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All Textbook Solutions for Essentials of Business Analytics (MindTap Course List)
A Wall Street Journal subscriber survey asked 46 questions about subscriber characteristics and interests. State whether each of the following questions provides categorical or quantitative data. a. What is your age? b. Are you male or female? c. When did you first start reading the WSJ? High school, college, early career, midcareer, late career, or retirement? d. How long have you been in your present job or position? e. What type of vehicle are you considering for your next purchase? Nine response categories include sedan, sports car, SUV, minivan, and so on.The following table contains a partial list of countries, the continents on which they are located, and their respective gross domestic products (GDP) in U.S. dollars. A list of 125 countries and their GDPs is contained in the file GDPlist. a. Sort the countries in GDPlist from largest to smallest GDP. What are the top 10 countries according to GDP? b. Filter the countries to display only the countries located in Africa. What are the top 5 countries located in Africa according to GDP? c. What are the top 5 countries by GDP that are located in Europe?Ohio Logistics manages the logistical activities for firms by matching companies that need products shipped with carriers that can provide the best rates and best service for the companies. Ohio Logistics is very concerned that its carriers deliver their customers material on time, so it carefully monitors the percentage of on-time deliveries. The following table contains a list of the carriers used by Ohio Logistics and the corresponding on-time percentages for the current and previous years. a. Sort the carriers in descending order by their current years percentage of on-time deliveries. Which carrier is providing the best service in the current year? Which carrier is providing the worst service in the current year? b. Calculate the change in percentage of on-time deliveries from the previous to the current year for each carrier. Use Excels conditional formatting to highlight the carriers whose on-time percentage decreased from the previous year to the current year. c. Use Excels conditional formatting tool to create data bars for the change in percentage of on-time deliveries from the previous year to the current year for each carrier calculated in part b. d. Which carriers should Ohio Logistics try to use in the future? Why?A partial relative frequency distribution is given. a. What is the relative frequency of class D? b. The total sample size is 200. What is the frequency of class D? c. Show the frequency distribution. d. Show the percent frequency distribution.In a recent report, the top five most-visited English-language web sites were google.com (GOOG), facebook.com (FB), youtube.com (YT), yahoo.com (YAH), and wikipedia.com (WIKI). The most-visited web sites for a sample of 50 Internet users are shown in the following table: a. Are these data categorical or quantitative? b. Provide frequency and percent frequency distributions. c. On the basis of the sample, which web site is most frequently the most-often-visited web site for Internet users? Which is second?In a study of how chief executive officers (CEOs) spend their days, it was found that CEOs spend an average of about 18 hours per week in meetings, not including conference calls, business meals, and public events. Shown here are the times spent per week in meetings (hours) for a sample of 25 CEOs: a. What is the least amount of time a CEO spent per week in meetings in this sample? The highest? b. Use a class width of 2 hours to prepare a frequency distribution and a percent frequency distribution for the data. c. Prepare a histogram and comment on the shape of the distribution.Consumer complaints are frequently reported to the Better Business Bureau. Industries with the most complaints to the Better Business Bureau are often banks, cable and satellite television companies, collection agencies, cellular phone providers, and new car dealerships. The results for a sample of 200 complaints are in the file BBB. a. Show the frequency and percent frequency of complaints by industry. b. Which industry had the highest number of complaints? c. Comment on the percentage frequency distribution for complaints.Reports have found that many U.S. adults would rather live in a different type of community than the one in which they are living now. A national survey of 2,260 adults asked: Where do you live now? and What do you consider to be the ideal community? Response options were City (C), Suburb (S), Small Town (T), or Rural (R). A representative portion of this survey for a sample of 100 respondents is as follows: a. Provide a percent frequency distribution and a histogram for each question. b. Where are most adults living now? c. Where do most adults consider the ideal community to be? d. What changes in living areas would you expect to see if people moved from where they currently live to their ideal community?Consider the following data: a. Develop a frequency distribution using classes of 1214, 1517, 1820, 2123, and 2426. b. Develop a relative frequency distribution and a percent frequency distribution using the classes in part a.Consider the following frequency distribution. Construct a cumulative frequency distribution.The owner of an automobile repair shop studied the waiting times for customers who arrive at the shop for an oil change. The following data with waiting times in minutes were collected over a 1-month period. 