American Black BearsUse the results from Black BearsThe

Problem 2E Total deviation = _____ deviation + _____ deviation

Problem 4E What is an influential observation?

Problem 1E The _____ _____ _____, R2, measures the proportion of total variation in the response variable that is explained by the leastsquares regression line.

Problem 3E A _____ _____ is a scatter diagram with the residuals on the vertical axis and the explanatory variable on the horizontal axis.

In Problems 5–8, analyze the residual plots and identify which, if any, of the conditions for an adequate linear model is not met.

In Problems 5–8, analyze the residual plots and identify which, if any, of the conditions for an adequate linear model is not met.

In Problems 5–8, analyze the residual plots and identify which, if any, of the conditions for an adequate linear model is not met.

In Problems 5–8, analyze the residual plots and identify which, if any, of the conditions for an adequate linear model is not met.

Match the coefficient of determination to the scatter diagram. The scales on the horizontal and vertical axis are the same for each scatter diagram. (a) \(R^{2}=0.58\) (b) \(R^{2}=0.90\) (c) \(R^{2}=1\) (d) \(R^{2}=0.12\) Equation Transcription: Text Transcription: R^2=0.58 R^2=0.90 R^2=1 R^2=0.12

Problem 10E Use the linear correlation coefficient given to determine the coefficient of determination, R2Interpret each R2 (a)r = -0.32 ________________ (b)r = 0.13 ________________ (c)r = 0.40 ________________ (d)r = 0.93

In Problems 11–14, a scatter diagram is given with one of the points drawn in blue. In addition, two least-squares regression lines are drawn: The line drawn in red is the least-squares regression line with the point in blue excluded. The line drawn in blue is the least-squares regression line with the point in blue included. On the basis of these graphs, do you think the point in blue is influential? Why?

In Problems 11–14, a scatter diagram is given with one of the points drawn in blue. In addition, two least-squares regression lines are drawn: The line drawn in red is the least-squares regression line with the point in blue excluded. The line drawn in blue is the least-squares regression line with the point in blue included. On the basis of these graphs, do you think the point in blue is influential? Why?

In Problems 11–14, a scatter diagram is given with one of the points drawn in blue. In addition, two least-squares regression lines are drawn: The line drawn in red is the least-squares regression line with the point in blue excluded. The line drawn in blue is the least-squares regression line with the point in blue included. On the basis of these graphs, do you think the point in blue is influential? Why?

In Problems 11–14, a scatter diagram is given with one of the points drawn in blue. In addition, two least-squares regression lines are drawn: The line drawn in red is the least-squares regression line with the point in blue excluded. The line drawn in blue is the least-squares regression line with the point in blue included. On the basis of these graphs, do you think the point in blue is influential? Why?

Concrete As concrete cures, it gains strength. The following data represent the 7-day and 28-day strength (in pounds per square inch) of a certain type of concrete. 1. A scatter diagram of the data is shown next. What type of relation appears to exist between 7-day strength and 28-day strength? 2. The following residual plot was obtained after finding the least-squares regression line. Does the residual plot confirm that the relation between 7-day and 28-day strength is linear? 3. The coefficient of determination is 57.5%. Provide an interpretation of this value.

The Other Old Faithful Perhaps you are familiar with the famous Old Faithful geyser in Yellowstone National Park. Another Old Faithful geyser is located in Calistoga in California’s Napa Valley. The following data represent the time (in minutes) between eruptions and the length of eruption for 11 randomly selected eruptions. 1. A scatter diagram of the data is shown next. What type of relation appears to exist between time between eruptions and length of eruption? 2. The following residual plot was obtained after finding the least-squares regression line. Does the residual plot confirm that the relation between time between eruptions and length of eruption is linear? 3. The coefficient of determination is 83.0%. Provide an interpretation of this value.

