Quiz 6: Simple Linear Regression

Which of the following situations is inconsistent? a. ? = 499 +.21 X and r =.75 b. ? = 100 +. 9 X and r = ?.70 c. ? = ?20 + 1 X and r =.40 d. ? = ?7 ? 4X and r = ?.90

AT T (American Telegraph and Telephone) earnings in billions of dollars are estimated using GNP (gross national product). The regression equation is ? =.078 +.06 X , where GNP is measured in billions of dollars. a. Interpret the slope. b. Interpret the Y -intercept.

Consider the data in Table P-3 where X = weekly advertising expenditures and Y = weekly sales. a. Does a significant relationship exist between advertising expenditures and sales? b. State the prediction equation. c. Forecast sales for an advertising expenditure of $50. d. What percentage of the variation in sales can be explained with the prediction equation? e. State the amount of unexplained variation. f. State the amount of total variation. TABLE P-3

The times required to check out customers in a supermarket and the corresponding values of purchases are shown in Table P-4. Answer parts a, b, e, and f of Problem 3 using these data. Give a point forecast and a 99% interval forecast for Y if X = 3.0. TABLE P-4

Lori Franz, maintenance supervisor for the Baltimore Transit Authority, would like to determine whether there is a positive relationship between the annual maintenance cost of a bus and its age. If a relationship exists, Lori feels that she can do a better job of forecasting the annual bus maintenance budget. She collects the data shown in Table P-5. a. Plot a scatter diagram. b. What kind of relationship exists between these two variables? c. Compute the correlation coefficient. d. Determine the least squares line. e. Test for the significance of the slope coefficient at the.05 significance level. Is the correlation significant? Explain. f. Forecast the annual maintenance cost for a five-year-old bus. TABLE P-5

Andrew Vazsonyi is the manager of the Spendwise supermarket chain. He would like to be able to forecast paperback book sales (books per week) based on the amount of shelf display space (feet) provided. Andrew gathers data for a sample of 11 weeks, as shown in Table P-6. a. Plot a scatter diagram. b. What kind of relationship exists between these two variables? c. Compute the correlation coefficient. d. Determine the least squares line. e. Test for the significance of the slope coefficient at the.10 significance level. Is the correlation significant? Explain. f. Plot the residuals versus the fitted values. Based on this plot, is the simple linear regression model appropriate for these data? g. Forecast the paperback book sales for a week during which four feet of shelf space is provided. TABLE P-6

Information supplied by a mail-order business for 12 cities is shown in Table P-7. a. Determine the fitted regression line. b. Calculate the standard error of the estimate. c. Determine the ANOVA table. d. What percentage of the variation in mail orders is explained by the number of catalogs distributed? e. Test to determine whether the regression slope coefficient is significantly different from zero. (Use the.01 significance level.) f. Test for the significance of the regression using the F statistic from the ANOVA table. (Use the.01 significance level.) Is the result consistent with the result in part e? Should it be? g. Forecast the number of mail orders received when 10,000 catalogs are distributed with a 90% prediction interval. TABLE P-7

In a regression of investment on the interest rate, the results in Table P-8 were observed over a 10-year period. a. Is the relationship between these variables significant? b. Can an effective prediction equation be developed? c. If the average interest rate is 4% five years from now, can yearly investment be forecast? d. Calculate and interpret r 2. e. Discuss correlation and causation in this example. TABLE P-8

The ABC Investment Company is in the business of making bids on investments offered by various firms that want additional financing. ABC has tabulated its bids on the last 25 issues in terms of their percentage of par value. The bids of ABC's major competitor, as a percentage of par value, are also tabulated on these issues. ABC now wonders if it is using the same rationale as its competitor in preparing bids. In other words, could ABC's bid be used to forecast the competitor's bid? If not, then the competitor must be evaluating issues differently. The data are given in Table P-9. TABLE P-9 a. To what extent are the two firms using the same rationale in preparing their bids? b. Forecast the competitor's bid if ABC bids 101% of par value. Give both a point forecast and an interval prediction. c. Under part b, what is the probability of ABC winning this particular bid (the lowest bid wins)?

Evaluate the following statements: a. A high r 2 means a significant regression. b. A very large sample size in a regression problem will always produce useful results.

Ed Bogdanski, owner of the American Precast Company, has hired you as a part-time analyst. Ed was extremely pleased when you determined that there is a positive relationship between the number of building permits issued and the amount of work available to his company. Now he wonders if it is possible to use knowledge of interest rates on first mortgages to predict the number of building permits that will be issued each month. You collect a random sample of nine months of data, as shown in Table P-11. TABLE P-11 a. Plot the data as a scatter diagram. b. Determine the fitted regression function. c. Test for the significance of the slope coefficient at the.05 significance level. d. When the interest rate increases by 1%, what is the average decrease in the number of building permits issued? e. Compute the coefficient of determination. f. Write a sentence that Ed can understand, interpreting the number computed in part e. g. Write Ed a memo explaining the results of your analysis.

Consider the population of 140 observations that are presented in Table P-12. The Marshall Printing Company wishes to estimate the relationship between the number of copies produced by an offset printing technique ( X ) and the associated direct labor cost ( Y ). Select a random sample of 20 observations. a. Construct a scatter diagram. b. Compute the sample correlation coefficient. c. Determine the fitted regression line. d. Plot the fitted line on the scatter diagram. e. Compute the standard error of estimate. f. Compute the coefficient of determination and interpret its value. g. Test the hypothesis that the slope, ? 1 , of the population regression line is zero. h. Calculate a point forecast and a 90% prediction interval for the direct labor cost if the project involves 250 copies. i. Examine the residuals. Does it appear as if a simple linear regression model is appropriate for these data? Explain. TABLE P-12

