Personality traits and job performance. Refer to the Journal of Applied Psychology (Jan. 2011) study of the relationship between task performance and conscientiousness, Exercise 12.94 (p. 722). Recall that y = task performance score (measured on a 30-point scale) was modeled as a function of \(x_1=\) conscientiousness score (measured on a scale of -3 to +3 ) and \(x_2\) = {1 if highly complex job, 0 if not} using the complete model \(E(y)=\beta_0+\beta_1 x_1+\beta_2\left(x_1\right)^2+\beta_3 x_2+\beta_4 x_1 x_2+\beta_5\left(x_1\right)^2 x_2\) a. Specify the null hypothesis for testing the overall adequacy of the model. b. Specify the null hypothesis for testing whether task performance score (y) and conscientiousness score \(\left(x_1\right)\) are curvilinearly related. c. Specify the null hypothesis for testing whether the curvilinear relationship between task performance score (y) and conscientiousness score \(\left(x_1\right)\) depends on job complexity \(\left(x_2\right)\). d. Explain how each of the tests, parts a-c, should be conducted (i.e., give the forms of the test statistic and the reduced model).

Problem 105E Reality TV and cosmetic surgery. Refer to the BodyImage: An International Journal of Research (March2010) study of the influence of reality TV shows onone’s desire to undergo cosmetic surgery, Exercise 12.17(p. 682). Recall that psychologists modeled desire tohave cosmetic surgery (y) as a function of gender (x1),self-esteem (x2), body satisfaction (x3), and impressionof reality TV (x4). The psychologists theorized that one’simpression of reality TV will “moderate” the impact thateach of the first three independent variables has on one’sdesire to have cosmetic surgery. If so, then x4 will interactwith each of the other independent variables. a. Give the equation of the model for E(y) that matchesthe theory. b. Fit the model, part a, to the simulated data savedin the file. Evaluate the overall utility of themodel. c. Give the null hypothesis for testing the psychologists’theory. d. Conduct a nested model F-test to test the theory.What do you conclude?

Study of supervisor-targeted aggression. “Moonlighters”are workers who hold two jobs at the same time. What are the factors that impact the likelihood of a moonlighting worker becoming aggressive toward his/her supervisor? This was the research question of interest in theJournal of Applied Psychology (July 2005). Completedquestionnaires were obtained from n = 105 moonlighters, and the data were used to fit several multiple regression models for supervisor-directed aggression score (y).Two of the models (with \(R^2\) values in parentheses) are given below: a. Interpret the \(R^2\) values for the models. b. Give the null and alternative hypotheses for comparing the fits of models 1 and 2. c. Are the two models nested? Explain. d. The nested F-test for comparing the two models resulted in F = 42.13 and p-value < .001. What can you conclude from these results? e. A third model was fit, one that hypothesizes all possible pairs of interactions between self-esteem, history of aggression, interactional injustice at primary job,and abusive supervisor at primary job. Give the equation of this model (model 3). f. A nested F-test to compare models 2 and 3 resulted in a p-value > .10. What can you conclude from this result?

Problem 103E Cooling method for gas turbines. Refer to the Journal ofEngineering for Gas Turbines and Power (Jan. 2005) studyof a high-pressure inlet fogging method for a gas turbineengine, Exercise 12.19 (p. 683). Consider a model for heatrate (kilojoules per kilowatt per hour) of a gas turbine asa function of cycle speed (revolutions per minute) andcycle pressure ratio. The data are saved in the file. a. Write a complete second-order model for heat rate (y). b. Give the null and alternative hypotheses for determining whether the curvature terms in the completesecond-order model are statistically useful for predicting heat rate (y). c. For the test in part b, identify the complete andreduced model. d. Portions of the Minitab printouts for the two modelsare shown below. Find the values of SSER, SSEC, andMSEC on the printouts. e. Compute the value of the test statistics for the test ofpart b. f. Find the rejection region for the test of part b using? = .10. g. State the conclusion of the test in the words of theproblem.

Agreeableness, gender, and wages. Refer to the Journal of Personality and Social Psychology (Feb. 2012) study of on-the-job agreeableness and wages, Exercise 12.92 (p. 722). The researchers modeled mean income, E(y), as a function of both agreeableness score (\(x_1\)) and a dummy variable for gender (\(x_2=1\) if male, 0 if female). Suppose the researchers theorize that for either gender, income will decrease at a decreasing rate as the agreeableness score increases. Consequently, they want to fit a second-order model. a. Consider the model, \(E(y)=\beta_0+\beta_1x_1+\beta_2(x_1)^2+\beta_3x_2\). If the researchers' belief is true, what is the expected sign of \(\beta_2\) in the model? b. Draw a sketch of the model, part a, showing how gender impacts the income-agreeableness score relationship. c. Write a complete second-order model for E(y) as a function of \(x_1\) and \(x_2\). d. Draw a sketch of the model, part c, showing how gender impacts the income-agreeableness score relationship. e. What null hypothesis would you test in order to compare the two models, parts a and c f. Fit the models to the sample data saved in the file and carry out the test, part e. What do you conclude? (Test using \(\alpha\ =\ .10\).)

Recently sold, single-family homes. Refer to the National Association of Realtors data on sales price (y), region NAR (NE, NW, S, or W), and sales volume for 28 recently sold, single-family homes, Exercise 12.95 (p. 723). You fit a complete second-order model for E(y) as a function of region and sales volume. a. Conduct a nested-model F-test to determine whether the quadratic terms in the model are statistically useful for predicting sales price (y). Use \(\alpha\) =.05 b. Based on the result, part a, which of the nested models (the complete or the reduced model) do you prefer to use in predicting sales price (y)? Explain. c. Refer to part b. Treat the preferred model as the complete model and conduct a nested model F-test to determine whether region and sales volume interact to affect sales price (y). Use \(\alpha\) =.05 d. Based on the result, part c, which of the nested models (the complete or the reduced model) do you prefer to use in predicting sales price (y)? Explain.

Glass as a waste encapsulant. Because glass is not subject to radiation damage, encapsulation of waste in glass is considered to be one of the most promising solutions to the problem of low-level nuclear waste in the environment. However, chemical reactions may weaken the glass. This concern led to a study undertaken jointly by theDepartment of Materials Science and Engineering at theUniversity of Florida and the U.S. Department of Energyto assess the utility of glass as a waste encapsulant.*Corrosive chemical solutions (called corrosion baths)were prepared and applied directly to glass samples containing one of three types of waste (TDS-3A, FE, andAL); the chemical reactions were observed over time. Afew of the key variables measured were y = Amount of silicon (in parts per million) found in solution at end of experiment. (This is both a measure of the degree of breakdown in the glass and a proxy for the amount of radioactive species released into the environment.) \(x_1\) = Temperature (\(^{\circ}C\)) of the corrosion bath \(x_2\) = 1 if waste type TDS-3A, 0 if not \(x_3\) = 1 if waste type FE, 0 if not (Waste type AL is the base level.) Suppose we want to model amount y of silicon as a function of temperature(\(x_1\)) and type of waste (\(x_2\), \(x_3\)). a. Write a model that proposes parallel straight-line relationships between amount of silicon and temperature, one line for each of the three waste types. b. Add terms for the interaction between temperature and waste type to the model of part a. c. Refer to the model of part b. For each waste type,give the slope of the line relating amount of silicon to temperature. d. Explain how you could test for the presence of temperature–waste type interaction.