Education and Teen Birthrates The scatter diagram represents the percentage of the population with a bachelor’s degree, x, and teen birthrates (births per 1000 15- to 19-year-olds), y, for the 50 states and Washington, DC, in 2009. Source: U.S. Census Bureau 1. The least-squares regression with Washington, DC, included is \(\hat{y}=-1.2688 x+76.0284\) The least-squares regression line with Washington, DC, excluded is \(\hat{y}=-1.9465 x+93.4464\). Do you think the point corresponding to Washington, DC, may be influential? Why? 2. Washington, DC, often appears as an influential observation in regression analysis. What is it about Washington, DC, that makes it influential? 3. Would you say that higher levels of education cause lower teen birthrates? Equation Transcription: Text Transcription: \hat{y}=-1.2688 x+76.0284 \hat{y}=-1.9465 x+93.4464

Calories versus Sugar The following data represent the number of calories per serving and the number of grams of sugar per serving for a random sample of high-fiber cereals. (a) Draw a scatter diagram of the data treating calories as the explanatory variable. What type of relation appears to exist between calories and sugar content? (b) Determine the correlation between calories and sugar content. Is there a linear relation between calories and sugar content? (c) Suppose that we add Kellogg’s All-Bran cereal, which has 80 calories and 6 grams of sugar per serving, to the data set. Redraw the scatter diagram and recompute the correlation between calories and sugar content. Is there a linear relation between calories and sugar content with All-Bran included in the data set? Is All-Bran an influential observation? (d) Explain why it is necessary to show a scatter diagram with the correlation coefficient when claiming that a linear relation exists between two variables.

Problem 19E An Unhealthy CommuteUse the results from Problem An Unhealthy CommuteThe Gallup Organization regularly surveys adult Americans regarding their commute time to work. In addition, they administer a Well-Being Survey. According to the Gallup Organization, “The Gallup-Healthways Well-Being Index Composite Score is comprised of six sub-indices: Life Evaluation, Emotional Health, Physical Health, Healthy Behavior, Work Environment and Basic Access.” A complete description of the index can be found athttp://www.well-beingindex.com/. The data in the following table are based on the results of the survey, which represent commute time to work (in minutes) and well-being index score. Commute Time (in minutes) Gallup-Healthways Well-Being Index Composite Score 5 69.2 15 68.3 25 67.5 35 67.1 50 66.4 72 66.1 105 63.9 The Gallup Organization (a)Which variable do you believe is likely the explanatory variable and which is the response variable? ________________ (b)Draw a scatter diagram of the data. ________________ (c)Determine the linear correlation coefficient between commute time and well-being index score. ________________ (d)Does a linear relation exist between the commute time and well-being index score? in Sectionand Problem An Unhealthy Commute(Refer to Problem An Unhealthy CommuteThe Gallup Organization regularly surveys adult Americans regarding their commute time to work. In addition, they administer a Well-Being Survey. According to the Gallup Organization, “The Gallup-Healthways Well-Being Index Composite Score is comprised of six sub-indices: Life Evaluation, Emotional Health, Physical Health, Healthy Behavior, Work Environment and Basic Access.” A complete description of the index can be found athttp://www.well-beingindex.com/. The data in the following table are based on the results of the survey, which represent commute time to work (in minutes) and well-being index score. Commute Time (in minutes) Gallup-Healthways Well-Being Index Composite Score 5 69.2 15 68.3 25 67.5 35 67.1 50 66.4 72 66.1 105 63.9 The Gallup Organization (a)Which variable do you believe is likely the explanatory variable and which is the response variable? ________________ (b)Draw a scatter diagram of the data. ________________ (c)Determine the linear correlation coefficient between commute time and well-being index score. ________________ (d)Does a linear relation exist between the commute time and well-being index score?, Section.) The following data represent commute times (in minutes) and score on a well-being survey. Commute Time (minutes), x Gallup-Healthways Well-Being Index Composite Score, y 5 69.2 15 68.3 25 67.5 35 67.1 50 66.4 72 66.1 105 63.9 (a)Find the least-squares regression line treating the commute time, x, as the explanatory variable and the index score, y, as the response variable. ________________ (b)Interpret the slope and y-intercept, if appropriate. ________________ (c)Predict the well-being index of a person whose commute is 30 minutes. ________________ (d)Suppose Barbara has a 20-minute commute and scores 67.3 on the survey. Is Barbara more “well-off” than the typical individual who has a 20-minute commute? in Sectionto: (a)Determine the coefficient of determination, R2 (b)Construct a residual plot to verify the model requirements of the least-squares regression model. ________________ (c)Interpret the coefficient of determination and comment on the adequacy of the linear model.