2 5 10 12 4 4 5 17 11 8 9 8 12 21 6 8 7 13 18 3 Using classes of 04, 59, and so on, show: a. The frequency distribution. b. The relative frequency distribution. c. The cumulative frequency distribution. d. The cumulative relative frequency distribution. e. The proportion of customers needing an oil change who wait 9 minutes or less.Approximately 1.65 million high school students take the Scholastic Aptitude Test (SAT) each year, and nearly 80 percent of the college and universities without open admissions policies use SAT scores in making admission decisions. The current version of the SAT includes three parts: reading comprehension, mathematics, and writing. A perfect combined score for all three parts is 2400. A sample of SAT scores for the combined three-part SAT are as follows: a. Show a frequency distribution and histogram. Begin with the first bin starting at 800, and use a bin width of 200. b. Comment on the shape of the distribution. c. What other observations can be made about the SAT scores based on the tabular and graphical summaries?Consider a sample with data values of 10, 20, 12, 17, and 16. a. Compute the mean and median. b. Consider a sample with data values 10, 20, 12, 17, 16, and 12. How would you expect the mean and median for these sample data to compare to the mean and median for part a (higher, lower, or the same)? Compute the mean and median for the sample data 10, 20, 12, 17, 16, and 12.Consider a sample with data values of 27, 25, 20, 15, 30, 34, 28, and 25. Compute the 20th, 25th, 65th, and 75th percentiles.Consider a sample with data values of 53, 55, 70, 58, 64, 57, 53, 69, 57, 68, and 53. Compute the mean, median, and mode.If an asset declines in value from 5,000 to 3,500 over nine years, what is the mean annual growth rate in the assets value over these nine years?Suppose that you initially invested 10,000 in the Stivers mutual fund and 5,000 in the Trippi mutual fund. The value of each investment at the end of each subsequent year is provided in the table: Which of the two mutual funds performed better over this time period?The average time that Americans commute to work is 27.7 minutes (Sterlings Best Places, April 13, 2012). The average commute times in minutes for 48 cities are as follows: a. What is the mean commute time for these 48 cities? b. What is the median commute time for these 48 cities? c. What is the mode for these 48 cities? d. What is the variance and standard deviation of commute times for these 48 cities? e. What is the third quartile of commute times for these 48 cities?Suppose that the average waiting time for a patient at a physicians office is just over 29 minutes. To address the issue of long patient wait times, some physicians offices are using wait-tracking systems to notify patients of expected wait times. Patients can adjust their arrival times based on this information and spend less time in waiting rooms. The following data show wait times (in minutes) for a sample of patients at offices that do not have a wait-tracking system and wait times for a sample of patients at offices with such systems. a. What are the mean and median patient wait times for offices with a wait-tracking system? What are the mean and median patient wait times for offices without a wait-tracking system? b. What are the variance and standard deviation of patient wait times for offices with a wait-tracking system? What are the variance and standard deviation of patient wait times for visits to offices without a wait tracking system? c. Create a box plot for patient wait times for offices without a wait-tracking system. d. Create a box plot for patient wait times for offices with a wait-tracking system. e. Do offices with a wait-tracking system have shorter patient wait times than offices without a wait-tracking system? Explain.According to the National Education Association (NEA), teachers generally spend more than 40 hours each week working on instructional duties. The following data show the number of hours worked per week for a sample of 13 high school science teachers and a sample of 11 high school English teachers. High school science teachers 53 56 54 54 55 58 49 61 54 54 52 53 54 High school English teachers 52 47 50 46 47 48 49 46 55 44 47 a. What is the median number of hours worked per week for the sample of 13 high school science teachers? b. What is the median number of hours worked per week for the sample of 11 high school English teachers? c. Create a box plot for the number of hours worked for high school science teachers. d. Create a box plot for the number of hours worked for high school English teachers. e. Comment on the differences between the box plots for science and English teachers.Return to the waiting times given for the physicians office in Problem 19. a. Considering only offices without a wait-tracking system, what is the z-score for the 10th patient in the sample (wait time 5 37 minutes)? b. Considering only offices with a wait-tracking system, what is the z-score for the 6th patient in the sample (wait time 5 37 minutes)? How does this z-score compare with the z-score you calculated for part a? c. Based on z-scores, do the data for offices without a wait-tracking system contain any outliers? Based on z-scores, do the data for offices without a wait-tracking system contain any outliers? 19. Suppose that the average waiting time for a patient at a physicians office is just over 29 minutes. To address the issue of long patient wait times, some physicians offices are using wait-tracking systems to notify patients of expected wait times. Patients can adjust their arrival times based on this information and spend less time in waiting rooms. The following data show wait times (in minutes) for a sample of patients at offices that do not have a wait-tracking system and wait times for a sample of patients at offices with such systems. a. What are the mean and median patient wait times for offices with a wait-tracking system? What are the mean and median patient wait times for offices without a wait-tracking system? b. What are the variance and standard deviation of patient wait times for offices with a wait-tracking system? What are the variance and standard deviation of patient wait times for