Emotional distress in firefighters. The Journal of HumanStress (Summer 1987) reported on a study of “psychological response of firefighters to chemical fire.” It is thought that the following complete second-order model will be adequate to describe the relationship between emotional distress and years of experience for two groups of firefighters—those exposed to a chemical fire and those exposed: \(E(y)=\beta_{0}+\beta_{1} x_{1}+\beta_{2} x_{1}^{2}+\beta_{3} x_{2}+\beta_{4} x_{1} x_{2}+\beta_{5} x_{1}^{2} x_{2}\) where y = Emotional distress \(x_1\) = Experience (years) \(x_2\) = 1 if exposed to chemical fire, 0 if not a. How would you determine whether the rate of increase of emotional distress with experience is different for the two groups of firefighters? b. How would you determine whether there are differences in mean emotional distress levels that are attributable to exposure group?

There are six independent variables, \(x_1\), \(x_2\), \(x_3\), \(x_4\), \(x_5\), and \(x_6\), that might be useful in predicting a response y. A total of n = 50 observations is available, and it is decided to employ stepwise regression to help in selecting the independent variables that appear to be useful. The softwarefits all possible one-variable models of the form \(E(y)=\beta_0+\beta_1x_1\) where \(x_i\) is the ith independent variable, i = 1, 2,... , 6.The information in the table is provided from the computer printout. a. Which independent variable is declared the best one-variable predictor of y? Explain. b. Would this variable be included in the model at this stage? Explain. c. Describe the next phase that a stepwise procedure would execute.

Teacher pay and pupil performance. In Economic Policy (January 2011), researchers from the London School of Economics conducted a cross-country analysis of the relationship between teacher's pay and pupils' performance. Data collected for 39 countries were used to model y = the country's average standardized score of its pupils. The independent variables under consideration were: \(x_1\) = country's total teaching staff as a percentage of the country's labor force, \(x_2\) = percentage of women on country's total teaching staff, \(x_3\) = country's pupil-teacher ratio, \(x_4\) = average teacher's salary after 15 years on staff, \(x_5\) = average teaching hours per year, \(x_6\) = GDP growth of country (%), \(x_7\) = country's educational spending per year, and \(x_8\) = percentile position of country's teachers' salaries after 15 years on staff. Consider a stepwise regression run on the data. a. What is the form of the model fit in step 1? How many models are fit? How is the "best" independent variable selected in this step? b. What is the form of the model fit in step 2 ? How many models are fit? How is the "best" independent variable selected in this step? c. What is the form of the model fit in step 3 ? How many models are fit? How is the "best" independent variable selected in this step? d. The variables \(x_3\), \(x_4\), and \(x_6\) were deemed the best variables for predicting y. How do you recommend the researchers proceed from here? Why?

Diet of ducks bred for broiling. Corn is high in starch content; consequently, it is considered excellent feed for domestic chickens. Does corn possess the same potential in feeding ducks bred for broiling? This was the subject of research published in Animal Feed Science andTechnology (April 2010). The objective of the study was to establish a prediction model for the true metabolizable energy (TME) of corn regurgitated from ducks. Theresearchers considered 11 potential predictors of TME:dry matter (DM), crude protein (CP), ether extract (EE),ash (ASH), crude fiber (CF), neutral detergent fiber(NDF), acid detergent fiber (ADF), gross energy (GE),amylose (AM), amylopectin (AP), and amylopectin/amylose (AMAP). Stepwise regression was used to find the best subset of predictors. The final stepwise model yielded the following results: \(\widehat{T M E}=7.70+2.14(\text { AMAP })+.16(\mathrm{NDF}), R^2=.988,\) \(s=.07, \text { Global } F p \text {-value }=.001\) a. Determine the number of t-tests performed in step 1of the stepwise regression. b. Determine the number of t-tests performed in step 2of the stepwise regression. c. Give a full interpretation of the final stepwise model regression results. d. Explain why it is dangerous to use the final stepwise model as the “best” model for predictingTME. e. Using the independent variables selected by the stepwise routine, write a complete second-order model forTME. f. Refer to part e. How would you determine if the terms in the model that allow for curvature are statistically useful for predicting TME?

Problem 113E Entry-level job preferences. Benefits Quarterly (FirstQuarter, 1995) published a study of entry-level job preferences. A number of independent variables were used tomodel the job preferences (measured on a 10-point scale)of 164 business school graduates. Suppose stepwise regression is used to build a model for job preference score(y) as a function of the following independent variables: a. How many models are fit to the data in step 1? Givethe general form of these models. b. How many models are fit to the data in step 2? Givethe general form of these models. c. How many models are fit to the data in step 3? Givethe general form of these models. d. Explain how the procedure determines when to stopadding independent variables to the model. e. Describe two major drawbacks to using the final stepwisemodel as the best model for job preference score y.

Problem 114E Accuracy of software effort estimates. Periodically, software engineers must provide estimates of their effort indeveloping new software. In the Journal of EmpiricalSoftware Engineering (Vol. 9, 2004), multiple regressionwas used to predict the accuracy of these effort estimates.The dependent variable, defined as the relative error inestimating effort, y = (Actual effort - Estimated effort)/(Actual effort) was determined for each in a sample of n = 49 softwaredevelopment tasks. Eight independent variables wereevaluated as potential predictors of relative error usingstepwise regression. Each of these was formulated as adummy variable, as shown in the table. a. In step 1 of the stepwise regression, how many different one-variable models are fit to the data? b. In step 1, the variable x1 is selected as the best one-variable predictor. How is this determined? c. In step 2 of the stepwise regression, how many different two-variable models (where x1 is one of thevariables) are fit to the data? d. The only two variables selected for entry into the stepwise regression model were x1 and x8. The stepwiseregression yielded the following prediction equation: Give a practical interpretation of the ? estimates multiplied by x1 and x8. e. Why should a researcher be wary of using the model,part d, as the final model for predicting effort (y)?

Reality TV and cosmetic surgery. Refer to the BodyImage: An International Journal of Research (March2010) study of the influence of reality TV shows on one's desire to undergo cosmetic surgery, Exercise 12.17 (p. 682).Recall that psychologists modeled desire to have cosmetic surgery (y) as a function of gender (\(x_1\)), self-esteem (\(x_2\)), body satisfaction (\(x_3\)), and impression of reality TV (\(x_4\)). Suppose you want to determine which subset of these four independent variables is best for predicting one's desire to have cosmetic surgery. Consequently, you will run a stepwise regression. a. In step 1 of the stepwise regression, how many t-tests will be performed? b. In step 2 of the stepwise regression, how many t-tests will be performed? c. Access the data saved in the file and run the stepwise regression. Which independent variables comprise the best subset of variables for predicting one’s desire to have cosmetic surgery? d. Do you recommend using the resulting stepwise regression model for predicting one’s desire to have cosmetic surgery or do you recommend further analysis? Explain your reasoning.

Adverse effects of hot-water runoff. A marine biologist was hired by the EPA to determine whether the hot-water runoff from a particular power plant located near a large gulf is having an adverse effect on the marine life in the area. The biologist’s goal is to acquire a prediction equation for the number of marine animals located at certain designated areas, or stations, in the gulf. Based on past experience, the EPA considered the following environmental factors as predictors for the number of animals at a particular station: \(x_1\) = Temperature of water (TEMP) \(x_2\) = Salinity of water (SAL) \(x_3\) = Dissolved oxygen content of water (DO) \(x_4\) = Turbidity index, a measure of the turbidity of the water (TI) \(x_5\) = Depth of the water at the station (ST_DEPTH) \(x_6\) = Total weight of seagrasses in the sampled area (TGRSWT) As a preliminary step in the construction of this model, the biologist used a stepwise regression procedure to identify the most important of these six variables. A total of 716 samples were taken at different stations in the gulf, producing the SPSS printout shown below. (The response measured was y, the logarithm of the number of marine animals found in the sampled area.) a. According to the SPSS printout, which of the six independent variables should be used in the model? (Use \(\alpha\ =\ .10\).) b. Are we able to assume that the marine biologist has identified all the important independent variables for the prediction of y? Why? c. Using the variables identified in part a, write the first-order model with interaction that may be used to predict y. d. How would the marine biologist determine whether the model specified in part c is better than the first-order model? e. Note the small value of \(R^2\). What action might the biologist take to improve the model?

