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BU 5710 Fall 2019 Homework 2 1. Page 100, Question 4.3 (a-d) 2. Pages 100-101, Question 4.5 (a-d) 3. Baseball Franchise Values The value of a sports franchise is directly related to the amount of revenue that a franchise can generate. The file BBValues represents the value in 2014 (in $millions) and the annual revenue (in Smillions) for the 30 Major League Baseball franchises. (Data extracted from www. forbes.com/mlb-valuations/list). Suppose you want to develop a simple linear regression model to predict franchise value based on annual revenue generated. a. Construct a scatterplot. b. Assuming a linear relationship, use the least squares method to determine the regression coefficients bo (intercept) and b (slope). c. Interpret the meaning of bo and b, in this problem d. Use (b) above to predict the mean value of a baseball franchise that generates $250 million in annual revenue. e. What would you tell a group considering an investment in a major league baseball team about the relationship between revenue and the value of a team? f. What is the coefficient of determination (R2) of the regression? Interpret its meaning. g. What is the standard error of the regression? What does it mean? h. How useful do you think this regression model is for predicting the value of a baseball franchise? i. Perform a residual analysis. Based on your results, evaluate whether the assumptions of regression have been seriously violated. 66 Chapter Three Summary Sretisti Histogram of the Diameter of Elevator Rails from Two Different Machines 0.45 Relative Frequency 0.12 0.05 11-11.99 12-12.99 13-13.99 14-14.99 15-15.99 16-16.99 17-17.99 18-18.99 19-19.99 20-20.99 21-21.99 22-22.99 23-2399 Elevator Rall Diameter % % 24-24.99 25-25.99 26-26.99 27-27.99 28-28.99 29-29,99 3.4 Se A human resource manager in concerned about employee job satisfaction. A sample of 26 employees was selected to complete a questionnaire designed to measure job satisfac- tion. The data are shown here. Higher scores indicate a greater degree of job satisfaction. 31 36 42 47 47 56 56 59 60 61 64 69 70 72 73 74 75 76 76 77 78 82 82 84 84 85 *-1,716, Z4 - 119,234, (5 7 * = 5978 a. Compute Pearson's coefficient of skewness, and explain what it tells you about the histogram for these data. b. Compute the summary statistic that best measures the central tendency or norm in these data. c. Compute the summary statistic(s) that best measures the variation in these data, 3.5 Annual sales, in millions of dollars, for 21 pharmaceutical companies follow. 8,408 1,374 1,872 8,879 2,459 11,413 14,138 6,452 1,850 2,818 1,356 10,498 7,478 4,019 4,341 739 2,127 3,653 5,794 8,305 a. Provide a five-number summary. b. Do these data contain any outliers? c. Johnson & Johnson's sales are the largest on the list at $14,138 million. Suppose a data entry error (a transposition) had been made and the sales had been entered as $41,138 million. Would the method of detecting outliers in part (c) identify this problem and allow for a correction of the data-entry error? 3.6 The manager of a local fast-food restaurant is interested in improving the service provided to customers who use the restaurant's drive-up window. The manager asks his assistant to record the time (in minutes) it takes to serve 200 customers at the final window in the facility's drive-up system. The following 50 customer service times are observed for an hour in the day. Chapter Four Simple Liner Regression 4.2 Consider the following scatterplot: WW - O 4.3 4.4 Using simple linear regression analysis, the equation for a line through the data is estimated to be y - 1.9 + 0.55x. For each of the x values, calculate the observed y, the predicted y, and the residual. For the following four regression equations, explain what the slope and intercept mean. a. wage = 2.05 + 1.32education, where wage is dollars earned per hour and educa. tion is number of years the person went to school. b. GPA = 1.14 + 0.23hours study, where GPA is measured in points and hours study is the number of hours spent studying in a week. c. sleep = 10.33 -0.44work, where sleep is hours spent sleeping per night and work is the number of hours worked per day. d. savings = 586 + 0.15 salary, where savings is dollars saved in the bank and the salary is the number of dollars earned in a year. Name the type of error that has occurred in each of the following circumstances. a. You specified the equation as wage = Bo + B, education + e, but wage is also determined by the ability of the person. b. You specified the equation as a linear function, but the scatterplot has a quadratic shape. c. When collecting data to determine the relationship between GPA and hours spent studying every week, most students do not know the exact hours they spent study ing each week and have to give an estimate. d. You eat lunch at the same cafeteria every day. Some days you choose to eat a hamburger and other days you choose to eat a taco, even though these choices cost the same amount and you have the same income to spend each day. For the following four regressions, predict the value of the y variable for the value of the x variable given. a. wage = 2.05 + 1.32education, where wage is dollars earned per hour and educa- tion is number of years the person went to school. Predict the wage if a person has 15 years of education. b. GPA = 