Problem 20E Credit ScoresUse the results from Problem Credit ScoresYour Fair Isaacs Corporation (FICO) credit score is used to determine your creditworthiness. It is used to help determine whether you qualify for a mortgage or credit and is even used to determine insurance rates. FICO scores have a range of 300 to 850, with a higher score indicating a better credit history. The given data represent the interest rate (in percent) a bank would offer on a 36-month auto loan for various FICO scores. Credit Score Interest Rate (percent) 545 18.982 595 17.967 640 12.218 675 8.612 705 6.680 750 5.150 www.myfico.com (a)Which variable do you believe is likely the explanatory variable and which is the response variable? ________________ (b)Draw a scatter diagram of the data. ________________ (c)Determine the linear correlation coefficient between FICO score and interest rate on a 36-month auto loan. ________________ (d)Does a linear relation exist between the FICO score and interest rate? in Sectionand Problem Credit Scores(Refer to Problem Credit ScoresYour Fair Isaacs Corporation (FICO) credit score is used to determine your creditworthiness. It is used to help determine whether you qualify for a mortgage or credit and is even used to determine insurance rates. FICO scores have a range of 300 to 850, with a higher score indicating a better credit history. The given data represent the interest rate (in percent) a bank would offer on a 36-month auto loan for various FICO scores. Credit Score Interest Rate (percent) 545 18.982 595 17.967 640 12.218 675 8.612 705 6.680 750 5.150 www.myfico.com (a)Which variable do you believe is likely the explanatory variable and which is the response variable? ________________ (b)Draw a scatter diagram of the data. ________________ (c)Determine the linear correlation coefficient between FICO score and interest rate on a 36-month auto loan. ________________ (d)Does a linear relation exist between the FICO score and interest rate?, Section.) An economist wants to determine the relation between one’s FICO score, x, and the interest rate of a 36-month auto loan, y. The given data represent the interest rate (in percent) a bank would offer on a 36-month auto loan for various FICO scores. Credit Score, x Interest Rate (percent), y 545 18.982 595 17.967 640 12.218 675 8.612 705 6.680 750 5.150 www.myfico.com (a)Find the least-squares regression line treating the FICO score, x, as the explanatory variable and the interest rate, y, as the response variable. ________________ (b)Interpret the slope and y-intercept, if appropriate. Note: Credit scores have a range of 300 to 850. ________________ (c)Predict the interest rate a person would pay if her FICO scores were the median score of 723. ________________ (d)Suppose Bob has a FICO score of 680 and he is offered an interest rate of 8.3%. Is this a good offer? Why? in Sectionto: (a)Determine the coefficient of determination, R2 (b)Construct a residual plot to verify the model requirements of the least-squares regression model. ________________ (c)Interpret the coefficient of determination and comment on the adequacy of the linear model.

American Black Bears Use the results from Problem 28 in Section 4.1 and Problem 20 in Section 4.2 to: (a) Compute the coefficient of determination, R2. (b) Construct a residual plot to verify the requirements of the least-squares regression model. (c) Interpret the coefficient of determination and comment on the adequacy of the linear model.

Bone Length Use the results from Problem 30 in Section 4.1 and Problem 22 in Section 4.2. (a) What proportion of the variability in the length of the right tibia is explained by the relation between the length of the right humerus and the length of the right tibia? (b) Construct a residual plot to verify the requirements of the least-squares regression model. (c) Interpret the coefficient of determination and comment on the adequacy of the linear model.

Height versus Head Circumference Use the results from Problem 27 in Section 4.1 and Problem 19 in Section 4.2 to: (a) Compute the coefficient of determination, R2. (b) Construct a residual plot to verify the requirements of the least-squares regression model. (c) Interpret the coefficient of determination and comment on the adequacy of the linear model.

Problem 25E Kepler’s Law of Planetary MotionThe time it takes for a planet to complete its orbit around the sun is called the planet’s sidereal year. In 1618, Johannes Kepler discovered that the sidereal year of a planet is related to the distance the planet is from the sun. The following data show the distances of the planets, and the dwarf planet Pluto, from the sun and their sidereal years. Planet Distance from Sun, x (millions of miles) Sidereal Year, y Mercury 36 0.24 Venus 67 0.62 Earth 93 1.00 Mars 142 1.88 Jupiter 483 11.9 Saturn 887 29.5 Uranus 1785 84.0 Neptune 2797 165.0 Pluto 3675 248.0 (a)Draw a scatter diagram of the data treating distance from the sun as the explanatory variable. ________________ (b)Determine the correlation between distance and sidereal year. Does this imply a linear relation between distance and sidereal year? ________________ (c)Compute the least-squares regression line. ________________ (d)Plot the residuals against the distance from the sun. ________________ (e)Do you think the least-squares regression line is a good model? Why?