Consider fitting the multiple regression model \(E(y)=\beta_{0}+\beta_{1} x_{1}+\beta_{2} x_{2}+\beta_{3} x_{3}+\beta_{4} x_{4}+\beta_{5} x_{5}\) A matrix of correlations for all pairs of independent variables is given below. Do you detect a multicollinearity problem? Explain.

Bus Rapid Transit study. Bus Rapid Transit (BRT) is a rapi -dly growing trend in the provision of public transportation in America. The Center for Urban Transportation Research(CUTR) at the University of South Florida conducted a survey of BRT customers in Miami (TransportationResearch Board Annual Meeting, Jan. 2003). Data onthe following variables (all measured on a 5-point scale,where 1 = very unsatisfied and 5 = very satisfied) werecollected for a sample of over 500 bus riders: overall satisfaction with BRT (y), safety on bus \(\left(x_1\right)\), seat availability \(\left(x_2\right)\), dependability \(\left(x_3\right)\), travel time \(\left(x_4\right)\), cost \(\left(x_5\right)\), information/maps \(\left(x_6\right)\), convenience of routes \(\left(x_7\right)\), traffic signals \(\left(x_8\right)\), safety at bus stops \(\left(x_9\right)\), hours of service \(\left(x_{10}\right)\), and frequency of service \(\left(x_{11}\right)\). CUTR analysts used stepwise regression to model overall satisfaction (y). a. How many models are fit at step 1 of the stepwise regression? b. How many models are fit at step 2 of the stepwise regression? c. How many models are fit at step 11 of the stepwise regression? d. The stepwise regression selected the following eight variables to include in the model (in order of selection): \(x_{11}, x_4, x_2, x_7, x_{10}, x_1, x_9\), and \(x_3\). Write the equation for E(y) that results from stepwise regression. e. The model, part d, resulted in \(R^2=.677\). Interpret this value. f. Explain why the CUTR analysts should be cautious in concluding that the best model for E(y) has been found.

Identify the problem(s) in each of the residual plot shown below.

State casket sales restrictions. Some states permit only licensed firms to sell funeral goods (e.g., caskets, urns) to the consumer, while other states have no restrictions. States with casket sales restrictions are being challenged in court to lift these monopolistic restrictions. A paper in the Journal of Law and Economics (Feb. 2008) used multiple regression to investigate the impact of lifting casket sales restrictions on the cost of a funeral. Data collected for a sample of 1,437 funerals were used to fit the model. A simpler version of the model estimated by the researchers is \(E(y)=\beta_{0}+\beta_{1} x_{1}+\beta_{2} x_{2}+\beta_{3} x_{1} x_{2}\), where y is the price (in dollars) of a direct burial, \(x_{1}=\{1\) if funeral home is in a restricted state, 0 if not}, and \(x_{2}=\) {1 if price includes a basic wooden casket, 0 if no casket }. The estimated equation (with standard errors in parentheses) is: \(\hat{y}=1,432+ 793 x_{1}-252 x_{2}+261 x_{1} x_{2}, R^{2}=.78\) \((70) \quad(134) \quad(109)\) a. Calculate the predicted price of a direct burial with an abasic wooden casket at a funeral home in a restricted state. b. The data include a direct burial funeral with a basic wooden casket at a funeral home in a restricted state that costs $2,200. Assuming the standard deviation of the model is $50, is this data value an outlier? c. The data also include a direct burial funeral with an abasic wooden casket at a funeral home in a restricted state that costs $2,500. Again, assume that the standard deviation of the model is $50. Is this data value outlier?

Personality traits and job performance. Refer to the Journal of Applied Psychology (Jan. 2011) study of the determinants of task performance, Exercise 12.94 (p. 722 ). In addition to \(x_1\) = conscientiousness score and \(x_2\) {1 if highly complex job, 0 if not} , the researchers also used \(x_3\) = emotional stability score, \(x_4\) = organizational citizenship behavior score, and \(x_5\) = counterproductive work behavior score to model y = task performance score. One of their concerns is the level of multicollinearity in the data. Below is a matrix of correlations for all possible pairs of independent variables. Based on this information, do you detect a moderate or high level of multicollinearity? If so, what are your recommendations?

Problem 123E Women in top management. Refer to the Journal ofOrganizational Culture, Communications and Conflict(July 2007) study on women in upper management positions at U.S. firms, Exercise 11.67 (p. 639). Monthly data(n = 252 months) were collected for several variables inan attempt to model the number of females in managerial positions (y). The independent variables included thenumber of females with a college degree (x1), the numberof female high school graduates with no college degree(x2), the number of males in managerial positions (x3),the number of males with a college degree (x4), and thenumber of male high school graduates with no collegedegree (x5). The correlations provided in Exercise 11.67are given in each part. Determine which of the correlations results in a potential multicollinearity problem forthe regression analysis. a. The correlation relating number of females in managerial positions and number of females with a collegedegree: r = .983. b. The correlation relating number of females in managerial positions and number of female high schoolgraduates with no college degree: r = .074. c. The correlation relating number of males in managerial positions and number of males with a collegedegree: r = .722. d. The correlation relating number of males in managerial positions and number of male high school graduates with no college degree: r = .528.

Accuracy of software effort estimates. Refer to theJournal of Empirical Software Engineering (Vol. 9, 2004)study of the accuracy of new software effort estimates,Exercise 12.114 (p. 738). Recall that stepwise regression was used to develop a model for the relative error in estimating effort (y) as a function of company role of estimator (\(x_1\) = 1 if developer, 0 if project leader) andprevious accuracy (\(x_8\) = 1 if more than 20% accurate,0 if less than 20% accurate). The stepwise regression yielded the prediction equation \(\hat{y}=.12-.28x_1+.27_{x_8}\).The researcher is concerned that the sign of the estimated \(\beta\) multiplied by \(x_1\) is the opposite from what i expected. (The researcher expects a project leader to have a smaller relative error of estimation than a developer.) Give at least one reason why this phenomenon occurred.

Problem 127E Arsenic in groundwater. Refer to the EnvironmentalScience & Technology (Jan. 2005) study of the reliabilityof a commercial kit to test for arsenic in groundwater,Exercise 12.16 (p. 682). Recall that you fit a first-ordermodel for arsenic level (y) as a function of latitude (x1),longitude (x2), and depth (x3) to data saved in the file.Conduct a residual analysis of the data. Based on theresults, comment on each of the following: a. assumption of mean error = 0 b. assumption of constant error variance c. outliers d. assumption of normally distributed errors e. multicollinearity

Factors identifying urban counties. The ProfessionalGeographer (Feb. 2000) published a study of urban and rural counties in the western United States. Six independent variables—total county population (\(x_1\)), population density (\(x_2\)), population concentration (\(x_3\)), population growth (\(x_4\)), proportion of county land in farms (\(x_5\)), and 5-year change in agricultural land base (\(x_6\))—were used to model the urban/rural rating (y) of a county, where rating was recorded on a scale of 1 (most rural) to 10 (mosturban). Prior to running the multiple regression analysis,the researchers were concerned about possible multicollinearity in the data. Below is a correlation matrix for data collected on n = 256 counties. a. Based on the correlation matrix, is there any evidence of extreme multicollinearity? b. The first-order model with all six independent variables was fit, and the results are shown in the next table. Based on the reported tests, is there any evidence of extreme multicollinearity?

