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introduction to operations research

Introduction To The Practice Of Statistics 10th Edition David S. Moore, George P. McCabe, Bruce A. Craig - Solutions

11.20 Discrimination at work? A survey of 457 engineers in Canada was performed to identify the relationship of race, language proficiency, and location of training in finding work in the engineering field. In addition, each participant completed the Workplace Prejudice and Discrimination
11.19 Game-day spending. Game-day spending (ticket sales and food and beverage purchases)is critical for the sustainability of many professional sports teams. In the National Hockey League(NHL), nearly half the franchises generate more than two-thirds of their annual income from game-day spending.
11.18 Comparing linear models. When the effect of one explanatory variable depends upon the value of another explanatory variable, we say the explanatory variables interact with each other. In a regression model, interaction can be included using the product of the two explanatory variables as an
11.17 Differences in means. Verify that the coefficient of x in each part of the previous exercise is equal to the mean for Group B minus the mean for Group A. Do you think that this will be true in general?Explain your answer.
11.16 Models with indicator variables. Suppose that x is an indicator variable with the value 0 for Group A and 1 for Group B. The following equations describe relationships between the value ofμy and membership in Group A or B. For each equation, give the value of the mean responseμy for Group A
11.15 Polynomial models. Multiple regression can be used to fit a polynomial curve of degree q, y=β0+β1x+β2x2+⋯+βqxq, through the creation of additional explanatory variables x2 , x3 , etc. Sketch each of the following equations for values of x between 0 and 4. Then describe the relationship
11.14 Another ANOVA table for multiple regression. Use the following information and the general form of the ANOVA table for multiple regression on page 543 to perform the ANOVA F test and compute R2 .Source Degrees of freedom Sum of squares Mean square F Model 4 17.5 Error Total 33 524
11.13 ANOVA table for multiple regression. Use the following information and the general form of the ANOVA table for multiple regression on page 543 to perform the ANOVA F test and compute R2 .Source Degrees of freedom Sum of squares Mean square F Model 3 90 Error Total 43 510
11.12 Compare the variability. In many multiple regression summaries, researchers report both sy , the standard deviation of the response variable, and s, the regression standard error from the model fit. If the regression model explains the response variable y well, describe what we should expect
11.11 Significance tests for regression coefficients. Refer to Check-in question 11.1 (page 566). The following table contains the estimated coefficients and standard errors of their multiple regression fit.Each explanatory variable is an average of several five-point Likert scale
11.10 More on constructing the ANOVA table. A multiple regression analysis of 57 cases was performed with four explanatory variables. Suppose that SSM=16.5 and SSE=100.8 .a. Find the value of the F statistic for testing the null hypothesis that the coefficients of all the explanatory variables are
11.9 Constructing the ANOVA table. Six explanatory variables are used to predict a response variable using multiple regression. There are 183 observations.a. Write the statistical model that is the foundation for this analysis. Also include a description of all assumptions.b. Outline the ANOVA
11.8 Significance tests for regression coefficients. For each of the settings in the previous exercise, test the null hypothesis that the coefficient of x1 is zero versus the twosided alternative.
11.7 95% confidence intervals for regression coefficients. In each of the following settings, give a 95% confidence interval for the coefficient of x1 .a. n=23 , y^=1.6+6.5x1+3.7x2 , SEb1=3.1b. n=33 , y^=1.6+6.5x1+3.7x2 , SEb1=3.2c. n=23 , y^=1.6+4.9x1+3.2x2+5.8x3, SEb1=2.4d. n=84 ,
11.6 The effect of inbreeding. Refer to the previous exercise. Say that you hold the average adult weight(x1) at a fixed value.a. What is the effect of the inbreeding coefficient increasing by 0.1, 0.25, and 0.5 on life expectancy?b. Given the results in part (a), do you think it is important to
11.5 Predicting life expectancy. Refer to Exercise 11.3. The fitted linear model is y^=10.8−1.78x1−0.06x2 where x1 is the common logarithm of the adult average weight and x2 is the inbreeding coefficient. In the data set, x1 varies between 0.36 and 1.89, and x2 varies between 0.02 and 0.52a.