Harry Daniels is a quality control engineer for the Specific Electric Corporation. Specific manufactures electric motors. One of the steps in the manufacturing process involves the use of an automatic milling machine to produce slots in the shafts of the motors. Each batch of shafts is tested, and all shafts that do not meet required dimensional tolerances are discarded. The milling machine must be readjusted at the beginning of each new batch because its cutter head wears slightly during production. Harry is assigned the job of forecasting how the size of a batch affects the number of defective shafts in the batch so that he can select the best batch size. He collects the data for the average batch size of 13 batches shown in Table P-13 and assigns you to analyze it. a. Plot the data as a scatter diagram. b. Fit a simple linear regression model. c. Test for the significance of the slope coefficient. d. Examine the residuals. e. Develop a curvilinear model by fitting a simple linear regression model to some transformation of the independent variable. f. Test for the significance of the slope coefficient of the transformed variable. g. Examine the residuals. h. Forecast the number of defectives for a batch size of 300. i. Which of the models in parts b and e do you prefer? j. Write Harry a memo summarizing your results. TABLE P-13

The data in Table P-14 were collected as part of a study of real estate property evaluation. The numbers are observations on X = assessed value (in thousands of dollars) on the city assessor's books and Y = market value (selling price in thousands of dollars) for n = 30 parcels of land that sold in a particular calendar year in a certain geographical area. TABLE P-14 a. Plot the market value against the assessed value as a scatter diagram. b. Assuming a simple linear regression model, determine the least squares line relating market value to assessed value. c. Determine r 2 and interpret its value. d. Is the regression significant? Explain. e. Predict the market value of a property with an assessed value of 90.5. Is there any danger in making this prediction? f. Examine the residuals. Can you identify any observations that have a large influence on the location of the least squares line?

Player costs ( X ) and operating expenses ( Y ) for n = 26 major league baseball teams for the 1990-1991 season are given in Table P-15. TABLE P-15 a. Assuming a simple linear regression model, determine the equation for the fitted straight line. b. Determine r 2 and comment on the strength of the linear relation. c. Test for the significance of the regression with a level of significance of.10. d. Can we conclude that, as a general rule, operating expenses are about twice player costs? Discuss. e. Forecast operating expenses, with a large-sample 95% prediction interval, if player costs are $30.5 million. f. Using the residuals as a guide, identify any unusual observations. That is, do some teams have unusually low or unusually high player costs as a component of operating expenses?

Table P-16 contains data for 23 cities on newsprint consumption ( Y ) and the number of families in the city ( X ) during a particular year. a. Plot newsprint consumption against number of families as a scatter diagram. b. Is a simple linear regression model appropriate for the data in Table P-16? Be sure your answer includes an analysis of the residuals. c. Consider a log transformation of newsprint consumption and a simple linear regression model relating Y = log (newsprint consumption) to X = number of families. Fit this model. d. Examine the residuals from the regression in part c. Which model, the one in part b or the one in part c, is better? Justify your answer. e. Using the fitted function in part c, forecast the amount of newsprint consumed in a year if a city has 10,000 families. f. Can you think of other variables that are likely to influence the amount of newsprint consumed in a year? TABLE P-16

Outback Steakhouse grew explosively during its first years of operation. The numbers of Outback Steakhouse locations for the period 1988-1993 are given below. Source: Outback Steakhouse. a. Does there appear to be linear or exponential growth in the number of steakhouses? b. Estimate the annual growth rate for Outback Steakhouse over the 1988-1993 period. c. Forecast the number of Outback locations for 2007. Does this number seem reasonable? Explain.

On The Double, a chain of campus copy centers, began operations with a single store in 1993. The number of copy centers in operation, Y , is recorded for 14 consecutive years in Table P-18. a. Plot number of copy centers versus year. Has On The Double experienced linear or exponential growth? b. Determine the annual growth rate for On The Double. c. Predict the number of copy centers in operation in 2012. Does this number seem reasonable? Why or why not? TABLE P-18

The number of employees ( X ) and profits per employee ( Y ) for n = 16 publishing firms are given in Table P-19. Employees are recorded in thousands and profits per employee in thousands of dollars. A portion of the Minitab output from a straight-line regression analysis is given below. TABLE P-19 *Dun and Bradstreet. a. Identify the least squares estimates of the slope and intercept coefficients. b. Test the hypothesis H 0 : ? 1 = 0 with ? =.10. Does there appear to be a relationship between profits per employee and number of employees? Explain. c. Identify and interpret r 2. d. Is this fitted regression function likely to be a good tool for forecasting profits per employee for a given number of publishing firm employees? Discuss.

Refer to Problem 19. Observation 12 corresponds to Dun and Bradstreet. Redo the straight-line regression analysis omitting this observation. Do your conclusions in parts b and d of Problem 19 change? What, if anything, does this imply about the influence of a single observation on a regression analysis when the number of observations is fairly small? Do you think it is reasonable to throw out Dun and Bradstreet? Discuss.

Data on the actual cost ( Y ) and the estimated cost ( X ), in millions of dollars, for n = 26 construction projects are given in Table 2-8 in the context of Example 2.13. Figure 2-17 shows the fitted line plot for these data. a. Reanalyze the cost data using a simple linear regression model. Construct a plot of the residuals versus the fitted values. b. Is the regression of actual cost on estimated cost significant? Justify your answer. c. Identify and interpret r 2. d. What are the appropriate values for the slope and intercept coefficients in the population regression function if estimated cost is a perfect predictor of expected actual cost? Are the estimated coefficients in the equation for your fitted straight line consistent with these values? Discuss. e. Consider the plot of the residuals versus the fitted values. Does it appear as if estimated costs are, in general, more accurate predictors of actual costs for relatively inexpensive projects than for expensive projects? Why or why not?

Refer to Problem 21. The cost data can be transformed by taking the natural logarithms of both estimated costs and actual costs. Consider a simple linear regression model for the transformed costs. A portion of the Minitab output for this analysis is shown below. a. Is this regression significant? Explain. b. Identify and interpret r 2. c. Are the sample data consistent with the population values ? 0 = 0 and ? 1 = 1? Explain. What is the significance of these particular choices for ? 0 and ? 1 ? d. Construct a forecast of actual cost when estimated cost is $24 million.