visits to offices without a wait tracking system? c. Create a box plot for patient wait times for offices without a wait-tracking system.The results of a national survey showed that on average, adults sleep 6.9 hours per night. Suppose that the standard deviation is 1.2 hours and that the number of hours of sleep follows a bell-shaped distribution. a. Use the empirical rule to calculate the percentage of individuals who sleep between 4.5 and 9.3 hours per day. b. What is the z-value for an adult who sleeps 8 hours per night? c. What is the z-value for an adult who sleeps 6 hours per night?Suppose that the national average for the math portion of the College Boards SAT is 515. The College Board periodically rescales the test scores such that the standard deviation is approximately 100. Answer the following questions using a bell-shaped distribution and the empirical rule for the math test scores. a. What percentage of students have an SAT math score greater than 615? b. What percentage of students have an SAT math score greater than 715? c. What percentage of students have an SAT math score between 415 and 515? d. What is the z-score for student with an SAT math score of 620? e. What is the z-score for a student with an SAT math score of 405?Five observations taken for two variables follow. a. Develop a scatter diagram with x on the horizontal axis. b. What does the scatter diagram developed in part a indicate about the relationship between the two variables? c. Compute and interpret the sample covariance. d. Compute and interpret the sample correlation coefficient.The scatter chart in the following figure was created using sample data for profits and market capitalizations from a sample of firms in the Fortune 500. a. Discuss what the scatter chart indicates about the relationship between profits and market capitalization? b. The data used to produce this are contained in the file Fortune500. Calculate the covariance between profits and market capitalization. Discuss what the covariance indicates about the relationship between profits and market capitalization? c. Calculate the correlation coefficient between profits and market capitalization. What does the correlations coefficient indicate about the relationship between profits and market capitalization?The economic downturn in 20082009 resulted in the loss of jobs and an increase in delinquent loans for housing. In projecting where the real estate market was headed in the coming year, economists studied the relationship between the jobless rate and the percentage of delinquent loans. The expectation was that if the jobless rate continued to increase, there would also be an increase in the percentage of delinquent loans. The following data show the jobless rate and the delinquent loan percentage for 27 major real estate markets. Source: The Wall Street Journal, January 27, 2009. a. Compute the correlation coefficient. Is there a positive correlation between the jobless rate and the percentage of delinquent housing loans? What is your interpretation? b. Show a scatter diagram of the relationship between the jobless rate and the percentage of delinquent housing loans.Heavenly Chocolates manufactures and sells quality chocolate products at its plant and retail store located in Saratoga Springs, New York. Two years ago, the company developed a web site and began selling its products over the Internet. Web-site sales have exceeded the company’s expectations, and management is now considering strategies to increase sales even further. To learn more about the web-site customers, a sample of 50 Heavenly Chocolate transactions was selected from the previous month’s sales. Data showing the day of the week each transaction was made, the type of browser the customer used, the time spent on the web site, the number of web pages viewed, and the amount spent by each of the 50 customers are contained in the file named Heavenly Chocolates. A portion of the data is shown in the table that follows:
Heavenly Chocolates would like to use the sample data to determine whether online shoppers who spend more time and view more pages also spend more money during their visit to the web site. The company would also like to investigate the effect that the day of the week and the type of browser have on sales.
Managerial Report
Use the methods of descriptive statistics to learn about the customers who visit the Heavenly Chocolates web site. Include the following in your report.
Graphical and numerical summaries for the length of time the shopper spends on the web site, the number of pages viewed, and the mean amount spent per transaction. Discuss what you learn about Heavenly Chocolates’ online shoppers from these numerical summaries.
Summarize the frequency, the total dollars spent, and the mean amount spent per transaction for each day of week. Discuss the observations you can make about Heavenly Chocolates’ business based on the day of the week?
Summarize the frequency, the total dollars spent, and the mean amount spent per transaction for each type of browser. Discuss the observations you can make about Heavenly Chocolates’ business based on the type of browser?
Develop a scatter diagram, and compute the sample correlation coefficient to explore the relationship between the time spent on the web site and the dollar amount spent. Use the horizontal axis for the time spent on the web site. Discuss your findings.
Develop a scatter diagram, and compute the sample correlation coefficient to explore the relationship between the number of web pages viewed and the amount spent. Use the horizontal axis for the number of web pages viewed. Discuss your findings.
Develop a scatter diagram, and compute the sample correlation coefficient to explore the relationship between the time spent on the web site and the number of pages viewed. Use the horizontal axis to represent the number of pages viewed. Discuss your findings.