Problem 26E Wind Chill FactorThe wind chill factor depends on wind speed and air temperature. The following data represent the wind speed (in mph) and wind chill factor at an air temperature of 15?Fahrenheit. Wind Speed, x (mph) Wind Chill Factor, y Wind Speed, x (mph) Wind Chill Factor, y 5 12 25 -22 10 -3 30 -25 15 -11 35 -27 20 -17 (a)Draw a scatter diagram of the data treating wind speed as the explanatory variable. ________________ (b)Determine the correlation between wind speed and wind chill factor. Does this imply a linear relation between wind speed and wind chill factor? ________________ (c)Compute the least-squares regression line. ________________ (d)Plot the residuals against the wind speed. ________________ (e)Do you think the least-squares regression line is a good model? Why?

Weight of a Car versus Miles per Gallon Use the results from Problem 29 in Section 4.1 and Problem 21 in Section 4.2. (a) What proportion of the variability in miles per gallon is explained by the relation between weight of the car and miles per gallon? (b) Construct a residual plot to verify the requirements of the least-squares regression model. Are there any problems with the model? (c) Interpret the coefficient of determination and comment on the adequacy of the linear model.

Weight of a Car versus Miles per Gallon Suppose that we add the Dodge Viper to the data in Problem 21 in Section 4.2. A Dodge Viper weighs 3425 pounds and gets 11 miles per gallon. 1. Compute the coefficient of determination of the expanded data set. What effect does the addition of the Viper to the data set have on \(R^{2}\)? 2. Is the point corresponding to the Dodge Viper influential? Is it an outlier? Equation Transcription: Text Transcription: R^{2}

Problem 28E American Black BearsSuppose that we find a bear that is 205 cm tall and weighs 187 kg and add the bear to the data in Problem 20 from Section. (a)Compute the coefficient of determination of the expanded data set. What effect does the additional bear have on R2? ________________ (b)Is the new bear influential? Is it an outlier?

Problem 31E Putting It Together: Building a Financial ModelIn Section. we introduced one measure of risk used by financial analysts, the standard deviation of rate of return. In this problem we will learn about a second measure of risk of a stock, the beta of a stock. The following data represent the annual rate of return of General Electric (GE) stock and the annual rate of return of the Standard & Poor’s 500 (S&P 500) Index for the past 15 years. Year Rate of Return of S&P 500 Rate of Return of GE 1996 0.203 0.402 1997 0.310 0.510 1998 0.267 0.410 1999 0.195 0.536 2000 -0.101 -0.060 2001 -0.130 -0.151 2002 -0.234 -0.377 2003 0.264 0.308 2004 0.090 0.207 2005 0.030 -0.014 2006 0128 0.093 2007 -0.035 0.027 2008 -0.385 -0.593 2009 0.235 -0.102 2010 0.067 0.053 (a)Draw a scatter diagram of the data treating the rate of return of the S&P 500 as the explanatory variable. ________________ (b)Determine the correlation coefficient between rate of return of the S&P 500 and GE stock. ________________ (c)Based on the scatter diagram and correlation coefficient, is there a linear relation between rate of return of the S&P 500 and GE? ________________ (d)Find the least-squares regression line treating the rate of return of the S&P 500 as the explanatory variable. ________________ (e)Predict the rate of return of GE stock if the rate of return of the S&P 500 is 0.10 (10%). ________________ (f )If the actual rate of return for GE was 13.2% when the rate of return of the S&P 500 was 10%, was GE’s performance above or below average among all years the S&P 500 returns were 10%? ________________ (g)Interpret the slope. ________________ (h)Interpret the intercept, if appropriate. ________________ (i)What proportion of the variability in GE’s rate of return is explained by the variability in the rate of return of the S&P 500? ________________ (j)Plot residuals against the rate of return of the S&P 500. Does the residual plot confirm that the relation between the rate of return of the S&P 500 and GE stock is linear? ________________ (k)Are there any years where GE’s rate of return was unusual?