TIGER TRANSPORT Tiger Transport Company is a trucking firm that moves household goods locally and across the country. Its current concern involves the price charged for moving small loads over long distances. It has rates it is happy with for full truckloads; these rates are based on the variable costs of driver, fuel, and maintenance, plus overhead and profit. When a truck is less than fully loaded, however, there is some question about the proper rate to be charged on goods needed to fill the truck. To forecast future fuel needs and prepare long-range budgets, Tiger would like to determine the cost of adding cargo to a partially filled truck. Tiger feels that the only additional cost incurred if extra cargo is added to the truck is the cost of additional fuel, since the miles per gallon of the truck would then be lowered. As one of the factors used to determine rates for small loads, the company would like to know its out-of-pocket fuel costs associated with additional cargo. You are a recent business school graduate working in the cost accounting department, and you are assigned the job of investigating this matter and advising top management on the considerations necessary for a sound rate decision. You begin by assuming that all trucks are the same; in fact, they are nearly identical in terms of size, gross-weight capacity, and engine size. You also assume that every driver will get the same truck mileage over a long trip. Tiger's chief accountant feels that these assumptions are reasonable. You are then left with only one variable that might affect the miles per gallon of long-haul trucks: cargo weight. You find that the accounting department has records for every trip made by a Tiger truck over the past several years. These records include the total cargo weight, the distance covered, and the number of gallons of diesel fuel used. A ratio of these last two figures is the miles per gallon for the trip. You select trips made over the past four years as your population; there are a total of 5,428 trips. You then select 40 random numbers from a random number table, and since the trips are recorded one after another, you assign the number 1 to the first recorded trip, 2 to the second, and so on. Your 40 random numbers thus produce a random selection of 40 trips to be examined. The cargo weight and miles per gallon for these trips are recorded and appear in Table 6-12. TABLE 6-12 Data for Trip Cargo Weight and Miles per Gallon for Tiger Transport TABLE 6-13 Regression Analysis Output for Tiger Transport Since your desktop computer has software with a regression analysis package, you fit a simple linear regression model to the data in Table 6-12. The resulting printout appears in Table 6-13. After studying the printout in Table 6-13, you decide that the sample data have produced a useful regression equation. This conclusion is based on a relatively high r 2 (76%), a large negative t value (?10.9), and a high F value (119). From the printout, you write down the equation of the fitted line where Y is measured in miles per gallon and X is measured in thousands of pounds. The slope of the regression equation (?.0604) is interpreted as follows: Each additional 1,000 pounds of cargo reduces the mileage of a truck by an average of.0604 mile per gallon. Tiger is currently paying approximately $2.55 per gallon for diesel fuel. You can therefore calculate the cost of hauling an additional 1,000 pounds of cargo 100 miles, as follows: Mean miles per gallon = 4.7(from Table 6-12) 100(2.55) Cost of same trip with an additional Thus, Incremental cost of 1,000 pounds carried 100 miles = $.70 You now believe you have completed part of your assignment, namely, determination of the out-of-pocket fuel costs associated with adding cargo weight to a less-than-full truck. You realize, of course, that other factors bear on a rate decision for small loads. Prepare a memo for Tiger's top management that summarizes the analysis. Include comments on the extent to which your work will improve forecasts for fuel needs and truck revenue.

BUTCHER PRODUCTS, INC. Gene Butcher is owner and president of Butcher Products, Inc., a small company that makes fiberglass ducting for electrical cable installations. Gene has been studying the number of duct units manufactured per day over the past two and a half years and is concerned about the wide variability in this figure. To forecast production output, costs, and revenues properly, Gene needs to establish a relationship between output and some other variable. Based on his experience with the company, Gene is unable to come up with any reason for the variability in output until he begins thinking about weather conditions. His reasoning is that the outside temperature may have something to do with the productivity of his workforce and the daily output achieved. He randomly selects several days from his records and records the number of ducting units produced for each of these days. He then goes to the local weather bureau and, for each of the selected days, records the high temperature for the day. He is then ready to run a correlation study between these two figures when he realizes that output would probably be related to deviation from an ideal temperature rather than to the temperature itself. That is, he thinks that a day that is either too hot or too cold would have a negative effect on production when compared with a day that has an ideal temperature. He decides to convert his temperature readings to deviations from 65 degrees Fahrenheit, a temperature he understands is ideal in terms of generating high worker output. His data appear as follows; Y represents the number of units produced, while X represents the absolute difference (negative signs eliminated) between the day's high temperature and 65 degrees: Gene performs a simple linear regression analysis using his company's computer and the Minitab software program. Gene is pleased to see the results of his regression analysis, as presented in Table 6-14. The t values corresponding to the estimated intercept and slope coefficients are large (in absolute value), and their p -values are very small. Both coefficients (552 and ?8.9) in the sample regression equation are clearly significant. Turning to r 2 , Gene is somewhat disappointed to find that this value, although satisfactory, is not as high as he had hoped (64.2%). However, he decides that it is high enough to begin thinking about ways to increase daily production levels. TABLE 6-14 Regression Analysis Output for Butcher Products, Inc. How many units would you forecast for a day in which the high temperature is 89 degrees?