A sales manager is trying to determine appropriate sales performance bonuses for her team this year. The following table contains the data relevant to determining the bonuses, but it is not easy to read and interpret. Reformat the table to improve readability and to help the sales manager make her decisions about bonuses.The following table shows an example of gross domestic product values for five countries over six years in equivalent U.S. dollars (). a. How could you improve the readability of this table? b. The file GDPyears contains sample data from the United Nations Statistics Division on 30 countries and their GDP values from Year 1 to Year 6 in US. Create a table that provides all these data for a user. Format the table to make it as easy to read as possible. Hint: It is generally not important for the user to know GDP to an exact dollar figure. It is typical to present GDP values in millions or billions of dollars.The following table provides monthly revenue values for Tedstar, Inc., a company that sells valves to large industrial firms. The monthly revenue data have been graphed using a line chart in the following figure. a. What are the problems with the layout and display of this line chart? b. Create a new line chart for the monthly revenue data at Tedstar, Inc. Format the chart to make it easy to read and interpret.In the file MajorSalary, data have been collected from 111 College of Business graduates on their monthly starting salaries. The graduates include students majoring in management, finance, accounting, information systems, and marketing. Create a PivotTable in Excel to display the number of graduates in each major and the average monthly starting salary for students in each major. a. Which major has the greatest number of graduates? b. Which major has the highest average starting monthly salary? c. Use the PivotTable to determine the major of the student with the highest overall starting monthly salary. What is the major of the student with the lowest overall starting monthly salary?Entrepreneur magazine ranks franchises. Among the factors that the magazine uses in its rankings are growth rate, number of locations, start-up costs, and financial stability. A recent ranking listed the top 20 U.S. franchises and the number of locations as follows: These data can be found in the file Franchises. Create a PivotTable to summarize these data using classes 09,999, 10,00019,999, 20,00029,999, 30,00039,999 to answer the following questions. (Hint: Use Number of U.S. Locations as the COLUMNS, and use Count of Number of U.S. Locations as the VALUES in the PivotTable.) a. How many franchises have between 0 and 9,999 locations? b. How many franchises have more than 30,000 locations?The file MutualFunds contains a data set with information for 45 mutual funds that are part of the Morningstar Funds 500. The data set includes the following five variables: Fund Type: The type of fund, labeled DE (Domestic Equity), IE (International Equity), and FI (Fixed Income) Net Asset Value (): The closing price per share Five-Year Average Return (%): The average annual return for the fund over the past five years Expense Ratio (%): The percentage of assets deducted each fiscal year for fund expenses Morningstar Rank: The risk adjusted star rating for each fund; Morningstar ranks go from a low of 1 Star to a high of 5 Stars. a. Prepare a PivotTable that gives the frequency count of the data by Fund Type (rows) and the five-year average annual return (columns). Use classes of 09.99, 1019.99, 2029.99, 3039.99, 4049.99, and 5059.99 for the Five-Year Average Return (%). b. What conclusions can you draw about the fund type and the average return over the past five years?The file TaxData contains information from federal tax returns filed in 2007 for all counties in the United States (3,142 counties in total). Create a PivotTable in Excel to answer the questions below. The PivotTable should have State Abbreviation as Row Labels. The Values in the PivotTable should be the sum of adjusted gross income for each state. a. Sort the PivotTable data to display the states with the smallest sum of adjusted gross income on top and the largest on the bottom. Which state had the smallest sum of adjusted gross income? What is the total adjusted gross income for federal tax returns filed in this state with the smallest total adjusted gross income? (Hint: To sort data in a PivotTable in Excel, right-click any cell in the PivotTable that contains the data you want to sort, and select Sort.) b. Add the County Name to the Row Labels in the PivotTable. Sort the County Names by Sum of Adjusted Gross Income with the lowest values on the top and the highest values on the bottom. Filter the Row Labels so that only the state of Texas is displayed. Which county had the smallest sum of adjusted gross income in the state of Texas? Which county had the largest sum of adjusted gross income in the state of Texas? c. Click on Sum of Adjusted Gross Income in the Values area of the PivotTable in Excel. Click Value Field Settings. Click the tab for Show Values As. In the Show values as box, choose % of Parent Row Total. Click OK. This displays the adjusted gross income reported by each county as a percentage of the total state adjusted gross income. Which county has the highest percentage adjusted gross income in the state of Texas? What is this percentage? d. Remove the filter on the Row Labels to display data for all states. What percentage of total adjusted gross income in the United States was provided by the state of New York?The file FDICBankFailures contains data on failures of federally insured banks between 2000 and 2012. Create a PivotTable in Excel to answer the following questions. The PivotTable should group the closing dates of the banks into yearly bins and display the counts of bank closures each year in columns of Excel. Row labels should include the bank locations and allow for grouping the locations into states or viewing by city. You should also sort the PivotTable so that the states with the greatest number of total bank failures between 2000 and 2012 appear at the top of the PivotTable. a. Which state had the greatest number of federally insured bank closings between 2000 and 2012? b. How many bank closings occurred in the state of Nevada (NV) in 2010? In what cities did these bank closings occur? c. Use the PivotTables filter capability to view only bank closings in California (CA), Florida (FL), Texas (TX), and New York (NY) for the years 2009 through 2012. What is the total number of bank closings in these states between 2009 and 2012? d. Using the filtered PivotTable from part c, what city in Florida had the greatest number of bank closings between 2009 and 2012? How many bank closings occurred in this city? e. Create a PivotChart to display a column chart that shows the total number of bank closings in each year 2000 through 2012 in the state of Florida. Adjust the formatting of this column chart so that it best conveys the data. What does this column chart suggest about bank closings between 2000 and 2012 in Florida? Discuss. (Hint: You may have to switch the row and column labels in the PivotChart to get the best presentation for your PivotChart.)The following 20 observations are for two quantitative variables, x and y. a. Create a scatter chart for these 20 observations. b. Fit a linear trendline to the 20 observations. What can you say about the relationship between the two quantitative variables?The file Fortune500 contains data for profits and market capitalizations from a recent sample of firms in the Fortune 500 a. Prepare a scatter diagram to show the relationship between the variables Market Capitalization and Profit in which Market Capitalization is on the vertical axis and Profit is on the horizontal axis. Comment on any relationship between the variables. b. Create a trendline for the relationship between Market Capitalization and