Problem 32E Putting It Together : Exam ScoresThe following data represent scores earned by students in Sullivan’s Elementary Algebra course for Chapter 2 (Linear Equations and Inequalities in One Variable) and Chapter 3 (Linear Equations and Inequalities in Two Variables). Completely summarize the relation between Chapter 2 and Chapter 3 exam scores, treating Chapter 2 exam scores as the explanatory variable. Write a report detailing the results of the analysis including the presence of any outliers or influential points. What does the relationship say about the role Chapter 2 plays in a student’s understanding of Chapter 3 Chapter 2 Score Chapter 3 Score 71.4 76.1 76.2 82.6 60.3 60.9 88.1 91.3 95.2 78.3 82.5 100 100 95.7 87.3 81.5 71.4 50.7 95.2 81.5 95.2 87.0 85.7 73.9 71.4 74.6 33.3 45.7 78.0 37.3 83.3 88.0 100 100 81.0 76.1 76.2 63.0

Problem 29E CO2 Emissions and EnergyThe following data represent the carbon dioxide emissions (in thousands of tons), y, and energy generated (thousands of megawatts), x, for a random sample of 14 power plants in Oregon. Is there anything unusual about the Boardman Power Plant, which produces 4618 thousand megawatts of energy and emits 4813 thousand tons of CO2? What about the Hermiston power plant, which produces 2636 thousand megawatts of energy and emits 1311 thousand tons of CO2? Energy, x Carbon, y 4618 4813 2636 1311 535 377 357 341 370 238 15 44 25 18 22 15 13 10 64 6 13 5 33 3 24 2 17 2 (www.carma.org)

The _____ _____ _____, R2 , measures the proportion of total variation in the response variable that is explained by the leastsquares regression line.

Total deviation = _____ deviation + _____ deviation

A _____ _____ is a scatter diagram with the residuals on the vertical axis and the explanatory variable on the horizontal axis.

What is an influential observation?

In Problems 58, analyze the residual plots and identify which, if any, of the conditions for an adequate linear model is not met.

In Problems 58, analyze the residual plots and identify which, if any, of the conditions for an adequate linear model is not met.

In Problems 58, analyze the residual plots and identify which, if any, of the conditions for an adequate linear model is not met.

In Problems 58, analyze the residual plots and identify which, if any, of the conditions for an adequate linear model is not met.

Match the coefficient of determination to the scatter diagram. The scales on the horizontal and vertical axis are the same for each scatter diagram. (a) R2 = 0.58 (b) R2 = 0.90 (c) R2 = 1 (d) R2 = 0.12 Response Explanatory (I) Response Explanatory (II)

Use the linear correlation coefficient given to determine the coefficient of determination, R2 . Interpret each R2 . (a) r = -0.32 (b) r = 0.13 (c) r = 0.40 (d) r = 0.93

In Problems 1114, a scatter diagram is given with one of the points drawn in blue. In addition, two least-squares regression lines are drawn: The line drawn in red is the least-squares regression line with the point in blue excluded. The line drawn in blue is the least-squares regression line with the point in blue included. On the basis of these graphs, do you think the point in blue is influential? Why?

In Problems 1114, a scatter diagram is given with one of the points drawn in blue. In addition, two least-squares regression lines are drawn: The line drawn in red is the least-squares regression line with the point in blue excluded. The line drawn in blue is the least-squares regression line with the point in blue included. On the basis of these graphs, do you think the point in blue is influential? Why?

In Problems 1114, a scatter diagram is given with one of the points drawn in blue. In addition, two least-squares regression lines are drawn: The line drawn in red is the least-squares regression line with the point in blue excluded. The line drawn in blue is the least-squares regression line with the point in blue included. On the basis of these graphs, do you think the point in blue is influential? Why?

In Problems 1114, a scatter diagram is given with one of the points drawn in blue. In addition, two least-squares regression lines are drawn: The line drawn in red is the least-squares regression line with the point in blue excluded. The line drawn in blue is the least-squares regression line with the point in blue included. On the basis of these graphs, do you think the point in blue is influential? Why?