BUTCHER PRODUCTS, INC. Gene Butcher is owner and president of Butcher Products, Inc., a small company that makes fiberglass ducting for electrical cable installations. Gene has been studying the number of duct units manufactured per day over the past two and a half years and is concerned about the wide variability in this figure. To forecast production output, costs, and revenues properly, Gene needs to establish a relationship between output and some other variable. Based on his experience with the company, Gene is unable to come up with any reason for the variability in output until he begins thinking about weather conditions. His reasoning is that the outside temperature may have something to do with the productivity of his workforce and the daily output achieved. He randomly selects several days from his records and records the number of ducting units produced for each of these days. He then goes to the local weather bureau and, for each of the selected days, records the high temperature for the day. He is then ready to run a correlation study between these two figures when he realizes that output would probably be related to deviation from an ideal temperature rather than to the temperature itself. That is, he thinks that a day that is either too hot or too cold would have a negative effect on production when compared with a day that has an ideal temperature. He decides to convert his temperature readings to deviations from 65 degrees Fahrenheit, a temperature he understands is ideal in terms of generating high worker output. His data appear as follows; Y represents the number of units produced, while X represents the absolute difference (negative signs eliminated) between the day's high temperature and 65 degrees: Gene performs a simple linear regression analysis using his company's computer and the Minitab software program. Gene is pleased to see the results of his regression analysis, as presented in Table 6-14. The t values corresponding to the estimated intercept and slope coefficients are large (in absolute value), and their p -values are very small. Both coefficients (552 and ?8.9) in the sample regression equation are clearly significant. Turning to r 2 , Gene is somewhat disappointed to find that this value, although satisfactory, is not as high as he had hoped (64.2%). However, he decides that it is high enough to begin thinking about ways to increase daily production levels. TABLE 6-14 Regression Analysis Output for Butcher Products, Inc. How many units would you forecast for a day in which the high temperature is 41 degrees?

BUTCHER PRODUCTS, INC. Gene Butcher is owner and president of Butcher Products, Inc., a small company that makes fiberglass ducting for electrical cable installations. Gene has been studying the number of duct units manufactured per day over the past two and a half years and is concerned about the wide variability in this figure. To forecast production output, costs, and revenues properly, Gene needs to establish a relationship between output and some other variable. Based on his experience with the company, Gene is unable to come up with any reason for the variability in output until he begins thinking about weather conditions. His reasoning is that the outside temperature may have something to do with the productivity of his workforce and the daily output achieved. He randomly selects several days from his records and records the number of ducting units produced for each of these days. He then goes to the local weather bureau and, for each of the selected days, records the high temperature for the day. He is then ready to run a correlation study between these two figures when he realizes that output would probably be related to deviation from an ideal temperature rather than to the temperature itself. That is, he thinks that a day that is either too hot or too cold would have a negative effect on production when compared with a day that has an ideal temperature. He decides to convert his temperature readings to deviations from 65 degrees Fahrenheit, a temperature he understands is ideal in terms of generating high worker output. His data appear as follows; Y represents the number of units produced, while X represents the absolute difference (negative signs eliminated) between the day's high temperature and 65 degrees: Gene performs a simple linear regression analysis using his company's computer and the Minitab software program. Gene is pleased to see the results of his regression analysis, as presented in Table 6-14. The t values corresponding to the estimated intercept and slope coefficients are large (in absolute value), and their p -values are very small. Both coefficients (552 and ?8.9) in the sample regression equation are clearly significant. Turning to r 2 , Gene is somewhat disappointed to find that this value, although satisfactory, is not as high as he had hoped (64.2%). However, he decides that it is high enough to begin thinking about ways to increase daily production levels. TABLE 6-14 Regression Analysis Output for Butcher Products, Inc. Based on the results of the regression analysis as shown earlier, what action would you advise Gene to take in order to increase daily output?

BUTCHER PRODUCTS, INC. Gene Butcher is owner and president of Butcher Products, Inc., a small company that makes fiberglass ducting for electrical cable installations. Gene has been studying the number of duct units manufactured per day over the past two and a half years and is concerned about the wide variability in this figure. To forecast production output, costs, and revenues properly, Gene needs to establish a relationship between output and some other variable. Based on his experience with the company, Gene is unable to come up with any reason for the variability in output until he begins thinking about weather conditions. His reasoning is that the outside temperature may have something to do with the productivity of his workforce and the daily output achieved. He randomly selects several days from his records and records the number of ducting units produced for each of these days. He then goes to the local weather bureau and, for each of the selected days, records the high temperature for the day. He is then ready to run a correlation study between these two figures when he realizes that output would probably be related to deviation from an ideal temperature rather than to the temperature itself. That is, he thinks that a day that is either too hot or too cold would have a negative effect on production when compared with a day that has an ideal temperature. He decides to convert his temperature readings to deviations from 65 degrees Fahrenheit, a temperature he understands is ideal in terms of generating high worker output. His data appear as follows; Y represents the number of units produced, while X represents the absolute difference (negative signs eliminated) between the day's high temperature and 65 degrees: Gene performs a simple linear regression analysis using his company's computer and the Minitab software program. Gene is pleased to see the results of his regression analysis, as presented in Table 6-14. The t values corresponding to the estimated intercept and slope coefficients are large (in absolute value), and their p -values are very small. Both coefficients (552 and ?8.9) in the sample regression equation are clearly significant. Turning to r 2 , Gene is somewhat disappointed to find that this value, although satisfactory, is not as high as he had hoped (64.2%). However, he decides that it is high enough to begin thinking about ways to increase daily production levels. TABLE 6-14 Regression Analysis Output for Butcher Products, Inc. Do you think Gene has developed an effective forecasting tool?

ACE MANUFACTURING Ace Manufacturing Company employs several thousand people in the manufacture of keyboards, equipment cases, and cables for the small-computer industry. The president of Ace has recently become concerned with the absentee rate among the company's employees and has asked the personnel department to look into this matter. Personnel realizes that an effective method of forecasting absenteeism would greatly strengthen its ability to plan properly. Bill McGone, the personnel director, decides to take a look at a few personnel folders in an attempt to size up the problem. He decides to randomly select 15 folders and record the number of absent days during the past fiscal year, along with employee age. After reading an article in a recent personnel journal, he believes that age may have a significant effect on absenteeism. If he finds that age and absent days show a good correlation in his small sample, he intends to take a sample of 200 or 300 folders and formulate a good prediction equation. The following table contains the data values collected in the initial sample. The number of absent days during the past fiscal year is represented by Y , while X represents employee age. How well are absent days and age correlated? Can this correlation be generalized to the entire workforce?