Profit. What does the trendline indicate about this relationship?The International Organization of Motor Vehicle Manufacturers (officially known as the Organisation Internationale des Constructeurs dAutomobiles, OICA) provides data on worldwide vehicle production by manufacturer. The following table shows vehicle production numbers for four different manufacturers for five recent years. Data are in millions of vehicles. a. Construct a line chart for the time series data for years 1 through 5 showing the number of vehicles manufactured by each automotive company. Show the time series for all four manufacturers on the same graph. b. What does the line chart indicate about vehicle production amounts from years 1 through 5? Discuss. c. Construct a clustered-bar chart showing vehicles produced by automobile manufacturer using the year 1 through 5 data. Represent the years of production along the horizontal axis, and cluster the production amounts for the four manufacturers in each year. Which company is the leading manufacturer in each year?12P13PThe total number of term life insurance contracts sold in Problem 13 is 199. The following pie chart shows the percentages of contracts sold by each salesperson. a. What are the problems with using a pie chart to display these data? b. What type of chart would be preferred for displaying the data in this pie chart? c. Use a different type of chart to display the percentage of contracts sold by each salesperson that conveys the data better than the pie chart. Format the chart and add data labels to improve the charts readability.An automotive company is considering the introduction of a new model of sports car that will be available in four-cylinder and six-cylinder engine types. A sample of customers who were interested in this new model were asked to indicate their preference for an engine type for the new model of automobile. The customers were also asked to indicate their preference for exterior color from four choices: red, black, green, and white. Consider the following data regarding the customer responses: a. Construct a clustered-column chart with exterior color as the horizontal variable. b. What can we infer from the clustered-bar chart in part a?Consider the following survey results regarding smartphone ownership by age: a. Construct a stacked-column chart to display the survey data on type of cell-phone ownership. Use Age Category as the variable on the horizontal axis. b. Construct a clustered column chart to display the survey data. Use Age Category as the variable on the horizontal axis. c. What can you infer about the relationship between age and smartphone ownership from the column charts in parts a and b? Which column chart (stacked or clustered) is best for interpreting this relationship? Why?The Northwest regional manager of Logan Outdoor Equipment Company has conducted a study to determine how her store managers are allocating their time. A study was undertaken over three weeks that collected the following data related to the percentage of time each store manager spent on the tasks of attending required meetings, preparing business reports, customer interaction, and being idle. The results of the data collection appear in the following table: a. Create a stacked-bar chart with locations along the vertical axis. Reformat the bar chart to best display these data by adding axis labels, a chart title, and so on. b. Create a clustered-bar chart with locations along the vertical axis and clusters of tasks. Reformat the bar chart to best display these data by adding axis labels, a chart title, and the like. c. Create multiple bar charts in which each location becomes a single bar chart showing the percentage of time spent on tasks. Reformat the bar charts to best display these data by adding axis labels, a chart title, and so forth. d. Which form of bar chart (stacked, clustered, or multiple) is preferable for these data? Why? e. What can we infer about the differences among how store managers are allocating their time at the different locations?The Ajax Company uses a portfolio approach to manage their research and development (RD) projects. Ajax wants to keep a mix of projects to balance the expected return and risk profiles of their RD activities. Consider a situation in which Ajax has six RD projects as characterized in the table. Each project is given an expected rate of return and a risk assessment, which is a value between 1 and 10, where 1 is the least risky and 10 is the most risky. Ajax would like to visualize their current RD projects to keep track of the overall risk and return of their RD portfolio. a. Create a bubble chart in which the expected rate of return is along the horizontal axis, the risk estimate is on the vertical axis, and the size of the bubbles represents the amount of capital invested. Format this chart for best presentation by adding axis labels and labeling each bubble with the project number. b. The efficient frontier of RD projects represents the set of projects that have the highest expected rate of return for a given level of risk. In other words, any project that has a smaller expected rate of return for an equivalent, or higher, risk estimate cannot be on the efficient frontier. From the bubble chart in part a, which projects appear to be located on the efficient frontier?Heat maps can be very useful for identifying missing data values in moderate to large data sets. The file SurveyResults contains the responses from a marketing survey: 108 individuals responded to the survey of 10 questions. Respondents provided answers of 1, 2, 3, 4, or 5 to each question, corresponding to the overall satisfaction on 10 different dimensions of quality. However, not all respondents answered every question. a. To find the missing data values, create a heat map in Excel that shades the empty cells a different color. Use Excels Conditional Formatting function to create this heat map. Hint: Click on Conditional Formatting in the Styles group in the Home tab. Select Highlight Cells Rules and click More Rules. Then enter Blanks in the Format only cells with: box. Choose a format for these blank cells that will make them obviously stand out. b. For each question, which respondents did not provide answers? Which question has the highest nonresponse rate?The following table shows monthly revenue for six different web development companies. a. Use Excel to create sparklines for sales at each company. b. Which companies have generally decreasing revenues over the six months? Which company has exhibited the most consistent growth over the six months? Which companies have revenues that are both increasing and decreasing over the six months? c. Use Excel to create a heat map for the revenue of the six companies. Do you find the heat map or the sparklines to be better at communicating the trend of revenues over the six months for each company? Why?21PAurora Radiological Services is a health care clinic that provides radiological imaging services (such as MRIs, X-rays, and CAT scans) to patients. It is part of Front Range Medical Systems that operates clinics throughout the state of Colorado. a. What type of key performance indicators and other information would be appropriate to display on a data dashboard to assist the Aurora clinics manager in making daily staffing decisions for the clinic? b. What type of key performance indicators and other information would be appropriate to display on a data dashboard for the CEO of Front Range Medical Systems who oversees the operation of multiple radiological imaging clinics?The motion picture industry is an extremely competitive business. Dozens of movie studios produce hundreds of movies each year, many of which cost hundreds of millions of dollars to produce and distribute. Some of these movies will go on to earn hundreds of millions of dollars in box office revenues, while others will earn much less than their production cost.