The Other Old Faithful Perhaps you are familiar with the famous Old Faithful geyser in Yellowstone National Park. Another Old Faithful geyser is located in Calistoga in Californias Napa Valley. The following data represent the time (in minutes) between eruptions and the length of eruption for 11 randomly selected eruptions. Time between Eruptions, x Length of Eruption, y Time between Eruptions, x Length of Eruption, y 12.17 1.88 11.70 1.82 11.63 1.77 12.27 1.93 12.03 1.83 11.60 1.77 12.15 1.83 11.72 1.83 11.30 1.70 Source: Ladonna Hansen, Park Curator (a) A scatter diagram of the data is shown next. What type of relation appears to exist between time between eruptions and length of eruption? (b) The following residual plot was obtained after finding the least-squares regression line. Does the residual plot confirm that the relation between time between eruptions and length of eruption is linear? (c) The coefficient of determination is 83.0%. Provide an interpretation of this value

Concrete As concrete cures, it gains strength. The following data represent the 7-day and 28-day strength (in pounds per square inch) of a certain type of concrete. 7-Day Strength, x 28-Day Strength, y 7-Day Strength, x 28-Day Strength, y 2300 4070 2480 4120 3390 5220 3380 5020 2430 4640 2660 4890 2890 4620 2620 4190 3330 4850 3340 4630 (a) A scatter diagram of the data is shown next. What type of relation appears to exist between 7-day strength and 28-day strength? (b) The following residual plot was obtained after finding the least-squares regression line. Does the residual plot confirm that the relation between 7-day and 28-day strength is linear? (c) The coefficient of determination is 57.5%. Provide an interpretation of this value

Calories versus Sugar The following data represent the number of calories per serving and the number of grams of sugar per serving for a random sample of high-f iber cereals. Calories, x Sugar, y Calories, x Sugar, y 200 18 210 23 210 23 210 16 170 17 210 17 190 20 190 12 200 18 190 11 180 19 200 11 Source: Consumer Reports (a) Draw a scatter diagram of the data treating calories as the explanatory variable. What type of relation appears to exist between calories and sugar content? (b) Determine the correlation between calories and sugar content. Is there a linear relation between calories and sugar content? (c) Suppose that we add Kelloggs All-Bran cereal, which has 80 calories and 6 grams of sugar per serving, to the data set. Redraw the scatter diagram and recompute the correlation between calories and sugar content. Is there a linear relation between calories and sugar content with All-Bran included in the data set? Is All-Bran an influential observation? (d) Explain why it is necessary to show a scatter diagram with the correlation coefficient when claiming that a linear relation exists between two variables.

Education and Teen Birthrates The scatter diagram represents the percentage of the population with a bachelors degree, x, and teen birthrates (births per 1000 15- to 19-year-olds), y, for the 50 states and Washington, DC, in 2009. Source: U.S. Census Bureau (a) The least-squares regression with Washington, DC, included is yn = -1.2688 x + 76.0284. The least-squares regression line with Washington, DC, excluded is yn = -1.9465 x + 93.4464. Do you think the point corresponding to Washington, DC, may be influential? Why? (b) Washington, DC, often appears as an influential observation in regression analysis. What is it about Washington, DC, that makes it influential? (c) Would you say that higher levels of education cause lower teen birthrates?

Problems 1924 use the results from Problems 2530 in Section 4.1 and Problems 1722 in Section 4.2 An Unhealthy Commute Use the results from Problem 25 in Section 4.1 and Problem 17 in Section 4.2 to: (a) Determine the coefficient of determination, R2 . (b) Construct a residual plot to verify the model requirements of the least-squares regression model. (c) Interpret the coefficient of determination and comment on the adequacy of the linear model.

Problems 1924 use the results from Problems 2530 in Section 4.1 and Problems 1722 in Section 4.3Credit Scores Use the results from Problem 26 in Section 4.1 and Problem 18 in Section 4.2 to: (a) Determine the coefficient of determination, R2 . (b) Construct a residual plot to verify the model requirements of the least-squares regression model. (c) Interpret the coefficient of determination and comment on the adequacy of the linear model.

Problems 1924 use the results from Problems 2530 in Section 4.1 and Problems 1722 in Section 4.4 Height versus Head Circumference Use the results from Problem 27 in Section 4.1 and Problem 19 in Section 4.2 to: (a) Compute the coefficient of determination, R2 . (b) Construct a residual plot to verify the requirements of the least-squares regression model. (c) Interpret the coefficient of determination and comment on the adequacy of the linear model.