11.4 Health behavior versus mindfulness among undergraduates. Researchers surveyed 357 undergraduates throughout the United States and quantified each student’s health behavior, mindfulness, subjective sleep quality (SSQ), and perceived stress level. Of interest was whether the relationship of
11.3 Describe the regression model. Is the adult life expectancy of a dog breed related to its level of inbreeding? To investigate this, researchers collected information on 168 breeds and fit a model using each breed’s autosomal inbreeding coefficient and the common logarithm of the adult male
11.2 What’s wrong? In each of the following situations, explain what is wrong and why.a. One of the assumptions for multiple regression is that the distribution of each explanatory variable is Normal.b. The null hypothesis H0: β3=0 in a multiple regression involving three explanatory variables
11.1 What’s wrong? In each of the following situations, explain what is wrong and why.a. A small P-value for the ANOVA F test implies that all explanatory variables are significantly different from zero.b. R2 is the proportion of variation explained by the collection of explanatory variables. It
10.68 Is the price right? Refer to the previous exercise. Zoey and Aiden are looking to buy a house in this midwestern city.a. When they first meet with you, they say they’re interested in an 1800-square-foot home. What price range would you tell them to expect?b. Suppose that, after looking
10.67 Size and selling price of a house. TABLE 10.4 summarizes an SRS of 30 houses sold in a midwestern city during a recent year. Can a simple linear regression model, using a house’s size, be used to predict its selling price?TABLE 10.4 Selling price and size of 30 houses Price ($1000) Size (sq
10.66 Sales price versus assessed value, continued. Refer to the previous exercise. Let’s consider linear regression analysis using just the 34 properties.a. Obtain the residuals and plot them versus assessed value. Is there anything unusual to report? If so, explain.b. Do the residuals appear to
10.65 Sales price versus assessed value. Real estate is typically reassessed annually for property tax purposes. This assessed value, however, is not necessarily the same as the fair market value of the property. Let’s examine an SRS of 35 homes recently sold in a midwestern city. Both variables
10.64 Significance tests and confidence intervals. The significance test for the slope in a simple linear regression gave a value t=2.08 with 18 degrees of freedom. Would the 95% confidence interval for the slope include the value zero? Give a reason for your answer.
10.63 Resting metabolic rate and exercise, continued. Refer to the previous exercise. It is tempting to conclude that there is a strong linear relationship for the women but no relationship for the men. Let’s look at this issue a little more carefully.a. Find the confidence interval for the slope
10.62 Resting metabolic rate and exercise. Metabolic rate, the rate at which the body consumes energy, is important in studies of weight gain, dieting, and exercise. The following table gives data on the lean body mass and resting metabolic rate for 12 women and 7 men who are subjects in a study of
10.61 A mechanistic explanation of popularity. Previous experimental work has suggested that the serotonin system plays an important and causal role in social status. In other words, genes may predispose individuals to be popular/likable. As part of a recent study on adolescents, an experimenter
10.60 Are the results consistent? A researcher surveyed n=214 hotel managers to assess the relationship between customer-relationship management (CRM) and organizational culture.Each variable was an average of more than 25 5-point Likert survey responses and, therefore, was treated as a
10.59 Matching standardized scores. Refer to the previous two exercises. An alternative to the least-squares method is based on matching standardized scores. Specifically, we set(y-y¯)sy=(x-x¯)sx and solve for y. Let’s use the notation y=a0+a1x for this line. The slope is a1=sy/sx , and the
10.58 SAT versus ACT, continued. Refer to the previous exercise. Find the predicted value of ACT for each observation in the data set.a. What is the mean of these predicted values? Compare it with the mean of the ACT scores.b. Compare the standard deviation of the predicted values with the standard
10.57 SAT versus ACT. The SAT and the ACT are the two major standardized tests that colleges use to evaluate candidates. Most students take just one of these tests. However, some students take both.Consider the scores of 60 students who did this. How can we relate the two tests?a. Plot the data
10.56 State and college binge drinking. Excessive consumption of alcohol is associated with numerous adverse consequences. In one study, researchers analyzed binge-drinking rates from two national surveys, the Harvard School of Public Health College Alcohol Study (CAS) and the Centers for Disease