Data from 50 of the top box-office-receipt-generating movies are provided in the file Top50Movies. The following table shows the first 10 movies contained in this data set. The categorical variables included in the data set for each movie are the rating and genre. Quantitative variables for the movie’s release year, inflation- and noninflation-adjusted box-office receipts in the United States, budget, and the world box-office receipts are also included.
Managerial Report
Use the data-visualization methods presented in this chapter to explore these data and discover relationships between the variables. Include the following in your report:
Create a scatter chart to examine the relationship between the year released and the inflation-adjusted U.S. box office receipts. Include a trendline for this scatter chart. What does the scatter chart indicate about inflation-adjusted U.S. box office receipts over time for these top 50 movies?
Create a scatter chart to examine the relationship between the budget and the noninflation-adjusted world box office receipts. (Note: You may have to adjust the data in Excel to ignore the missing budget data values to create your scatter chart. You can do this by first sorting the data using Budget and then creating a scatter chart using only the movies that include data for Budget.) What does this scatter chart indicate about the relationship between the movie’s budget and the world box office receipts?
Create a frequency distribution, percent frequency distribution, and histogram for inflation-adjusted U.S. box office receipts. Use bin sizes of $100 million. Interpret the results. Do any data points appear to be outliers in this distribution?
Create a PivotTable for these data. Use the PivotTable to generate a crosstabulation for movie genre and rating. Determine which combinations of genre and rating are most represented in the top 50 movie data. Now filter the data to consider only movies released in 1980 or later. What combinations of genre and rating are most represented for movies after 1980? What does this indicate about how the preferences of moviegoers may have changed over time?
Use the PivotTable to display the average inflation-adjusted U.S. box-office receipts for each genre–rating pair for all movies in the data set. Interpret the results.
On-time arrivals, lost baggage, and customer complaints are three measures that are typically used to measure the quality of service being offered by airlines. Suppose that the following values represent the on-time arrival percentage, amount of lost baggage, and customer complaints for 10 U.S. airlines. a. Based on the data above, if you randomly choose a Delta Air Lines flight, what is the probability that this individual flight will have an on-time arrival? b. If you randomly choose 1 of the 10 airlines for a follow-up study on airline quality ratings, what is the probability that you will choose an airline with less than two mishandled baggage reports per 1,000 passengers? c. If you randomly choose 1 of the 10 airlines for a follow-up study on airline quality ratings, what is the probability that you will choose an airline with more than one customer complaint per 1,000 passengers? d. What is the probability that a randomly selected AirTran Airways flight will not arrive on time?Consider the random experiment of rolling a pair of dice. Suppose that we are interested in the sum of the face values showing on the dice. a. How many outcomes are possible? b. List the outcomes. c. What is the probability of obtaining a value of 7? d. What is the probability of obtaining a value of 9 or greater?Suppose that for a recent admissions class, an Ivy League college received 2,851 applications for early admission. Of this group, it admitted 1,033 students early, rejected 854 outright, and deferred 964 to the regular admission pool for further consideration. In the past, this school has admitted 18% of the deferred early admission applicants during the regular admission process. Counting the students admitted early and the students admitted during the regular admission process, the total class size was 2,375. Let E, R, and D represent the events that a student who applies for early admission is admitted early, rejected outright, or deferred to the regular admissions pool.
Use the data to estimate P(E), P(R), and P(D).
Are events E and D mutually exclusive? Find .
For the 2,375 students who were admitted, what is the probability that a randomly selected student was accepted during early admission?
Suppose a student applies for early admission. What is the probability that the student will be admitted for early admission or be deferred and later admitted during the regular admission process?