Problems 1924 use the results from Problems 2530 in Section 4.1 and Problems 1722 in Section 4.5American Black Bears Use the results from Problem 28 in Section 4.1 and Problem 20 in Section 4.2 to: (a) Compute the coefficient of determination, R2 . (b) Construct a residual plot to verify the requirements of the least-squares regression model. (c) Interpret the coefficient of determination and comment on the adequacy of the linear model.

Problems 1924 use the results from Problems 2530 in Section 4.1 and Problems 1722 in Section 4.6Weight of a Car versus Miles per Gallon Use the results from Problem 29 in Section 4.1 and Problem 21 in Section 4.2. (a) What proportion of the variability in miles per gallon is explained by the relation between weight of the car and miles per gallon? (b) Construct a residual plot to verify the requirements of the leastsquares regression model. Are there any problems with the model? (c) Interpret the coefficient of determination and comment on the adequacy of the linear model.

Problems 1924 use the results from Problems 2530 in Section 4.1 and Problems 1722 in Section 4.7Bone Length Use the results from Problem 30 in Section 4.1 and Problem 22 in Section 4.2. (a) What proportion of the variability in the length of the right tibia is explained by the relation between the length of the right humerus and the length of the right tibia? (b) Construct a residual plot to verify the requirements of the least-squares regression model. (c) Interpret the coefficient of determination and comment on the adequacy of the linear model

Keplers Law of Planetary Motion The time it takes for a planet to complete its orbit around the sun is called the planets sidereal year. In 1618, Johannes Kepler discovered that the sidereal year of a planet is related to the distance the planet is from the sun. The following data show the distances of the planets, and the dwarf planet Pluto, from the sun and their sidereal years. Planet Distance from Sun, x (millions of miles) Sidereal Year, y Mercury 36 0.24 Venus 67 0.62 Earth 93 1.00 Mars 142 1.88 Jupiter 483 11.9 Saturn 887 29.5 Uranus 1785 84.0 Neptune 2797 165.0 Pluto 3675 248.0 (a) Draw a scatter diagram of the data treating distance from the sun as the explanatory variable. (b) Determine the correlation between distance and sidereal year. Does this imply a linear relation between distance and sidereal year? (c) Compute the least-squares regression line. (d) Plot the residuals against the distance from the sun. (e) Do you think the least-squares regression line is a good model? Why?

Wind Chill Factor The wind chill factor depends on wind speed and air temperature. The following data represent the wind speed (in mph) and wind chill factor at an air temperature of 15 Fahrenheit. Wind Speed, x (mph) Wind Chill Factor, y Wind Speed, x (mph) Wind Chill Factor, y 5 12 25 -22 10 -3 30 -25 15 -11 35 -27 20 -17 Source: Information Please Almanac (a) Draw a scatter diagram of the data treating wind speed as the explanatory variable. (b) Determine the correlation between wind speed and wind chill factor. Does this imply a linear relation between wind speed and wind chill factor? (c) Compute the least-squares regression line. (d) Plot the residuals against the wind speed. (e) Do you think the least-squares regression line is a good model? Why?

Weight of a Car versus Miles per Gallon Suppose that we add the Dodge Viper to the data in Problem 21 in Section 4.2. A Dodge Viper weighs 3425 pounds and gets 11 miles per gallon. (a) Compute the coefficient of determination of the expanded data set. What effect does the addition of the Viper to the data set have on R2 ? (b) Is the point corresponding to the Dodge Viper influential? Is it an outlier?

American Black Bears Suppose that we find a bear that is 205 cm tall and weighs 187 kg and add the bear to the data in Problem 20 from Section 4.2. (a) Compute the coefficient of determination of the expanded data set. What effect does the additional bear have on R2 ? (b) Is the new bear influential? Is it an outlier?