10.55 Significance test of the correlation. A study reported a correlation r=0.5 based on a sample size of n=15 ; another reported the same correlation based on a sample size of n=25 . For each, perform the test of the null hypothesis thatρ=0 . Describe the results and explain why the conclusions
10.54 Does a math pretest predict success? Can a pretest on mathematics skills predict success in a statistics course? The 62 students in an introductory statistics class took a pretest at the beginning of the semester. The least-squares regression line for predicting the score y on the final exam
10.53 Predicting the lean in 2021. Refer to the previous two exercises.a. How would you code the explanatory variable for the year 2021?b. The engineers working on the Leaning Tower of Pisa were most interested in how much the tower would lean if no corrective action were taken. Use the
10.52 More on the Leaning Tower of Pisa. Refer to the previous exercise.a. In 1918 the lean was 2.9071 meters. (The coded value is 71.) Using the least-squares equation for the years 1975 to 1987, calculate a predicted value for the lean in 1918. (Note that you must use the coded value 18 for
10.51 Leaning Tower of Pisa. The Leaning Tower of Pisa is an architectural wonder. Engineers concerned about the tower’s stability have done extensive studies of its increasing tilt. Measurements of the lean of the tower over time provide much useful information. The following table gives
10.50 CEO pay and gross profits. Publicly traded companies must disclose their workers’ median pay and the compensation ratio between a worker and the company’s CEO. Does this ratio say something about the performance of the company? CNBC collected this ratio and the gross profits per employee
10.49 Compare the predictions. Refer to Exercise 10.46. Another way to compare analyses is to compare predictions. Consider Case 37 in Table 10.3 (8th row, 2nd column). For this case, the area is 10 km2 , and the percent forest is 63%. Calculate the predicted index of biotic integrity based on area
10.48 Predicting water quality for an area of 40 km2 . Refer to Exercise 10.44.a. Find a 95% confidence interval for the mean response corresponding to an area of 40 km2 .b. Find a 95% prediction interval for a future response corresponding to an area of 40 km2 .c. Write a short paragraph
10.47 How an outlier can affect statistical significance. Consider the data in Table 10.3 and the relationship between IBI and the percent of watershed area that was forest. The relationship between these two variables is almost significant at the 0.05 level. In this exercise you will demonstrate
10.46 Comparing the analyses. In Exercises 10.44 and 10.45, you used two different explanatory variables to predict IBI. Summarize the two analyses and compare the results. If you had to choose between the two explanatory variables for predicting IBI, which one would you prefer? Give reasons for
10.45 More on predicting water quality. The researchers who conducted the study described in the previous exercise also recorded the percent of the watershed area that was forest for each of the streams. These data are also given in Table 10.3. Analyze these data using the questions in the previous
10.44 Predicting water quality. The index of biotic integrity (IBI) is a measure of the water quality in streams. IBI and land use measures for a collection of streams in the Ozark Highland ecoregion of Arkansas were collected as part of a study. TABLE 10.3 gives the data for IBI, the percent of
10.43 Gambling and alcohol use by first-year college students. Gambling and alcohol use are problematic behaviors for many college students. One study looked at 908 first-year students from a large northeastern university. Each participant was asked to fill out the 10-item Alcohol Use Disorders
10.42 Are female CEOs older? A pair of researchers looked at the age and sex of a large sample of CEOs. To investigate the relationship between these two variables, they fit a regression model with age as the response variable and sex as the explanatory variable. The explanatory variable was coded
10.41 Studying the residuals. Refer to the previous two exercises. Using the residuals from the model fits in Exercise 10.39 and 10.40, who are the top three players to outperform their bonus percent, and who are the top three players to underperform their bonus percent? Does the choice of response
10.40 Performance bonuses, continued. Refer to the previous exercise.a. Now run the simple linear regression for the variable’s square root of the performance rating and percent of salary devoted to incentive payments.b. Obtain the residuals and assess whether the assumptions for the linear
10.39 Incentive pay and job performance. In the National Football League (NFL), performance bonuses now account for roughly 25% of player compensation. Does tying a player’s salary to performance bonuses result in better individual or team success on the field? Focusing on linebackers, let’s
10.38 Draw the fitted line. Suppose you fit 10 pairs of (x, y) data using least squares. Draw the fitted line if x¯=4 , y¯=9 , and the residual for the pair (2, 4) is -1 .