1.14 + 0.23hours study, where GPA is measured in points and hours study is the number of hours spent studying in a week. Predict the GPA for a student who studies 11 hours a week. c. sleep = 10.33 -0.44work, where sleep is hours spent sleeping per night and work is the number of hours worked per day. Predict the amount of sleep a person gets a night if they work 7 hours a day. 4.5 Chapter Four Simple Liner Regression d avings = 586 +0.13salary where savings 18 dollars saved in the bank and the salary is the number of dollars earned in year. Predict the amount of savings a person has if they earn $25,000 a year. Grinfield Service Company's marketing director is interested in analyzing the rela- cionship between her company's sales and the advertising dollars spent. In the course of her analysis, she selected a random sample of 20 weeks and recorded the sales for cach week and the amount spent on advertising. The summary statistics are for 20 observations Sample mean Standard deviation Sales (5) 3,353 1,408 Advertising (5) 298 133 Cowx, y) = 170,436.10 a. Find the correlation coefficient between sales and advertising. What does it mean? b. Find the least squares line that shows how weekly sales (y) is related to advertis- ing dollars (1) c. According to the least squares line, how is weekly sales related to advertising dollars? What does the intercept mean? What is the standard error of the regression model in the following scenarios? a. SSUnexplained = 83, n - 2 = 58. b. SS Total = 1234, SSExplained = 865, n = 145. c. SSExplained - 56, SSTotal = 78, n = 78. 4.7 Exercises E4.1 For each of the following three data sets, create a scatterplot, calculate the regression line, determine the R', and explain what the slope means a. X b. X DOO VOUWN- ono ovawan ova W * N E4.2 Consider the two data sets listed in In the first data set, the prices of houses are listed in dollars and the square footage is listed in feet, while the second data set lists the prices of houses in hundreds of dollars and the square footage in yards. Calculate the covariance and the correlation coefficient for both data sets, and comment on the similarities and differences between your findings. Is this what you expected? E4.3 A business owner is interested in how the cost of shipping a package is related to the distance a package is shipped. To this end, data have been collected and are contained in ship. a. Calculate a correlation coefficient, and explain what it means about the relation ship between shipping cost and distance. b. Now calculate the estimated sample regression function. What do the slope and intercept mean? c. If the company is going to ship a package 1.200 miles, how much do you estimate it will cost? BU 5710 Fall 2019 Homework 2 1. Page 100, Question 4.3 (a-d) 2. Pages 100-101, Question 4.5 (a-d) 3. Baseball Franchise Values The value of a sports franchise is directly related to the amount of revenue that a franchise can generate. The file BBValues represents the value in 2014 (in $millions) and the annual revenue (in Smillions) for the 30 Major League Baseball franchises. (Data extracted from www. forbes.com/mlb-valuations/list). Suppose you want to develop a simple linear regression model to predict franchise value based on annual revenue generated. a. Construct a scatterplot. b. Assuming a linear relationship, use the least squares method to determine the regression coefficients bo (intercept) and b (slope). c. Interpret the meaning of bo and b, in this problem d. Use (b) above to predict the mean value of a baseball franchise that generates $250 million in annual revenue. e. What would you tell a group considering an investment in a major league baseball team about the relationship between revenue and the value of a team? f. What is the coefficient of determination (R2) of the regression? Interpret its meaning. g. What is the standard error of the regression? What does it mean? h. How useful do you think this regression model is for predicting the value of a baseball franchise? i. Perform a residual analysis. Based on your results, evaluate whether the assumptions of regression have been seriously violated. 66 Chapter Three Summary Sretisti Histogram of the Diameter of Elevator Rails from Two Different Machines 0.45 Relative Frequency 0.12 0.05 11-11.99 12-12.99 13-13.99 14-14.99 15-15.99 16-16.99 17-17.99 18-18.99 19-19.99 20-20.99 21-21.99 22-22.99 23-2399 Elevator Rall Diameter % % 24-24.99 25-25.99 26-26.99 27-27.99 28-28.99 29-29,99 3.4 Se A human resource manager in concerned about employee job satisfaction. A sample of 26 employees was selected to complete a questionnaire designed to measure job satisfac- tion. The data are shown here. Higher scores indicate a greater degree of job satisfaction. 31 36 42 47 47 56 56 59 60 61 64 69 70 72 73 74 75 76 76 77 78 82 82 84 84 85 *-1,716, Z4 - 119,234, (5 7 * = 5978 a. Compute Pearson's coefficient of skewness, and explain what it tells you about the histogram for these data. b. Compute the summary statistic that best measures the central tendency or norm in these data. c. Compute the summary statistic(s) that best measures the variation in these data, 3.5 Annual sales, in millions of dollars, for 21 pharmaceutical companies follow. 