Failure times of silicon wafer microchips. Refer to the National Semiconductor study of manufactured silicon wafer integrated circuit chips, Exercise 12.63 (p. 706). Recall that the failure times of the microchips (in hours) was determined at different solder temperatures (degrees Celsius). The data are repeated in the table in the next column. a. Fit the straight-line model \(E(y)=\beta_0+\beta_1 x\) to the data, where y = failure time and x = solder temperature. b. Compute the residual for a microchip manufactured at a temperature of \(149^{\circ} \mathrm{C}\). c. Plot the residuals against solder temperature (x). Do you detect a trend? d. In Exercise 12.63c, you determined that failure time(y) and solder temperature (x) were curvilinearly related. Does the residual plot, part c, support this conclusion?

Cooling method for gas turbines. Refer to the Journal Of Engineering for Gas Turbines and Power (Jan. 2005)study of a high-pressure inlet fogging method for a gas turbine engine, Exercise 12.103 (p. 729). Now consider the interaction model for heat rate (y) of a gas turbine as a function of cycle speed (\(x_1\)) and cycle pressure ratio (\(x_2\)), \(E(y)=\beta_0+\beta_1x_1+\beta_2x_2+\beta_3x_1x_2\). Use the data saved in the file to conduct a complete residual analysis for the model. Do you recommend making model modifications?

Problem 128E Contamination from a plant’s discharge. Refer to theU.S. Army Corps of Engineers data on fish contaminatedfrom the toxic discharges of a chemical plant locatedon the banks of the Tennessee River in Alabama. InExercise 12.18 (p. 683), you fit the first-order model,E(y) = ?0 + ?1x1 + ?2x2 + ?3x3, where y = DDT levelin captured fish, x1 = miles captured upstream, x2 = fishlength, and x3 = fish weight. Conduct a complete residual analysis for the model. Do you recommend anymodel modifications be made? Explain.

Agreeableness, gender, and wages. Refer to the Journal Of Personality and Social Psychology (Feb. 2012) study of on-the-job agreeableness and wages, Exercise 12.107(p. 730). Recall that the researchers modeled mean income,E(y), as a function of both agreeableness score (x1) and a dummy variable for gender (x2 = 1 if male, 0 if female).Use the data saved in the file to analyze the residuals of the model, \(E(y)=\beta_0+\beta_1 x_1+\beta_2\left(x_1\right)^2+\beta_3 x_2\) What model modifications, if any, do you recommend?

Suppose you have developed a regression model to explain the relationship between y and \(x_1\), \(x_2\), and \(x_3\). The ranges of the variables you observed were as follows: \(10\ \leq\ y\ \leq\ 100,\ 5\ \leq\ x_1\ \leq\ 55,\ .5\ \leq\ x_2\ \leq\ 1,\ \text{and}\ 1,000\ \leq\ x_3\ \leq\ 2,000\). Will The error of prediction be smaller when you use the least squares equation to predict y when \(x_1\) = 30, \(x_2\) = .6,and \(x_3\) = 1,300, or when \(x_1\) = 60, \(x_2\) = .4, and \(x_3\) = 900?Why?

Reality TV and cosmetic surgery. Refer to the BodyImage: An International Journal of Research (March2010) study of the influence of reality TV shows on one's desire to undergo cosmetic surgery, Exercise12.17 (p. 682). Simulated data for the study is saved in the file. In Exercise 12.17, you fit the first-order model, \(E(y)=\beta_0+\beta_1x_1+\beta_2x_2+\beta_3x_3+\beta_4x_4\), where y = desire to have cosmetic surgery, \(x_1\) is a dummy variable for gender, \(x_2\) = level of self-esteem, \(x_3\) = level of body satisfaction,and \(x_4\) = impression of reality TV. a. Check the data for multicollinearity. If you detect multicollinearity, what modifications to the model do you recommend? b. Conduct a complete residual analysis for the model. Doyou detect any violations of the assumptions? If so, what modifications to the model do you recommend?

When a multiple regression model is used for estimating the mean of the dependent variable and for predicting a new value of y, which will be narrower—the confidence interval for the mean or the prediction interval for the new y value? Why?

Suppose you fit the model \(y=\beta_{0}+\beta_{1} x_{1}+\beta_{2} x_{1}^{2}+\beta_{3} x_{2}+\beta_{4} x_{1} x_{2}+\varepsilon\) to n = 25 data points with the following results: \(\hat{\beta}_{10}=1.26\ \ \ \quad \hat{\beta}_{1}=-2.43\ \ \ \quad \hat{\beta}_{2}=.05\ \ \ \quad \hat{\beta}_{3}=.62\ \ \ \quad \hat{\beta}_{4}=1.81\) \(s_{\hat{\beta}_{1}}=1.21\ \ \ \quad s_{\hat{\beta}_{2}}=.16\ \ \ \quad s_{\hat{\beta}_{3}}=.26\ \ \ \quad s_{\hat{\beta}_{4}}=1.49\) \(\mathrm{SSE}=.41\ \ \ R^{2}=.83\) a. Is there sufficient evidence to conclude that at least one of the parameters \(\beta_1\), \(\beta_2\), \(\beta_3\), or \(\beta_4\) is nonzero? Test using \(\alpha\ =\ .05\). b. Test \(H_0:\ \beta_1\ =\ 0\) against \(H_a:\ \beta_1\ <\ 0\). Use \(\alpha\ =\ .05\). c. Test \(H_0:\ \beta_2\ =\ 0\) against \(H_a:\ \beta_2\ >\ 0\). Use \(\alpha\ =\ .05\). d. Test \(H_0:\ \beta_3\ =\ 0\) against \(H_a:\ \beta_3\ ?\ 0\). Use \(\alpha\ =\ .05\).

Problem 135SE Suppose you used Minitab to fit the model y = ?0 + ?1x1 + ?2x2 + ? to n = 15 data points and obtained the printout shownbelow. a. What is the least squares prediction equation? b. Find R2 and interpret its value. c. Is there sufficient evidence to indicate that the modelis useful for predicting y? Conduct an F-test using? = .05. d. Test the null hypothesis H0: ?1 = 0 against the alternative hypothesis Ha: ?1 ? 0. Test using ? = .05. Drawthe appropriate conclusions. e. Find the standard deviation of the regression modeland interpret it.

The first-order model \(E(y)=\beta_{0}+\beta_{1} x_{1}\) was fit to n = 19data points. A residual plot for the model is provided below. Is the need for a quadratic term in the model evident from the residual plot? Explain.

Write a model relating E(y) to one qualitative independent variable that is at four levels. Define all the terms in your model.

Problem 138SE It is desired to relate E(y) to a quantitative variable x1and a qualitative variable at three levels. a. Write a first-order model. b. Write a model that will graph as three differentsecond-order curves—one for each level of the qualitative variable.

Problem 139SE Explain why stepwise regression is used. What is its valuein the model-building process?

Suppose you fit the regression model \(E(y)=\beta_{0}+\beta_{1} x_{1}+\beta_{2} x_{2}+\beta_{3} x_{2}^{2}+\beta_{4} x_{1} x_{2}+\beta_{5} x_{1} x_{2}^{2}\) to n = 35 data points and wish to test the null hypothesis \(H_0:\ \beta_4\ =\ \beta_5\ =\ 0\). a. State the alternative hypothesis. b. Explain in detail how to compute the F-statistic needed to test the null hypothesis. c. What are the numerator and denominator degrees of freedom associated with the F-statistic in part b? d. Give the rejection region for the test if \(\alpha\ =\ .05\).