10.37 Predicting public university tuition: 2008 versus 2018. Refer to Exercise 10.35. The data file also includes the in-state undergraduate tuition for the year 2008.a. Run the simple linear regression using year 2008 in place of year 2014. What is the least-squares line?b. Obtain the residuals
10.36 Even more on public university tuition. Refer to the previous two exercises.a. The tuition at Skinflint U was $9800 in 2014. What is the predicted tuition in 2018?b. The tuition at I.O.U. was $17,800 in 2014. What is the predicted tuition in 2018?c. Discuss the appropriateness of using the
10.35 More on public university tuition. Refer to the previous exercise. We’ll now move forward with inference using the model fit to the data without the unusual observations identified in part (e) of the previous exercise.a. Give the null and alternative hypotheses for examining if there is a
10.34 Public university tuition: 2014 versus 2018. TABLE 10.2 shows the in-state undergraduate tuition in 2014 and 2018 for 33 public universities.TABLE 10.2 In-state tuition and fees (in dollars) for 33 public universities University Year2014 Year2018 University Year2014 Year2018 University
10.33 Interpreting the results. Refer to the previous two exercises. You are a math teacher whose pay raise is based on your students’ academic performance. Suppose your class has 40 students, 20 of each sex. Explain how you might use the model results of the previous two exercises to determine
10.32 Temperature and academic performance, continued. Refer to the previous exercise. Repeat parts (a)–(e) using the female average score, Fave, as the response variable.
10.31 Temperature and academic performance. Does temperature affect academic performance? If yes, does the relationship vary by sex? To study these questions, researchers from Berlin, Germany, divided 543 students into 24 sessions. In each session, students were presented with 50 similar arithmetic
10.30 The relationship between cell phone use and academic performance. College students are the most rapid adopters of cell phone technology. They use the phone to surf the Internet, watch videos, listen to music, email, and play video games. Because a cell phone is almost always nearby,
10.29 Interpreting a residual plot. FIGURE 10.18 shows four plots of residuals versus x. For each plot, comment on the regression model conditions necessary for inference. Which plots suggest a reasonable fit to the linear regression model?FIGURE 10.18 Four plots of residual versus x, Exercise
10.28 School budget and number of students. Suppose that there is a linear relationship between the number of students x in a school system and the annual budget y. Write a population regression model to describe this relationship.a. Which parameter in your model is the fixed cost in the budget
10.27 Confidence intervals for the slope and intercept. Refer to the previous two exercises.The mean and standard deviation of the S&P 500 returns for these years are 12.11% and 17.61%, respectively. From this and your work in the previous two exercises:a. Find the standard error for the
10.26 Interpreting statistical software output. Refer to the previous exercise. What are the values of the estimated model standard error s and the squared correlation r2 ?