ACE MANUFACTURING Ace Manufacturing Company employs several thousand people in the manufacture of keyboards, equipment cases, and cables for the small-computer industry. The president of Ace has recently become concerned with the absentee rate among the company's employees and has asked the personnel department to look into this matter. Personnel realizes that an effective method of forecasting absenteeism would greatly strengthen its ability to plan properly. Bill McGone, the personnel director, decides to take a look at a few personnel folders in an attempt to size up the problem. He decides to randomly select 15 folders and record the number of absent days during the past fiscal year, along with employee age. After reading an article in a recent personnel journal, he believes that age may have a significant effect on absenteeism. If he finds that age and absent days show a good correlation in his small sample, he intends to take a sample of 200 or 300 folders and formulate a good prediction equation. The following table contains the data values collected in the initial sample. The number of absent days during the past fiscal year is represented by Y , while X represents employee age. What is the forecasting equation for absent days using age as a predictor variable?

ACE MANUFACTURING Ace Manufacturing Company employs several thousand people in the manufacture of keyboards, equipment cases, and cables for the small-computer industry. The president of Ace has recently become concerned with the absentee rate among the company's employees and has asked the personnel department to look into this matter. Personnel realizes that an effective method of forecasting absenteeism would greatly strengthen its ability to plan properly. Bill McGone, the personnel director, decides to take a look at a few personnel folders in an attempt to size up the problem. He decides to randomly select 15 folders and record the number of absent days during the past fiscal year, along with employee age. After reading an article in a recent personnel journal, he believes that age may have a significant effect on absenteeism. If he finds that age and absent days show a good correlation in his small sample, he intends to take a sample of 200 or 300 folders and formulate a good prediction equation. The following table contains the data values collected in the initial sample. The number of absent days during the past fiscal year is represented by Y , while X represents employee age. What percentage of the variability in absent days can be explained through knowledge of age?

ACE MANUFACTURING Ace Manufacturing Company employs several thousand people in the manufacture of keyboards, equipment cases, and cables for the small-computer industry. The president of Ace has recently become concerned with the absentee rate among the company's employees and has asked the personnel department to look into this matter. Personnel realizes that an effective method of forecasting absenteeism would greatly strengthen its ability to plan properly. Bill McGone, the personnel director, decides to take a look at a few personnel folders in an attempt to size up the problem. He decides to randomly select 15 folders and record the number of absent days during the past fiscal year, along with employee age. After reading an article in a recent personnel journal, he believes that age may have a significant effect on absenteeism. If he finds that age and absent days show a good correlation in his small sample, he intends to take a sample of 200 or 300 folders and formulate a good prediction equation. The following table contains the data values collected in the initial sample. The number of absent days during the past fiscal year is represented by Y , while X represents employee age. Is there a significant relation between absent days and age? In answering this question, use proper statistical procedures to support your answer.

ACE MANUFACTURING Ace Manufacturing Company employs several thousand people in the manufacture of keyboards, equipment cases, and cables for the small-computer industry. The president of Ace has recently become concerned with the absentee rate among the company's employees and has asked the personnel department to look into this matter. Personnel realizes that an effective method of forecasting absenteeism would greatly strengthen its ability to plan properly. Bill McGone, the personnel director, decides to take a look at a few personnel folders in an attempt to size up the problem. He decides to randomly select 15 folders and record the number of absent days during the past fiscal year, along with employee age. After reading an article in a recent personnel journal, he believes that age may have a significant effect on absenteeism. If he finds that age and absent days show a good correlation in his small sample, he intends to take a sample of 200 or 300 folders and formulate a good prediction equation. The following table contains the data values collected in the initial sample. The number of absent days during the past fiscal year is represented by Y , while X represents employee age. Suppose a newly hired person is 24 years old. How many absent days would you forecast for this person during the fiscal year?

ACE MANUFACTURING Ace Manufacturing Company employs several thousand people in the manufacture of keyboards, equipment cases, and cables for the small-computer industry. The president of Ace has recently become concerned with the absentee rate among the company's employees and has asked the personnel department to look into this matter. Personnel realizes that an effective method of forecasting absenteeism would greatly strengthen its ability to plan properly. Bill McGone, the personnel director, decides to take a look at a few personnel folders in an attempt to size up the problem. He decides to randomly select 15 folders and record the number of absent days during the past fiscal year, along with employee age. After reading an article in a recent personnel journal, he believes that age may have a significant effect on absenteeism. If he finds that age and absent days show a good correlation in his small sample, he intends to take a sample of 200 or 300 folders and formulate a good prediction equation. The following table contains the data values collected in the initial sample. The number of absent days during the past fiscal year is represented by Y , while X represents employee age. Should Bill McGone proceed to take a large sample of company employees based on the preliminary results of his sample?

ACE MANUFACTURING Ace Manufacturing Company employs several thousand people in the manufacture of keyboards, equipment cases, and cables for the small-computer industry. The president of Ace has recently become concerned with the absentee rate among the company's employees and has asked the personnel department to look into this matter. Personnel realizes that an effective method of forecasting absenteeism would greatly strengthen its ability to plan properly. Bill McGone, the personnel director, decides to take a look at a few personnel folders in an attempt to size up the problem. He decides to randomly select 15 folders and record the number of absent days during the past fiscal year, along with employee age. After reading an article in a recent personnel journal, he believes that age may have a significant effect on absenteeism. If he finds that age and absent days show a good correlation in his small sample, he intends to take a sample of 200 or 300 folders and formulate a good prediction equation. The following table contains the data values collected in the initial sample. The number of absent days during the past fiscal year is represented by Y , while X represents employee age. Has an effective forecasting method been developed?