Suppose that we have two events, A and B, with P(A) = 0.50, P(B) = 0.60, and P(A B) = 0.40. a. Find P(A | B). b. Find P(B | A). c. Are A and B independent? Why or why not?Students taking the Graduate Management Admissions Test (GMAT) were asked about their undergraduate major and intent to pursue their MBA as a full-time or part-time student. A summary of their responses follows. a. Develop a joint probability table for these data. b. Use the marginal probabilities of undergraduate major (business, engineering, or other) to comment on which undergraduate major produces the most potential MBA students. c. If a student intends to attend classes full-time in pursuit of an MBA degree, what is the probability that the student was an undergraduate engineering major? d. If a student was an undergraduate business major, what is the probability that the student intends to attend classes full-time in pursuit of an MBA degree? e. Let F denote the event that the student intends to attend classes full-time in pursuit of an MBA degree, and let B denote the event that the student was an undergraduate business major. Are events F and B independent? Justify your answer.More than 40 million Americans are estimated to have at least one outstanding student loan to help pay college expenses (40 Million Americans Now Have Student Loan Debt, CNNMoney, September 2014). Not all of these graduates pay back their debt in satisfactory fashion. Suppose that the following joint probability table shows the probabilities of student loan status and whether or not the student had received a college degree. a. What is the probability that a student with a student loan had received a college degree? b. What is the probability that a student with a student loan had not received a college degree? c. Given that the student has received a college degree, what is the probability that the student has a delinquent loan? d. Given that the student has not received a college degree, what is the probability that the student has a delinquent loan? e. What is the impact of dropping out of college without a degree for students who have a student loan?The Human Resources Manager for Optilytics LLC is evaluating applications for the position of Senior Data Scientist. The file OptilyticsLLC presents summary data of the applicants for the position. a. Use a PivotTable in Excel to create a joint probability table showing the probabilities associated with a randomly selected applicants sex and highest degree achieved. Use this joint probability table to answer the questions below. b. What are the marginal probabilities? What do they tell you about the probabilities associated with the sex of applicants and highest degree completed by applicants? c. If the applicant is female, what is the probability that the highest degree completed by the applicant is a PhD? d. If the highest degree completed by the applicant is a bachelors degree, what is the probability that the applicant is male? e. What is the probability that a randomly selected applicant will be a male whose highest completed degree is a PhD?As was discussed in the Analytics in Action from Chapter 2, the U.S. Census Bureau is a leading source of quantitative data related to the people and economy of the United States. The crosstabulation below represents the number of households (1,000s) and the household income by the highest level of education for the head of household (U.S. Census Bureau web site, 2013). Use this crosstabulation to answer the following questions. a. Develop a joint probability table. b. What is the probability the head of one of these households has a masters degree or higher education? c. What is the probability a household is headed by someone with a high school diploma earning 100,000 or more? d. What is the probability one of these households has an income below 25,000? e. What is the probability a household is headed by someone with a bachelors degree earning less than 25,000? f. Are household income and educational level independent?Cooper Realty is a small real estate company located in Albany, New York, that specializes primarily in residential listings. The company recently became interested in determining the likelihood of one of its listings being sold within a certain number of days. An analysis of company sales of 800 homes in previous years produced the following data. a. If A is defined as the event that a home is listed for more than 90 days before being sold, estimate the probability of A. b. If B is defined as the event that the initial asking price is under 150,000, estimate the probability of B. c. What is the probability of AB? d. Assuming that a contract was just signed to list a home with an initial asking price of less than 150,000, what is the probability that the home will take Cooper Realty more than 90 days to sell? e. Are events A and B independent?10PA local bank reviewed its credit-card policy with the intention of recalling some of its credit cards. In the past, approximately 5% of cardholders defaulted, leaving the bank unable to collect the outstanding balance. Hence, management established a prior probability of 0.05 that any particular cardholder will default. The bank also found that the probability of missing a monthly payment is 0.20 for customers who do not default. Of course, the probability of missing a monthly payment for those who default is 1.
Given that a customer missed a monthly payment, compute the posterior probability that the customer will default.
The bank would like to recall its credit card if the probability that a customer will default is greater than 0.20. Should the bank recall its credit card if the customer misses a monthly payment? Why or why not?
RunningWithTheDevil.com created a web site to market running shoes and other running apparel. Management would like a special pop-up offer to appear for female web-site visitors and a different special pop-up offer to appear for male web-site visitors. From a sample of past web-site visitors, RunningWithTheDevil’s management learns that 60% of the visitors are male and 40% are female.
What is the probability that a current visitor to the web site is female?
Suppose that 30% of RunningWithTheDevil’s female visitors previously visited LetsRun.com and 10% of male customers previously visited LetsRun.com. If the current visitor to RunningWithTheDevil’s web site previously visited LetsRun.com, what is the revised probability that the current visitor is female? Should the RunningWithTheDevil’s web site display the special offer that appeals to female visitors or the special offer that appeals to male visitors?