10.25 Completing an ANOVA table. How are returns on common stocks in overseas markets related toreturns in U.S. markets? Consider measuring U.S. returns by the annual rate of return on the Standard &Poor’s 500 stock index and overseas returns by the annual rate of return on the Morgan Stanley
10.24 Grade inflation. The average undergraduate GPA for American colleges and universities was estimated based on a sample of institutions that published this information. Here are the data for public schools in that report:Year 1992 1996 2002 2007 GPA 2.85 2.90 2.97 3.01 Do the following by hand
10.23 Correlation between the prevalences of adult binge drinking and underage drinking. A group of researchers compiled data on the prevalence of adult binge drinking and the prevalence of underage drinking in 42 states. A correlation of 0.32 was reported.a. Test the null hypothesis that the
10.22 Food neophobia. Food neophobia is a personality trait associated with avoiding unfamiliar foods.In one study of 564 children who were two to six years of age, the degree of food neophobia and the frequency of consumption of different types of food were measured. Here is a summary of the
10.21 Research and development spending. The National Science Foundation collects data on research and development spending by universities and colleges in the United States. Here are the data for spending in the years 2013–2016 that was nonfederally funded:Year 2013 2014 2015 2016 Spending
10.20 What’s wrong? For each of the following, explain what is wrong and why.a. In simple linear regression, the standard error for a future observation is s, the measure of spread about the regression line.b. In an ANOVA table, SSE is the sum of the deviations.c. There is a close connection
10.19 What’s wrong? For each of the following, explain what is wrong and why.a. In simple linear regression, the null hypothesis of the ANOVA F test is H0: β0=0 .b. In an ANOVA table, the mean squares add. In other words, MST=MSM+MSE .c. The smaller the P-value for the ANOVA F test, the greater
10.18 Alternative tornado model. Refer to Exercise 10.15. Most of the largest positive and negative deviations occur later in time. This suggests that there may not be constant variance. Because the response variable is a count, one can argue the variance is not constant (e.g., see the Poisson
10.17 Computer memory. The capacity of memory commonly sold at retail has increased rapidly over time.a. Make a scatterplot of the data. The growth is much faster than linear.b. Compute the logarithm of capacity and plot it against year. Are these points closer to a straight line?c. Fit the simple
10.16 Annual increase? Refer to the previous exercise. Let’s proceed with inference regardless of your confidence level.a. Do these data support a linear trend in the number of tornadoes? Justify your answer.b. Construct a 95% confidence interval for the average annual increase in the number of
10.15 Is the number of tornadoes increasing? The Storm Prediction Center of the National Oceanic and Atmospheric Administration maintains a database of tornadoes, floods, and other weather phenomena. TABLE 10.1 summarizes the annual number of tornadoes in the United States between 1953 and 2019.
10.14 Are the two fuel-efficiency measurements similar? Refer to Exercise 7.18 (page 407). In addition to the computer calculating miles per gallon (mpg), the driver also measured mpg by dividing the miles driven by the number of gallons at fill-up. The driver wants to determine if these
10.13 Complete check of the residuals, continued. Refer to the previous exercise. In Example 10.12(page 533), we checked model assumptions using a scatterplot (Figure 10.10) after log transforming the response variable.a. Repeat parts (a) through (c) of the previous exercise using LogInc and
10.12 Complete check of the residuals. In Example 10.11 (page 532), we checked model assumptions using a scatterplot (Figure 10.9). Let’s consider assessing the model assumptions using the residuals.a. Fit the (Educ, Inc) data using least-squares regression and obtain the residuals. Write down
10.11 Predicting college debt: Other measures. Refer to Exercise 10.6. Let’s look at AveDebt and its relationship with the other explanatory variables in the data set. In addition to the in-state cost after aid(InCostAid), there is the admittance rate (Admit), the four-year graduation rate
10.10 Impact of an unusual observation. Refer to Exercise 10.6 (page 536). Colorado School of Mines was removed from this analysis because it was considered an influential observation. Is that the case?Let’s investigate its impact on the fit.a. Refit the model using the entire sample of 27
10.9 More on predicting college debt. Refer to the previous exercise. James Madison University has an in-state cost of $15,659, and University of Wisconsin–Madison has an in-state cost of $11,507.a. Using your answer to part (a) of the previous exercise, what is the predicted average debt for a
10.8 Predicting college debt. Refer to Exercise 10.6. Colorado School of Mines has a much larger instate cost than the other schools in the sample. FIGURE 10.13 contains JMP output for the simple linear regression of AveDebt on InCostAid with this case removed.a. State the least-squares regression
10.7 Can we consider this an SRS? Refer to the previous exercise. The report states that Kiplinger’s rankings focus on traditional four-year public colleges with broad-based curricula and on-campus housing. Each year, Kiplinger starts with more than 500 schools and then narrows down the list to
10.6 College debt versus adjusted in-state costs. Kiplinger’s “Best Values in Public Colleges” provides a ranking of U.S. public colleges based on a combination of various measures of academics and affordability. Let’s focus on the relationship between the average debt in dollars at
10.5 Importance of Normal model deviations? A general form of the central limit theorem tells us that the sampling distributions of b0 and b1 will be approximately Normal even if the model deviations are not Normally distributed. Using this fact, explain why the Normal distribution assumption is
10.4 Predicting BMI. In Example 10.2, Subject 13 averaged 9114 steps and has a BMI of 29.9. Using the least-squares regression equation in Example 10.3, find the predicted BMI and the residual for this individual.