8,408 1,374 1,872 8,879 2,459 11,413 14,138 6,452 1,850 2,818 1,356 10,498 7,478 4,019 4,341 739 2,127 3,653 5,794 8,305 a. Provide a five-number summary. b. Do these data contain any outliers? c. Johnson & Johnson's sales are the largest on the list at $14,138 million. Suppose a data entry error (a transposition) had been made and the sales had been entered as $41,138 million. Would the method of detecting outliers in part (c) identify this problem and allow for a correction of the data-entry error? 3.6 The manager of a local fast-food restaurant is interested in improving the service provided to customers who use the restaurant's drive-up window. The manager asks his assistant to record the time (in minutes) it takes to serve 200 customers at the final window in the facility's drive-up system. The following 50 customer service times are observed for an hour in the day. Chapter Four Simple Liner Regression 4.2 Consider the following scatterplot: WW - O 4.3 4.4 Using simple linear regression analysis, the equation for a line through the data is estimated to be y - 1.9 + 0.55x. For each of the x values, calculate the observed y, the predicted y, and the residual. For the following four regression equations, explain what the slope and intercept mean. a. wage = 2.05 + 1.32education, where wage is dollars earned per hour and educa. tion is number of years the person went to school. b. GPA = 1.14 + 0.23hours study, where GPA is measured in points and hours study is the number of hours spent studying in a week. c. sleep = 10.33 -0.44work, where sleep is hours spent sleeping per night and work is the number of hours worked per day. d. savings = 586 + 0.15 salary, where savings is dollars saved in the bank and the salary is the number of dollars earned in a year. Name the type of error that has occurred in each of the following circumstances. a. You specified the equation as wage = Bo + B, education + e, but wage is also determined by the ability of the person. b. You specified the equation as a linear function, but the scatterplot has a quadratic shape. c. When collecting data to determine the relationship between GPA and hours spent studying every week, most students do not know the exact hours they spent study ing each week and have to give an estimate. d. You eat lunch at the same cafeteria every day. Some days you choose to eat a hamburger and other days you choose to eat a taco, even though these choices cost the same amount and you have the same income to spend each day. For the following four regressions, predict the value of the y variable for the value of the x variable given. a. wage = 2.05 + 1.32education, where wage is dollars earned per hour and educa- tion is number of years the person went to school. Predict the wage if a person has 15 years of education. b. GPA = 1.14 + 0.23hours study, where GPA is measured in points and hours study is the number of hours spent studying in a week. Predict the GPA for a student who studies 11 hours a week. c. sleep = 10.33 -0.44work, where sleep is hours spent sleeping per night and work is the number of hours worked per day. Predict the amount of sleep a person gets a night if they work 7 hours a day. 4.5 Chapter Four Simple Liner Regression d avings = 586 +0.13salary where savings 18 dollars saved in the bank and the salary is the number of dollars earned in year. Predict the amount of savings a person has if they earn $25,000 a year. Grinfield Service Company's marketing director is interested in analyzing the rela- cionship between her company's sales and the advertising dollars spent. In the course of her analysis, she selected a random sample of 20 weeks and recorded the sales for cach week and the amount spent on advertising. The summary statistics are for 20 observations Sample mean Standard deviation Sales (5) 3,353 1,408 Advertising (5) 298 133 Cowx, y) = 170,436.10 a. Find the correlation coefficient between sales and advertising. What does it mean? b. Find the least squares line that shows how weekly sales (y) is related to advertis- ing dollars (1) c. According to the least squares line, how is weekly sales related to advertising dollars? What does the intercept mean? What is the standard error of the regression model in the following scenarios? a. SSUnexplained = 83, n - 2 = 58. b. SS Total = 1234, SSExplained = 865, n = 145. c. SSExplained - 56, SSTotal = 78, n = 78. 4.7 Exercises E4.1 For each of the following three data sets, create a scatterplot, calculate the regression line, determine the R', and explain what the slope means a. X b. X DOO VOUWN- ono ovawan ova W * N E4.2 Consider the two data sets listed in In the first data set, the prices of houses are listed in dollars and the square footage is listed in feet, while the second data set lists the prices of houses in hundreds of dollars and the square footage in yards. Calculate the covariance and the correlation coefficient for both data sets, and comment on the similarities and differences between your findings. Is this what you expected? E4.3 A business owner is interested in how the cost of shipping a package is related to the distance a package is shipped. To this end, data have been collected and are contained in ship. a. Calculate a correlation coefficient, and explain what it means about the relation ship between shipping cost and distance. b. Now calculate the estimated sample regression function. What do the slope and intercept mean? c. If the company is going to ship a package 1.200 miles, how much do you estimate it will cost