Problem 141SE To model the relationship between y, a dependent variable, and x, an independent variable, a researcher hastaken one measurement on y at each of three differentx values. Drawing on his mathematical expertise, theresearcher realizes that he can fit the second-order model E(y) = ?0 + ?1x + ?2x2 and it will pass exactly through all three points, yieldingSSE = 0. The researcher, delighted with the excellentfit of the model, eagerly sets out to use it to make inferences. What problems will he encounter in attempting tomake inferences?

Problem 140SE Consider relating E(y) to two quantitative independentvariables x1 and x2. a. Write a first-order model for E(y). b. Write a complete second-order model for E(y).

GPAs of business students. Research scientists at theEducational Testing Service (ETS) used multiple regression analysis to model y, the final grade point average(GPA) of business and management doctoral students. A List of the potential independent variables measured foreach doctoral student in the study follows: (1) Quantitative Graduate Management Aptitude Test(GMAT) score (2) Verbal GMAT score (3) Undergraduate GPA (4) First-year graduate GPA (5) Student cohort (i.e., year in which student entered doctoral program: year 1, year 3, or year 5) a. Identify the variables as quantitative or qualitative. b. For each quantitative variable, give your opinion on whether the variable is positively or negatively related to final GPA. c. For each of the qualitative variables, set up the appropriate dummy variable. d. Write a first-order, main effects model relating finalGPA, y, to the five independent variables. e. Interpret the ?’s in the model, part d. f. Write a first-order model for final GPA, y, that allows for a different slope for each student cohort. g. For each quantitative independent variable in the model, part f, give the slope of the line (in terms of the \(\beta\)‘s) for the year 1 cohort.

Comparing private and public college tuition. According To the Chronicle of Higher Education Almanac, 4-year private colleges charge, on average, five times as much of tuition and fees than 4-year public colleges. In order to estimate the true difference in the mean amount charged for an academic year, random samples of 40 private colleges and 40 public colleges were contacted and questioned about their tuition structures. a. Which of the procedures described in Chapter 8 could be used to estimate the difference in mean charges between private and public colleges? b. Propose a regression model involving the qualitative independent variable type of college that could be used to investigate the difference between the means.Be sure to specify the coding scheme for the dummy variable in the model. c. Explain how the regression model you developed in part b could be used to estimate the difference between the population means.

Problem 145SE Comparing two orange juice extractors. The Florida CitrusCommission is interested in evaluating the performanceof two orange juice extractors, brand A and brand B. Itis believed that the size of the fruit used in the test mayinfluence the juice yield (amount of juice per pound oforanges) obtained by the extractors. The commission wantsto develop a regression model relating the mean juice yieldE(y) to the type of orange juice extractor (brand A orbrand B) and the size of orange (diameter), x1. a. Identify the independent variables as qualitative orquantitative. b. Write a model that describes the relationship betweenE(y) and size of orange as two parallel lines, one foreach brand of extractor. c. Modify the model of part b to permit the slopes of thetwo lines to differ. d. Sketch typical response lines for the model of part b.Do the same for the model of part c. Carefully labelyour graphs. e. Specify the null and alternative hypotheses you woulduse to determine whether the model in part c provides more information for predicting yield than doesthe model in part b. f. Explain how you would obtain the quantities necessary to compute the F-statistic that would be used intesting the hypotheses you described in part e.

Global warming and foreign investments. Scientists believe that a major cause of global warming is higher levels of carbon dioxide \(\left(\mathrm{CO}_2\right)\) in the atmosphere. In the Journal of World-Systems Research (Summer 2003), sociologists examined the impact of foreign investment dependence on \(\mathrm{CO}_2\) emissions in n = 66 developing countries. In particular, the researchers modeled the level of \(\mathrm{CO}_2\) emissions in 1996 based on foreign investments made 16 years earlier and several other independent variables. The variables and the model results are listed in the table below. a. Interpret the value of \(R^2\). b. Use the value of \(R^2\) to test the null hypothesis, \(H_0: \beta_1=\beta_2=\cdots=\beta_7=0\), at \(\alpha=.01\). c. What null hypothesis would you test to determine if foreign investments in 1980 is a statistically useful predictor of \(\mathrm{CO}_2\) emissions in 1996 ? d. Conduct the test, part c, at \(\alpha=.05\).

Problem 147SE Global warming and foreign investments (cont’d). Referto Exercise 12.146. A matrix giving the correlation (r)for each pair of independent variables is shown above.Identify the independent variables that are highly correlated. What problems may result from including thesehighly correlated variables in the regression model?

Urban population estimation using satellite images. Can the population of an urban area be estimated without taking a census? In Geographical Analysis (January, 2007) geography professors at the University of Wisconsin-Milwaukee and Ohio State University demonstrated the use of satellite image maps for estimating urban population. A portion of Columbus, Ohio, was partitioned into n = 125 census block groups, and satellite imagery was obtained. For each census block, the following variables were measured: population density (y), proportion of block with low-density residential areas \(\left(x_1\right)\), and proportion of block with high-density residential areas \(\left(x_2\right)\). A first-order model for y was fit to the data with the following results: \(\hat{y}=-.0304+2.006 x_1+5.006 x_2, R^2=.686\) a. Give a practical interpretation of each \(\beta\) estimate in the model. b. Give a practical interpretation of the coefficient of determination, \(R^2\). c. State \(H_0\) and \(H_{\mathrm{a}}\) for a test of overall model adequacy. d. Refer to part c. Compute the value of the test statistic. e. Refer to parts c and d. Make the appropriate conclusion at \(\alpha=.01\).

Trust in e-retailers. Electronic commerce (or “e-commerce”) describes the use of electronic networks to simplify a business operation. With e-commerce, retailers now can advertise and sell their products easily over the Web. In Internet Research: Electronic NetworkingApplications and Policy (Vol. 11, 2001), Canadian researchers investigated the factors that impact the level of trustin Web e-retailers. Five quantitative independent variables were used to model level of trust (y): \(x_1\) = ease of navigation on the Web site \(x_2\) = consistency of the Web site \(x_3\) = ease of learning the Web interface \(x_4\) = perception of the interface design \(x_5\) = level of support available to the user a. Write a first-order model for level of trust as a function of the five independent variables. b. The model, part a, was fit to data collected from = 66 visitors to e-retailers’ Web sites and yielded a coefficient of determination of \(R^2\) = .58. Interpretthis result. c. Compute the F-statistic used to test the global utility of the model. d. Using \(\alpha\ =\ .10\), give the appropriate conclusion for the test, part c.

A CEO’s impact on corporate profits. Can a corporation’s annual profit be predicted from information about the company’s CEO? Each year Forbes publishes data on company profit (in $ millions), CEO’s annual income(in $ thousands), and percentage of the company’s stock owned by the CEO. Consider a model relating company profit (y) to CEO income (\(x_1\)) and stock percentage (\(x_2\)).Explain what it means to say that “CEO income \(x_1\) and stock percentage \(x_2\) interact to affect company profit y.”