10.3 U.S. versus overseas stock returns. Returns on common stocks in the United States and overseas appear to be growing more closely correlated as economies become more interdependent. Suppose that the following population regression line connects the total annual returns (in percent) on two
10.2 What’s wrong? For each of the following statements, explain what is wrong and why.a. The slope describes the change in x for a change in y.b. The population regression line is y=b0+b1x .c. A 95% confidence interval for the mean response is the same width, regardless of x.
10.1 What’s wrong? For each of the following statements, explain what is wrong and why.a. The parameters of the simple linear regression model are b0 , b1 , and s.b. To test H0: b1=0 , use a t test with n-2 degrees of freedom.c. For a particular value of the explanatory variable x, the confidence
9.40 Population estimates. Refer to the previous exercise. One reason to do an audit such as this is to estimate the number of claims that would not be allowed if all claims in a population were examined by experts. We have estimates of the proportions of such claims from each stratum based on our
9.39 Health care fraud. Most errors in billing insurance providers for health care services involve honest mistakes by patients, physicians, or others involved in the health care system.However, fraud is a serious problem. The National Health Care Anti-Fraud Association estimates that tens of
9.38 Titanic! In 1912, the luxury liner Titanic, on its first voyage, struck an iceberg and sank. Some passengers got off the ship in lifeboats, but many died. Think of the Titanic disaster as an experiment in how the people of that time behaved when faced with death in a situation where only some
9.37 McNemar’s test. In Exercise 9.23 (page 511), you examined the relationship between being harassed online and being harassed in person for a sample of 1002 girls. An additional question can be asked about these data. Suppose we wanted to compare the proportions of girls who were harassed
9.36 Suspicious results? An instructor who assigned an exercise similar to the one described in the previous exercise received homework from a student who reported a P-value of 0.999. The instructor suspected that the student did not use the computer for the assignment but just made up some numbers
9.35 More on goodness-of-fit to the uniform distribution.Refer to the previous exercise. Use software to generate your own sample of 500 uniform random variables on the interval from 0 to 1 and perform the goodness-of-fit test. Choose a different set of intervals than the ones used in the previous
9.34 Goodness-of-fit to the uniform distribution. Computer software generated 500 random numbers that should look as if they are from the uniform distribution on the interval 0 to 1 (see page 229). They are categorized into five groups: (1) less than or equal to 0.2, (2) greater than 0.2 and less
9.33 Are Mexican Americans less likely to be selected as jurors? Refer to Exercise 8.74 (page 485) concerning Castaneda v. Partida, the case where the Supreme Court review used the phrase “two or three standard deviations” as a criterion for statistical significance. Recall that there were
9.32 Government loans for Canadian students in private career colleges. Refer to the previous exercise. The survey also asked about how these college students paid for their education. A major source of funding was government loans. Here are the survey percents of Canadian private students who used
9.31 When do Canadian students enter private career colleges? A survey of 13,364 Canadian students who enrolled in private career colleges was conducted to understand student participation in the private postsecondary educational system. In one part of the survey, students were asked about their
9.30 Lying to a teacher. One of the questions in a survey of high school students asked about lying to teachers. The following table gives the numbers of students who said that they had lied to a teacher at least once during the past year, classified by sex.Lied at least once Sex Male Female Yes
9.29 DFW rates. One measure of student success for colleges and universities is the percent of admitted students who graduate.Studies indicate that a key issue in retaining students is their performance in so-called gateway courses. These are courses that serve as prerequisites for other key

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