TUX John Mosby has heard that regression analysis is often used to forecast time series variables, and since he has a personal computer with a regression software package, he decides to give it a try. The monthly sales volumes for Mr. Tux for the years 1998 through 2005 are the dependent variable. As a first try, he decides to use the period number as the predictor, or X , variable. His first Y sales value, $6,028, will have an X value 1, the second will have an X value of 2, and so on. His reasoning is that the upward trend he knows exists in his data will be accounted for by using an ever-rising X value to explain his sales data. After he performs the regression analysis on his computer, John records the following values: The high t value indicates to John that his fitted regression line slope (2,729.2) is significant; that is, he rejects the null hypothesis that the slope of the population regression line is zero. The high F value is consistent with this result (John recalls that F = t 2 for straight-line regression), and the null hypothesis that the regression is not significant must be rejected. John is disappointed with the relatively low r 2 (56.3%). He had hoped for a higher value so that his simple regression equation could be used to accurately forecast his sales. He realizes that this low value must be due to the seasonality of his monthly sales, a fact he knew about before he began his forecasting efforts. Considerable seasonal variation would result in monthly data points that do not cluster about the linear regression line, resulting in an unsatisfactory value. Another thing troubles John about his regression results. On the printout is this statement: Durbin-Watson =.99. He does not understand this statement and calls the statistics professor he had in college. After he describes the values on his regression printout, the professor says, "I have a class to teach right now, but your low Durbin-Watson statistic means that one of the assumptions of regression analysis does not hold." Comment on John's belief that his monthly sales are highly seasonal and therefore lead to a "low" r 2 value.

TUX John Mosby has heard that regression analysis is often used to forecast time series variables, and since he has a personal computer with a regression software package, he decides to give it a try. The monthly sales volumes for Mr. Tux for the years 1998 through 2005 are the dependent variable. As a first try, he decides to use the period number as the predictor, or X , variable. His first Y sales value, $6,028, will have an X value 1, the second will have an X value of 2, and so on. His reasoning is that the upward trend he knows exists in his data will be accounted for by using an ever-rising X value to explain his sales data. After he performs the regression analysis on his computer, John records the following values: The high t value indicates to John that his fitted regression line slope (2,729.2) is significant; that is, he rejects the null hypothesis that the slope of the population regression line is zero. The high F value is consistent with this result (John recalls that F = t 2 for straight-line regression), and the null hypothesis that the regression is not significant must be rejected. John is disappointed with the relatively low r 2 (56.3%). He had hoped for a higher value so that his simple regression equation could be used to accurately forecast his sales. He realizes that this low value must be due to the seasonality of his monthly sales, a fact he knew about before he began his forecasting efforts. Considerable seasonal variation would result in monthly data points that do not cluster about the linear regression line, resulting in an unsatisfactory value. Another thing troubles John about his regression results. On the printout is this statement: Durbin-Watson =.99. He does not understand this statement and calls the statistics professor he had in college. After he describes the values on his regression printout, the professor says, "I have a class to teach right now, but your low Durbin-Watson statistic means that one of the assumptions of regression analysis does not hold." What is your opinion regarding the adequacy of John's forecasting method?

TUX John Mosby has heard that regression analysis is often used to forecast time series variables, and since he has a personal computer with a regression software package, he decides to give it a try. The monthly sales volumes for Mr. Tux for the years 1998 through 2005 are the dependent variable. As a first try, he decides to use the period number as the predictor, or X , variable. His first Y sales value, $6,028, will have an X value 1, the second will have an X value of 2, and so on. His reasoning is that the upward trend he knows exists in his data will be accounted for by using an ever-rising X value to explain his sales data. After he performs the regression analysis on his computer, John records the following values: The high t value indicates to John that his fitted regression line slope (2,729.2) is significant; that is, he rejects the null hypothesis that the slope of the population regression line is zero. The high F value is consistent with this result (John recalls that F = t 2 for straight-line regression), and the null hypothesis that the regression is not significant must be rejected. John is disappointed with the relatively low r 2 (56.3%). He had hoped for a higher value so that his simple regression equation could be used to accurately forecast his sales. He realizes that this low value must be due to the seasonality of his monthly sales, a fact he knew about before he began his forecasting efforts. Considerable seasonal variation would result in monthly data points that do not cluster about the linear regression line, resulting in an unsatisfactory value. Another thing troubles John about his regression results. On the printout is this statement: Durbin-Watson =.99. He does not understand this statement and calls the statistics professor he had in college. After he describes the values on his regression printout, the professor says, "I have a class to teach right now, but your low Durbin-Watson statistic means that one of the assumptions of regression analysis does not hold." How do John's data violate one of the assumptions of regression analysis?

CONSUMER CREDIT COUNSELING The Consumer Credit Counseling (CCC) operation was described in Case 1-2. The executive director, Marv Hamishfeger, concluded that the most important variable that CCC needed to forecast was the number of new clients that would be seen for the rest of 1993. Marv provided Dorothy Mercer with monthly data for the number of new clients seen by CCC for the period from January 1985 through March 1993 (see Case 3-3). In Case 3-3, Dorothy used autocorrelation analysis to explore the data pattern. In Case 4-3, she used moving average and exponential smoothing methods to forecast the remaining months of 1993. Dorothy wondered if regression analysis could be used to develop a good forecasting model. She asked Marv if he could think of any potential predictor variables. Marv felt that the number of people on food stamps might be related to the number of new clients seen. Dorothy could find data for the number of people on food stamps only from January 1989 to December 1992. These data follow. Marv was also acquainted with a business activity index computed for the county by the local Economic Development Council. The business activity index was an indicator of the relative changes in overall business conditions for the region. The data for this index are at the top of the page. Determine whether there is a significant relationship between the number of new clients seen and the number of people on food stamps and/or the business activity index. Don't forget the possibility of data transformations.