CO2 Emissions and Energy The following data represent the carbon dioxide emissions (in thousands of tons), y, and energy generated (thousands of megawatts), x, for a random sample of 14 power plants in Oregon. Is there anything unusual about the Boardman Power Plant, which produces 4618 thousand megawatts of energy and emits 4813 thousand tons of CO2? What about the Hermiston power plant, which produces 2636 thousand megawatts of energy and emits 1311 thousand tons of CO2? Energy, x Carbon, y 4618 4813 2636 1311 535 377 357 341 370 238 15 44 25 18 22 15 13 10 64 6 13 5 33 3 24 2 17 2 Source: CARMA (www.carma.org)

Age versus Study Time Professor Katula feels that there is a relation between the number of hours a statistics student studies each week and the students age. She conducts a survey in which 26 statistics students are asked their age and the number of hours they study statistics each week. She obtains the following results: Age, x Hours Studying, y Age, x Hours Studying, y Age, x Hours Studying, y 18 4.2 19 5.1 22 2.1 18 1.1 19 2.3 22 3.6 18 4.6 20 1.7 24 5.4 18 3.1 20 6.1 25 4.8 18 5.3 20 3.2 25 3.9 18 3.2 20 5.3 26 5.2 19 2.8 21 2.5 26 4.2 19 2.3 21 6.4 35 8.1 19 3.2 21 4.2 (a) Draw a scatter diagram of the data. Comment on any potential influential observations. (b) Find the least-squares regression line using all the data points. (c) Find the least-squares regression line with the data point (35, 8.1) removed. (d) Draw each least-squares regression line on the scatter diagram obtained in part (a). (e) Comment on the influence that the point (35, 8.1) has on the regression line.

Putting It Together: Building a Financial Model In Section 3.2, we introduced one measure of risk used by financial analysts, the standard deviation of rate of return. In this problem we will learn about a second measure of risk of a stock, the beta of a stock. The following data represent the annual rate of return of General Electric (GE) stock and the annual rate of return of the Standard & Poors 500 (S&P 500) Index for the past 15 years.Year Rate of Return of S&P 500 Rate of Return of GE 1996 0.203 0.402 1997 0.310 0.510 1998 0.267 0.410 1999 0.195 0.536 2000 -0.101 -0.060 2001 -0.130 -0.151 2002 -0.234 -0.377 2003 0.264 0.308 2004 0.090 0.207 2005 0.030 -0.014 2006 0.128 0.093 2007 -0.035 0.027 2008 -0.385 -0.593 2009 0.235 -0.102 2010 0.067 0.053 Source: finance.yahoo.com (a) Draw a scatter diagram of the data treating the rate of return of the S&P 500 as the explanatory variable. (b) Determine the correlation coefficient between rate of return of the S&P 500 and GE stock. (c) Based on the scatter diagram and correlation coefficient, is there a linear relation between rate of return of the S&P 500 and GE? (d) Find the least-squares regression line treating the rate of return of the S&P 500 as the explanatory variable. (e) Predict the rate of return of GE stock if the rate of return of the S&P 500 is 0.10 (10%). (f ) If the actual rate of return for GE was 13.2% when the rate of return of the S&P 500 was 10%, was GEs performance above or below average among all years the S&P 500 returns were 10%? (g) Interpret the slope. (h) Interpret the intercept, if appropriate. (i) What proportion of the variability in GEs rate of return is explained by the variability in the rate of return of the S&P 500? ( j) Plot residuals against the rate of return of the S&P 500. Does the residual plot confirm that the relation between the rate of return of the S&P 500 and GE stock is linear? (k) Are there any years where GEs rate of return was unusual?

Putting It Together : Exam Scores The following data represent scores earned by students in Sullivans Elementary Algebra course for Chapter 2 (Linear Equations and Inequalities in One Variable) and Chapter 3 (Linear Equations and Inequalities in Two Variables). Completely summarize the relation between Chapter 2 and Chapter 3 exam scores, treating Chapter 2 exam scores as the explanatory variable. Write a report detailing the results of the analysis including the presence of any outliers or influential points. What does the relationship say about the role Chapter 2 plays in a students understanding of Chapter 3? Chapter 2 Score Chapter 3 Score 71.4 76.1 76.2 82.6 60.3 60.9 88.1 91.3 95.2 78.3 82.5 100 100 95.7 87.3 81.5 71.4 50.7 95.2 81.5 95.2 87.0 85.7 73.9 71.4 74.6 33.3 45.7 78.0 37.3 83.3 88.0 100 100 81.0 76.1 76.2 63.0 Source: Michael Sullivan