Accuracy of software effort estimates. Refer to theJournal of Empirical Software Engineering (Vol.9, 2004) study of the accuracy of software effort estimates, Exercise 12.73 (p. 712). Recall that the dependent variable (y) is measured as relative error of the effort estimate of a software development task. A total of eight independent variables were evaluated as potential predictors of relative error. Each of these was formulated as a dummy variable, as shown in the table below. a. The eight independent variables were entered into a stepwise regression, and the following two variables were selected for entry into the model: \(x_1\) and \(x_8\).Write the main effects model for E(y) as a function of these two variables. b. The stepwise regression yielded the following prediction equation: \(\hat{y}=.12-.28 x_{1}+.27 x_{8}\) Give a practical interpretation of the \(\beta\) estimates multiplied by \(x_1\) and \(x_8\). c. The researcher is concerned that the sign of \(\hat{\beta}_1\) in the model is the opposite from what is expected. (Theresearcher expects a project leader to have a smaller relative error of estimation than a developer.) Give atleast one reason why this phenomenon occurred.

“Sun safety” study. Numerous “sun safety” products exist on the market to prevent excessive exposure to solar radiation. But many people do not practice “sun safety” or recognize the effectiveness of these products. A group ofUniversity of Arizona researchers examined the feasibility of educating preschool (4- to 5-year-old) children about sun safety (American Journal of Public Health, July 1995).A sample of 122 preschool children was divided into two groups, the control group and the intervention group.Children in the intervention group received a Be Sun Safe Curriculum in preschool, while the control group did not.All children were tested for their knowledge, comprehension, and application of sun safety at two points in time:prior to the sun safety curriculum (pretest, \(x_1\)) and 7 weeks following the curriculum (posttest, y). a. Write a first-order model for mean posttest score,E(y), as a function of pretest score, \(x_1\), and group.Assume that no interaction exists between pretest score and group. b. For the model, part a, show that the slope of the line relating posttest score to pretest score is the same for both groups of children. c. Repeat part a but assume that pretest score and groups interact. d. For the model, part c, show that the slope of the line relating posttest score to pretest score differs for the two groups of children.

Prototyping new software. To meet the increasing demand for new software products, many systems development experts have adopted a prototyping methodology. The Effects of prototyping on the system development life cycle(SDLC) was investigated in the Journal of ComputerInformation Systems (Spring 1993). A survey of 500randomly selected corporate-level MIS managers was conducted. Three potential independent variables were(1) importance of prototyping to each phase of theSDLC; (2) degree of support prototyping provides for theSDLC; and (3) degree to which prototyping replaces each phase of the SDLC. The table below gives the pairwise correlations of the three variables in the survey data for one particular phase of the SDLC. Use this information to assess the degree of multicollinearity in the survey data. Would you recommend using all three independent variables in a regression analysis? Explain.

Promotion of supermarket vegetables. A supermarket chain is interested in exploring the relationship between the sales of its store-brand canned vegetables (y), the amount spent on promotion of the vegetables in local newspapers (\(x_1\)), and the amount of shelf space allocated to the brand (\(x_2\)). One of the chain’s supermarkets was randomly selected, and over a 20-week period, \(x_1\) and \(x_2\) were varied, as reported in the table. a. Fit the following model to the data: \(y=\beta_{0}+\beta_{1} x_{1}+\beta_{2} x_{2}+\beta_{3} x_{1} x_{2}+\varepsilon\) b. Conduct an F-test to investigate the overall usefulness of this model. Use \(\alpha\ =\ .05\). c. Test for the presence of interaction between advertising expenditures and shelf space. Use \(\alpha\ =\ .05\). d. Explain what it means to say that advertising expenditures and shelf since interact. e. Explain how you could be misled by using a first-order model instead of an interaction model to explain how advertising expenditures and shelf space influence sales. f. Based on the type of data collected, comment on the assumption of independent errors.

Problem 156SE Optimizing semiconductor material processing. Fluorocarbon plasmas are used in the production ofsemiconductor materials. In the Journal of AppliedPhysics (Dec. 1, 2000), electrical engineers at NagoyaUniversity (Japan) studied the kinetics of fluorocarbonplasmas in order to optimize material processing. In oneportion of the study, the surface production rate of fluorocarbon radicals emitted from the production processwas measured at various points in time (in milliseconds)after the radio frequency power was turned off. The dataare given in the next table. Consider a model relating surface production rate (y) to time (x). Rate Time Rate Time 1.00 0.1 0.00 1.7 0.80 0.3 -0.10 1.9 0.40 0.5 -0.15 2.1 0.20 0.7 -0.05 2.3 0.05 0.9 -0.13 2.5 0.00 1.1 -0.08 2.7 -0.05 1.3 0.00 2.9 -0.02 1.5 a. Graph the data in a scatterplot. What trend do youobserve? b. Fit a quadratic model to the data. Give the leastsquares prediction equation. c. Is there sufficient evidence of upward curvature inthe relationship between surface production rate andtime after turnoff? Use ? = .05.

Downtime of a production process. Downtime of a production process. An operations manager is interested in modeling E(y), the expected length of time per month (in hours) that a machine will be shut down for repairs, as a function of the type of machine (001 or 002) and the age of the machine (in years). The manager has proposed the following model: E(y) = ?0 + ?1x1 + ?2x12 + ?3x2 where x1 = Age of machine x2 = 1 if machine type 001, 0 if machine type 002 a. Use the data obtained on n = 20 machine breakdowns, shown below, to estimate the parameters of this model. b. Do these data provide sufficient evidence to conclude that the second-orcler term (x12) in the model proposed by the operations manager is necessary? Test using ? = .05. c. Test the null hypothesis that ?1 = ?2 = 0 using ? = .10. Interpret the results of the test in the context of the problem

Impact of advertising on market share. The audience for a product’s advertising can be divided into four segments according to the degree of exposure received as a result of the advertising. These segments are groups of consumers who receive very high (VH), high (H), medium (M),or low (L) exposure to the advertising. A company is interested in exploring whether its advertising effort affects its product’s market share. Accordingly, the company identifies 24 sample groups of consumers who have been exposed to its advertising, six groups at each exposure level. Then, the company determines its product's market share within each group. a. Write a regression model that expresses the company’s market share as a function of advertising exposure level. Define all terms in your model and list any assumptions you make about them. b. Did you include interaction terms in your model?Why or why not? c. The data in the table above were obtained by the company. Fit the model in part a to the data. d. Is there evidence to suggest that the firm’s expected market share differs for different levels of advertising exposure? Test using \(\alpha\ =\ .05\).

Problem 155SE Yield strength of steel alloy. Industrial engineers at theUniversity of Florida used regression modeling as atool to reduce the time and cost associated with developing new metallic alloys (Modelling and Simulationin Materials Science and Engineering, Vol. 13, 2005).To illustrate, the engineers built a regression modelfor the tensile yield strength (y) of a new steel alloy. The potentially important predictors of yield strength arelisted in the accompanying table. x 1 = Carbon amount (% weight) x 2 = Manganese amount (% weight) x 3 = Chromium amount (% weight) x 4 = Nickel amount (% weight) x 5 = Molybdenum amount (% weight) x 6 = Copper amount (% weight) x 7 = Nitrogen amount (% weight) x 8 = Vanadium amount (% weight) x 9 = Plate thickness (millimeters) x 10 = Solution treating (milliliters) x 11 = Aging temperature (degrees, Celsius) a. The engineers discovered that the variable Nickel (x4)was highly correlated with the other potential independent variables. Consequently, Nickel was droppedfrom the model. Do you agree with this decision?Explain. b. The engineers used stepwise regression on theremaining 10 potential independent variables in orderto search for a parsimonious set of predictor variables.Do you agree with this decision? Explain. c. The stepwise regression selected the following independent variables: x1 = Carbon, x2 = Manganese,x3 = Chromium, x5 = Molybdenum, x6 = Copper,x8 = Vanadium, x9 = Plate thickness, x10 = Solutiontreating, and x11 = Aging temperature. All thesevariables were statistically significant in the stepwise model, with R2 = .94. Consequently, theengineers used the estimated stepwise model to predict yield strength. Do you agree with this decision?Explain