CONSUMER CREDIT COUNSELING The Consumer Credit Counseling (CCC) operation was described in Case 1-2. The executive director, Marv Hamishfeger, concluded that the most important variable that CCC needed to forecast was the number of new clients that would be seen for the rest of 1993. Marv provided Dorothy Mercer with monthly data for the number of new clients seen by CCC for the period from January 1985 through March 1993 (see Case 3-3). In Case 3-3, Dorothy used autocorrelation analysis to explore the data pattern. In Case 4-3, she used moving average and exponential smoothing methods to forecast the remaining months of 1993. Dorothy wondered if regression analysis could be used to develop a good forecasting model. She asked Marv if he could think of any potential predictor variables. Marv felt that the number of people on food stamps might be related to the number of new clients seen. Dorothy could find data for the number of people on food stamps only from January 1989 to December 1992. These data follow. Marv was also acquainted with a business activity index computed for the county by the local Economic Development Council. The business activity index was an indicator of the relative changes in overall business conditions for the region. The data for this index are at the top of the page. Develop a regression equation and use it to forecast the number of new clients for the first three months of 1993.

CONSUMER CREDIT COUNSELING The Consumer Credit Counseling (CCC) operation was described in Case 1-2. The executive director, Marv Hamishfeger, concluded that the most important variable that CCC needed to forecast was the number of new clients that would be seen for the rest of 1993. Marv provided Dorothy Mercer with monthly data for the number of new clients seen by CCC for the period from January 1985 through March 1993 (see Case 3-3). In Case 3-3, Dorothy used autocorrelation analysis to explore the data pattern. In Case 4-3, she used moving average and exponential smoothing methods to forecast the remaining months of 1993. Dorothy wondered if regression analysis could be used to develop a good forecasting model. She asked Marv if he could think of any potential predictor variables. Marv felt that the number of people on food stamps might be related to the number of new clients seen. Dorothy could find data for the number of people on food stamps only from January 1989 to December 1992. These data follow. Marv was also acquainted with a business activity index computed for the county by the local Economic Development Council. The business activity index was an indicator of the relative changes in overall business conditions for the region. The data for this index are at the top of the page. Compare the results of your forecast with the actual observations for the first three months of 1993.

CONSUMER CREDIT COUNSELING The Consumer Credit Counseling (CCC) operation was described in Case 1-2. The executive director, Marv Hamishfeger, concluded that the most important variable that CCC needed to forecast was the number of new clients that would be seen for the rest of 1993. Marv provided Dorothy Mercer with monthly data for the number of new clients seen by CCC for the period from January 1985 through March 1993 (see Case 3-3). In Case 3-3, Dorothy used autocorrelation analysis to explore the data pattern. In Case 4-3, she used moving average and exponential smoothing methods to forecast the remaining months of 1993. Dorothy wondered if regression analysis could be used to develop a good forecasting model. She asked Marv if he could think of any potential predictor variables. Marv felt that the number of people on food stamps might be related to the number of new clients seen. Dorothy could find data for the number of people on food stamps only from January 1989 to December 1992. These data follow. Marv was also acquainted with a business activity index computed for the county by the local Economic Development Council. The business activity index was an indicator of the relative changes in overall business conditions for the region. The data for this index are at the top of the page. Would the business activity index be a good predictor of the number of new clients?

CONSUMER CREDIT COUNSELING The Consumer Credit Counseling (CCC) operation was described in Case 1-2. The executive director, Marv Hamishfeger, concluded that the most important variable that CCC needed to forecast was the number of new clients that would be seen for the rest of 1993. Marv provided Dorothy Mercer with monthly data for the number of new clients seen by CCC for the period from January 1985 through March 1993 (see Case 3-3). In Case 3-3, Dorothy used autocorrelation analysis to explore the data pattern. In Case 4-3, she used moving average and exponential smoothing methods to forecast the remaining months of 1993. Dorothy wondered if regression analysis could be used to develop a good forecasting model. She asked Marv if he could think of any potential predictor variables. Marv felt that the number of people on food stamps might be related to the number of new clients seen. Dorothy could find data for the number of people on food stamps only from January 1989 to December 1992. These data follow. Marv was also acquainted with a business activity index computed for the county by the local Economic Development Council. The business activity index was an indicator of the relative changes in overall business conditions for the region. The data for this index are at the top of the page. The data consist of a time series. Does this mean the independence assumption has been violated?

CONSUMER CREDIT COUNSELING The Consumer Credit Counseling (CCC) operation was described in Case 1-2. The executive director, Marv Hamishfeger, concluded that the most important variable that CCC needed to forecast was the number of new clients that would be seen for the rest of 1993. Marv provided Dorothy Mercer with monthly data for the number of new clients seen by CCC for the period from January 1985 through March 1993 (see Case 3-3). In Case 3-3, Dorothy used autocorrelation analysis to explore the data pattern. In Case 4-3, she used moving average and exponential smoothing methods to forecast the remaining months of 1993. Dorothy wondered if regression analysis could be used to develop a good forecasting model. She asked Marv if he could think of any potential predictor variables. Marv felt that the number of people on food stamps might be related to the number of new clients seen. Dorothy could find data for the number of people on food stamps only from January 1989 to December 1992. These data follow. Marv was also acquainted with a business activity index computed for the county by the local Economic Development Council. The business activity index was an indicator of the relative changes in overall business conditions for the region. The data for this index are at the top of the page. Assume that you developed a good regression equation. Would you be able to use this equation to forecast for the rest of 1993? Explain your answer.