Problem 157SE Modeling peak-hour roadway traffic. Traffic forecastersat the Minnesota Department of Transportation(MDOT) use regression analysis to estimate weekdaypeak-hour traffic volumes on existing and proposedroadways. In particular, they model y, the peak-hourvolume (typically, the volume between 7:00 and 8:00a.m.), as a function of x1, the road’s total volume for theday. For one project involving the redesign of a section ofInterstate 494, the forecasters collected n = 72 observations of peak-hour traffic volume and 24-hour weekdaytraffic volume using electronic sensors that count vehicles. The data are saved in the file. (The first and last fiveobservations are listed in the table.) a. Construct a scatterplot for the data, plotting peak-hour volume y against 24-hour volume x1. Note theisolated group of observations at the top of the scatterplot. Investigators discovered that all of these datapoints were collected at the intersection of Interstate35W and 46th Street. (These are observations 55–72in the table.) While all other locations in the samplewere three-lane highways, this location was uniquein that the highway widens to four lanes just north ofthe electronic sensor. Consequently, the forecastersdecided to include a dummy variable to account fora difference between the I-35W location and all otherlocations. b. Knowing that peak-hour traffic volumes have a theoretical upper bound, the forecasters hypothesized thata second-order model should be used to explainthe variation in y. Propose a complete second-ordermodel for E(y) as a function of 24-hour volume x1 andthe dummy variable for location. c. Using an available statistical software package, fitthe model of part b to the data. Interpret the results.Specifically, is the curvilinear relationship betweenpeak-hour volume and 24-hour volume different atthe two locations? d. Conduct a residual analysis of the model, part b.Evaluate the assumptions of normality and constanterror variance and determine whether any outliers exist.

Forecasting daily admission of a water park. To determine whether extra personnel are needed for the day,the owners of a water adventure park would like to find a model that would allow them to predict the day's attendance each morning before opening based on the day of the week and weather conditions. The model is of the form \(E(y)=\beta_{0}+\beta_{1} x_{1}+\beta_{2} x_{2}+\beta_{3} x_{3}\) where \(\begin{array}{l} y=\text { Daily admission } \\ x_{1}=\left\{\begin{array}{ll} 1 & \text { if weekend } \\ 0 & \text { otherwise } \end{array} \quad\right. \text { (dummy variable) } \\ x_{2}=\left\{\begin{array}{ll} 1 & \text { if sunny } \\ 0 & \text { if overcast } \quad \text { (dummy variable) } \end{array}\right. \\ x_{3}=\text { predicted daily high temperature }\left({ }^{\circ} \mathrm{F}\right) \end{array}\) These data were recorded for a random sample of 30 days, and a regression model was fitted to the data.The least squares analysis produced the following results: \(\hat{y}=-105+25 x_{1}+100 x_{2}+10 x_{3}\) with \(s_{\hat{\beta}_{1}}=10\ \ \ \quad s_{\hat{\beta}_{2}}=30\ \ \ \quad s_{\hat{\beta}_{3}}=4 \quad R^{2}=.65\) a. Interpret the estimated model coefficients. b. Is there sufficient evidence to conclude that this model is useful for the prediction of daily attendance?Use \(\alpha\ =\ .05\). c. Is there sufficient evidence to conclude that the mean attendance increases on weekends? Use \(\alpha\ =\ .10\). d. Use the model to predict the attendance on a sunny weekday with a predicted high temperature of \(95^{\circ}F\). e. Suppose the 90% prediction interval for part d is (645,1,245). Interpret this interval.

Problem 158SE Improving Math SAT scores. Refer to the Chance(Winter 2001) study of students who paid a privatetutor (or coach) to help them improve their ScholasticAssessment Test (SAT) scores, Exercise 2.88 (p. 86).Multiple regression was used to estimate theeffect of coaching on SAT–Mathematics scores. Dataon 3,492 students (573 of whom were coached) wereused to fit the model, E(y) = ?0 + ?1x1 + ?2x2, wherey = SAT9Math score, x1 = score on PSAT, andx2 = {1 if student was coached, 0 if not}. a. The fitted model had an adjusted R2 value of .76.Interpret this result. b. The estimate of ?2 in the model was 19, with astandard error of 3. Use this information to forma 95% confidence interval for ?2. Interpret theinterval. c. Based on the interval, part b, what can you say aboutthe effect of coaching on SAT–Math scores? d. As an alternative model, the researcher added several “control” variables, including dummy variablesfor student ethnicity (x3, x4, and x5), a socioeconomicstatus index variable (x6), two variables that measured high school performance (x7 and x8), the number of math courses taken in high school (x9), andthe overall GPA for the math courses (x10). Write thehypothesized equation for E(y) for the alternativemodel. e. Give the null hypothesis for a nested modelF-test comparing the initial and alternativemodels. f. The nested model F-test, part e, was statistically significant at ? = .05. Practically interpret this result. g. The alternative model, part d, resulted in = .79, = 14, and = 3. Interpret the value of . h. Refer to part g. Find and interpret a 95% confidenceinterval for ?2. i. The researcher concluded that “the estimated effectof SAT coaching decreases from the baseline modelwhen control variables are added to the model.” Doyou agree? Justify your answer. j. As a modification to the model of part d, the researcher added all possible interactions between thecoaching variable (x2) and the other independentvariables in the model. Write the equation for E(y) forthis modified model. k. Give the null hypothesis for comparing the models, parts d and j. How would you perform thistest?

Sale prices of apartments. A Minneapolis, Minnesota, real estate appraiser used regression analysis to explore the relationship between the sale prices of apartment buildings and various characteristics of the buildings. The file contains data for a random sample of 25 apartment buildings. Note: Physical condition of each apartment building is coded E (excellent), G (good), or F (fair). Data for selected observations are shown in the table below. a. Write a model that describes the relationship between sale price and number of apartment units as three parallel lines, one for each level of physical condition. Be sure to specify the dummy variable coding scheme you use. b. Plot y against \(x_1\) (number of apartment units) for all buildings in excellent condition. On the same graph,plot y against \(x_1\) for all buildings in good condition.Do this again for all buildings in fair condition. Doesit appear that the model you specified in part a is appropriate? Explain. c. Fit the model from part a to the data. Report the least squares prediction equation for each of the three building condition levels. d. Plot the three prediction equations of part c on a scatterplot of the data. e. Do the data provide sufficient evidence to conclude that the relationship between sale price and number of units differs depending on the physical condition of the apartments? Test using \(\alpha\ =\ .05\). f. Check the data set for multicollinearity. How does this impact your choice of independent variables to use in a model for sale price? g. Conduct a complete residual analysis for the model to check the assumptions on \(\epsilon\).

Forecasting daily admission of a water park (cont’d). Refer to Exercise 12.161. The owners of the water adventure park are advised that the prediction model could probably be improved if interaction terms were added. In Particular, it is thought that the rate at which mean attendance increases as predicted high temperature increases will be greater on weekends than on weekdays. The following model is therefore proposed: \(E(y)=\beta_{0}+\beta_{1} x_{1}+\beta_{2} x_{2}+\beta_{3} x_{3}+\beta_{4} x_{1} x_{3}\) The same 30 days of data used in Exercise 12.161 areagain used to obtain the least squares model \(\hat{y}=250-700 x_{1}+100 x_{2}+5 x_{3}+15 x_{1} x_{3}\) with \(s_{\hat{\beta}_{4}}=3.0\ \ \ \quad R^{2}=.96\) a. Graph the predicted day’s attendance, y, against the day’s predicted high temperature, \(x_3\), for a sunny weekday and for a sunny weekend day. Plot both on the same graph for \(x_3\) between \(70^{\circ}F\) and \(100^{\circ}F\). Note the increase in slope for the weekend day. Interpret this. b. Do the data indicate that the interaction term is a useful addition to the model? Use \(\alpha\ =\ .05\). c. Use this model to predict the attendance for a sunny weekday with a predicted high temperature of \(95^{\circ}F\). d. Suppose the 90% prediction interval for part c is (800, 850). Compare this result with the prediction interval for the model without interaction in Exercise 12.161,part e. Do the relative widths of the confidence intervals support or refute your conclusion about the utility of the interaction term (part b)? e. The owners, noting that the coefficient \(\hat{\beta}_1=-700\), conclude the model is ridiculous because it seems to imply that the mean attendance will be 700 less on weekends than on weekdays. Explain why this is not the case.

Forecasting a job applicant’s merit rating. A large research and development firm rates the performance of each member of its technical staff on a scale of 0 to 100,and this merit rating is used to determine the size of the person's pay raise for the coming year. The firm’s personnel department is interested in developing a regression model to help them forecast the merit rating that an applicant for a technical position will receive after being employed 3 years. The firm proposes to use the following second-order model to forecast the merit ratings of applicants who have just completed their graduate studies and have no prior related job experience: \(E(y)=\beta_{0}+\beta_{1} x_{1}+\beta_{2} x_{2}+\beta_{3} x_{1} x_{2}+\beta_{4} x_{1}^{2}+\beta_{5} x_{2}^{2}\) where y = Applicant’s merit rating after 3 years \(x_1\) = Applicant’s GPA in graduate school \(x_2\) = Applicant’s total score (verbal plus quantitative)on the Graduate Record Examination (GRE) The model, fit to data collected for a random sample of n = 40 employees, resulted in SSE = 1,830.44 andSS(model) = 4,911.5. The reduced model \(E(y)=\beta_{0}+\beta_{1} x_{1}+\beta_{2} x_{2}\) is also fit to the same data, resulting in SSE = 3,197.16. a. Identify the appropriate null and alternative hypotheses to test whether the complete (second-order)model contributes information for the prediction of y. b. Conduct the test of hypothesis given in part a. Test Using \(\alpha\ =\ .05\). Interpret the results in the context of this problem. c. Identify the appropriate null and alternative hypotheses to test whether the complete model contributes more information than the reduced (first-order)model for the prediction of y. d. Conduct the test of hypothesis given in part c. Test using \(\alpha\ =\ .05\). Interpret the results in the context of this problem. e. Which model, if either, would you use to predict y?Explain.

Household food consumption. The data in the table on the next page were collected for a random sample of 26 households in Washington, D.C. An economist wants to relate household food consumption, y, to household income, \(x_1\), and household size, \(x_2\), with the first-order model \(E(y)=\beta_{0}+\beta_{1} x_{1}+\beta_{2} x_{2}\) a. Fit the model to the data. Do you detect any signs of multicollinearity in the data? Explain. b. Is there visual evidence (from a residual plot) that second-order model may be more appropriate for predicting household food consumption? Explain. c. comment on the assumption of constant error variance, using a residual plot. Does it appear to be satisfied? d. Are there any outliers in the data? If so, identify them. e. Based on a graph of the residuals, does the assumption of normal errors appear to be reasonably satisfied? Explain.

Problem 164SE Light output of a bulb. A firm that has developed anew type of lightbulb is interested in evaluating its performance in order to decide whether to market it. It isknown that the light output of the bulb depends on thecleanliness of its surface area and the length of time thebulb has been in operation. Use the data in the next tableand the procedures you learned in this chapter to build aregression model that relates drop in light output to bulbsurface cleanliness and length of operation. Be sure toconduct a residual analysis also.

Modeling monthly collision claims. A medium-sized automobile insurance company is interested in developing a regression model to help predict the monthly collision claims of its policyholders. A company analyst has proposed modeling monthly collision claims (y) in the middle Atlantic States as a function of the percentage of claims by drivers under age 30 (\(x_1\)) and the average daily temperature during the month (\(x_2\)). She believes that as the percentage of claims by drivers under age 30 increases, claims will rise because younger drivers are usually involved in more serious accidents than older drivers. She also believes that claims will rise as the average daily temperature decreases because lower temperatures are associated with icy, hazardous driving conditions. In order to develop a preliminary model, data were collected for the state of New Jersey overa 3-year period. The data are saved in the file. (The first and last five observations are listed in the table below.) a. Use a statistical software package to fit the complete second-order model \(E(y)=\beta_{0}+\beta_{1} x_{1}+\beta_{2} x_{2}+\beta_{3} x_{1} x_{2}+\beta_{4} x_{1}^{2}+\beta_{5} x_{2}^{2}\) b. Test the hypothesis \(H_0:\ \beta_4=\beta_5=0\) using \(\alpha\ =\ .05\). Interpret the results in practical terms. c. Do the results support the analysts’ beliefs? Explain.(You may need to conduct further tests of hypotheses to answer this question.)

Developing a model for college GPA. Many colleges and universities develop regression models for predicting theGPA of incoming freshmen. This predicted GPA can then be used to make admission decisions. Although most models use many independent variables to predict GPA,we will illustrate by choosing two variables: \(x_1\) = Verbal score on college entrance examination(percentile) \(x_2\) = Mathematics score on college entrance examination (percentile) The file contains data on these variables for a random sample of 40 freshmen at one college. (Selected observations are shown in the table.) Use the data to develop a useful prediction equation for college freshman GPA (y). Be sure to conduct a residual analysis for the model.

State casket sales restrictions. Refer to the Journal of Law and Economics (Feb. 2008) study of the impact of lifting casket sales restrictions on the cost of a funeral, Exercise 12.121 (p. 758). Recall that data collected for a sample of 1,437 funerals were used to fit the model, \(E(y)=\beta_{0}+\beta_{1} x_{1}+\beta_{2} x_{2}+\beta_{3} x_{1} x_{2}\), where y is the price (in dollars) of a direct burial, \(x_{1}=\{1\) if funeral home is in a restricted state, 0 if not \(\}\), and \(x_{2}=\{1\) if price includes a basic wooden casket, 0 if no casket \(\}\). The estimated equation (with standard errors in parentheses) is: \(\hat{y}=1,432+793 x_{1}-252 x_{2}+261 x_{1} x_{2}, R^{2}=.78\) (70) (134) (109) a. Interpret the reported value of \(R^{2}\). b. Use the value of \(R^{2}\) to compute the F-statistic for testing the overall adequacy of the model. Test at \(\alpha=.05\). c. Compute the predicted price of a direct burial with a basic wooden casket for a funeral home in a restrictive state. d. Estimate the difference between the mean price of a direct burial with a basic wooden casket and the mean price of a burial with no casket for a funeral home in a restrictive state. e. Estimate the difference between the mean price of a direct burial with a basic wooden casket and the mean price of a burial with no casket for a funeral home in a nonrestrictive state. f. Is there sufficient evidence to indicate that the difference between the mean price of a direct burial with a basic wooden casket and the mean price of a burial with no casket depends on whether the funeral home is in a restrictive state? Test using \(\alpha=.05\). Text Transcription: E(y) = beta_0 + beta_{1} x_{1} + beta_{2} x_{2} + beta_{3} x_{1} x_{2} x_{1} = 1 } x_{2} = 1 hat{y} = 1,432 + 793 x_{1} - 252 x_{2} + 261 x_{1} x_{2}, R^{2} = .78 R^2 alpha = .05