AAA WASHINGTON An overview of AAA Washington was provided in Case 5-5 when students were asked to prepare a time series decomposition of the emergency road service calls received by the club over five years. The time series decomposition performed in Case 5-5 showed that the pattern Michael DeCoria had observed in emergency road service call volume was probably somewhat cyclical in nature. Michael would like to be able to predict emergency road service call volume for future years. Other research done by the club identified several factors that have an impact on emergency road service call volume. Among these factors are average daily temperature and the amount of rainfall received in a day. This research has shown that emergency road service calls increase as rainfall increases and as average daily temperature falls. The club also believes that the total number of emergency road service calls it receives is dependent on the number of members in the club. Finally, Michael feels that the number of calls received is related to the general economic cycle. The unemployment rate for Washington State is used as a good surrogate measurement for the general state of Washington's economy. Data on the unemployment rate, average monthly temperature, monthly rainfall, and number of members in the club have been gathered and are presented in Table 6-15. A conversation with the manager of the emergency road service call center has led to two important observations: (1) Automakers seem to design cars to operate best at 65 degrees Fahrenheit and (2) call volume seems to increase more sharply when the average temperature drops a few degrees from an average temperature in the 30s than it does when a similar drop occurs with an average temperature in the 60s. This information suggests that the effect of temperature on emergency road service is nonlinear. TABLE 6-15 Data for AAA Washington Run four simple linear regression models using total number of emergency road service calls as the dependent variable and unemployment rate, temperature, rainfall, and number of members as the four independent variables. Would any of these independent variables be useful for predicting the total number of emergency road service calls?

AAA WASHINGTON An overview of AAA Washington was provided in Case 5-5 when students were asked to prepare a time series decomposition of the emergency road service calls received by the club over five years. The time series decomposition performed in Case 5-5 showed that the pattern Michael DeCoria had observed in emergency road service call volume was probably somewhat cyclical in nature. Michael would like to be able to predict emergency road service call volume for future years. Other research done by the club identified several factors that have an impact on emergency road service call volume. Among these factors are average daily temperature and the amount of rainfall received in a day. This research has shown that emergency road service calls increase as rainfall increases and as average daily temperature falls. The club also believes that the total number of emergency road service calls it receives is dependent on the number of members in the club. Finally, Michael feels that the number of calls received is related to the general economic cycle. The unemployment rate for Washington State is used as a good surrogate measurement for the general state of Washington's economy. Data on the unemployment rate, average monthly temperature, monthly rainfall, and number of members in the club have been gathered and are presented in Table 6-15. A conversation with the manager of the emergency road service call center has led to two important observations: (1) Automakers seem to design cars to operate best at 65 degrees Fahrenheit and (2) call volume seems to increase more sharply when the average temperature drops a few degrees from an average temperature in the 30s than it does when a similar drop occurs with an average temperature in the 60s. This information suggests that the effect of temperature on emergency road service is nonlinear. TABLE 6-15 Data for AAA Washington Create a new temperature variable and relate it to emergency road service. Remember that temperature is a relative scale and that the selection of the zero point is arbitrary. If vehicles are designed to operate best at 65 degrees Fahrenheit, then every degree above or below 65 degrees should make vehicles operate less reliably. To accomplish a transformation of the temperature data that simulates this effect, begin by subtracting 65 from the average monthly temperature values. This repositions "zero" to 65 degrees Fahrenheit. Should absolute values of this new temperature variable be used?

AAA WASHINGTON An overview of AAA Washington was provided in Case 5-5 when students were asked to prepare a time series decomposition of the emergency road service calls received by the club over five years. The time series decomposition performed in Case 5-5 showed that the pattern Michael DeCoria had observed in emergency road service call volume was probably somewhat cyclical in nature. Michael would like to be able to predict emergency road service call volume for future years. Other research done by the club identified several factors that have an impact on emergency road service call volume. Among these factors are average daily temperature and the amount of rainfall received in a day. This research has shown that emergency road service calls increase as rainfall increases and as average daily temperature falls. The club also believes that the total number of emergency road service calls it receives is dependent on the number of members in the club. Finally, Michael feels that the number of calls received is related to the general economic cycle. The unemployment rate for Washington State is used as a good surrogate measurement for the general state of Washington's economy. Data on the unemployment rate, average monthly temperature, monthly rainfall, and number of members in the club have been gathered and are presented in Table 6-15. A conversation with the manager of the emergency road service call center has led to two important observations: (1) Automakers seem to design cars to operate best at 65 degrees Fahrenheit and (2) call volume seems to increase more sharply when the average temperature drops a few degrees from an average temperature in the 30s than it does when a similar drop occurs with an average temperature in the 60s. This information suggests that the effect of temperature on emergency road service is nonlinear. TABLE 6-15 Data for AAA Washington Develop a scatter diagram. Is there a linear relationship between calls and the new temperature variable?

AAA WASHINGTON An overview of AAA Washington was provided in Case 5-5 when students were asked to prepare a time series decomposition of the emergency road service calls received by the club over five years. The time series decomposition performed in Case 5-5 showed that the pattern Michael DeCoria had observed in emergency road service call volume was probably somewhat cyclical in nature. Michael would like to be able to predict emergency road service call volume for future years. Other research done by the club identified several factors that have an impact on emergency road service call volume. Among these factors are average daily temperature and the amount of rainfall received in a day. This research has shown that emergency road service calls increase as rainfall increases and as average daily temperature falls. The club also believes that the total number of emergency road service calls it receives is dependent on the number of members in the club. Finally, Michael feels that the number of calls received is related to the general economic cycle. The unemployment rate for Washington State is used as a good surrogate measurement for the general state of Washington's economy. Data on the unemployment rate, average monthly temperature, monthly rainfall, and number of members in the club have been gathered and are presented in Table 6-15. A conversation with the manager of the emergency road service call center has led to two important observations: (1) Automakers seem to design cars to operate best at 65 degrees Fahrenheit and (2) call volume seems to increase more sharply when the average temperature drops a few degrees from an average temperature in the 30s than it does when a similar drop occurs with an average temperature in the 60s. This information suggests that the effect of temperature on emergency road service is nonlinear. TABLE 6-15 Data for AAA Washington If a nonlinear relationship exists between calls and the new temperature variable, develop the best model.