An oil company purchased an option on land in Alaska. Preliminary geologic studies assigned the following prior probabilities. P(high-qualityoil)=0.50P(medium-qualityoil)=0.20P(nooil)=0.30 a. What is the probability of finding oil? b. After 200 feet of drilling on the first well, a soil test is taken. The probabilities of finding the particular type of soil identified by the test are as follows. P(soil|high-qualityoil)=0.20P(soilmedium-qualityoil)=0.80P(soilnooil)=0.20 How should the firm interpret the soil test? What are the revised probabilities, and what is the new probability of finding oil?Suppose the following data represent the number of persons unemployed for a given number of months in Killeen, Texas. The values in the first column show the number of months unemployed and the values in the second column show the corresponding number of unemployed persons.
Let x be a random variable indicating the number of months a randomly selected person is unemployed.
Use the data to develop an empirical discrete probability distribution for x.
Show that your probability distribution satisfies the conditions for a valid discrete probability distribution.
What is the probability that a person is unemployed for two months or less? Unemployed for more than two months?
What is the probability that a person is unemployed for more than six months?
The percent frequency distributions of job satisfaction scores for a sample of information systems (IS) senior executives and middle managers are as follows. The scores range from a low of 1 (very dissatisfied) to a high of 5 (very satisfied).
Develop a probability distribution for the job satisfaction score of a randomly selected senior executive.
Develop a probability distribution for the job satisfaction score of a randomly selected middle manager.
What is the probability that a randomly selected senior executive will report a job satisfaction score of 4 or 5?
What is the probability that a randomly selected middle manager is very satisfied?
Compare the overall job satisfaction of senior executives and middle managers.
The following table provides a probability distribution for the random variable y. a. Compute E(y). b. Compute Var(y) and .The probability distribution for damage claims paid by the Newton Automobile Insurance Company on collision insurance follows.
Use the expected collision payment to determine the collision insurance premium that would enable the company to break even.
The insurance company charges an annual rate of $520 for the collision coverage. What is the expected value of the collision policy for a policyholder? (Hint: It is the expected payments from the company minus the cost of coverage.) Why does the policyholder purchase a collision policy with this expected value?
The J.R. Ryland Computer Company is considering a plant expansion to enable the company to begin production of a new computer product. The companys president must determine whether to make the expansion a medium- or large-scale project. Demand for the new product is uncertain, which for planning purposes may be low demand, medium demand, or high demand. The probability estimates for demand are 0.20, 0.50, and 0.30, respectively. Letting x and y indicate the annual profit in thousands of dollars, the firms planners developed the following profit forecasts for the medium-and large-scale expansion projects. a. Compute the expected value for the profit associated with the two expansion alternatives. Which decision is preferred for the objective of maximizing the expected profit? b. Compute the variance for the profit associated with the two expansion alternatives. Which decision is preferred for the objective of minimizing the risk or uncertainty?19PMany companies use a quality control technique called acceptance sampling to monitor incoming shipments of parts, raw materials, and so on. In the electronics industry, component parts are commonly shipped from suppliers in large lots. Inspection of a sample of n components can be viewed as the n trials of a binomial experiment. The outcome for each component tested (trial) will be that the component is classified as good or defective. Reynolds Electronics accepts a lot from a particular supplier if the defective components in the lot do not exceed 1%. Suppose a random sample of five items from a recent shipment is tested.
Assume that 1% of the shipment is defective. Compute the probability that no items in the sample are defective.
Assume that 1% of the shipment is defective. Compute the probability that exactly one item in the sample is defective.
What is the probability of observing one or more defective items in the sample if 1% of the shipment is defective?
Would you feel comfortable accepting the shipment if one item was found to be defective? Why or why not?
Consider a Poisson distribution with .
Write the appropriate Poisson probability mass function.
Compute f(2).
Compute f(1).
Compute .
Emergency 911 calls to a small municipality in Idaho come in at the rate of one every two minutes. Assume that the number of 911 calls is a random variable that can be described by the Poisson distribution. a. What is the expected number of 911 calls in one hour? b. What is the probability of three 911 calls in five minutes? c. What is the probability of no 911 calls during a five-minute period?A regional director responsible for business development in the state of Pennsylvania is concerned about the number of small business failures. If the mean number of small business failures per month is 10, what is the probability that exactly 4 small businesses will fail during a given month? Assume that the probability of a failure is the same for any two months and that the occurrence or nonoccurrence of a failure in any month is independent of failures in any other month.The random variable x is known to be uniformly distributed between 10 and 20. a. Show the graph of the probability density function. b. Compute P(x15). c. Compute P(12x18) d. Compute E(x) e. Compute Var(x)26PSuppose we are interested in bidding on a piece of land and we know one other bidder is interested.1 The seller announced that the highest bid in excess of 10,000 will be accepted. Assume that the competitors bid x is a random variable that is uniformly distributed between 10,000 and 15,000. a. Suppose you bid 12,000. What is the probability that your bid will be accepted? b. Suppose you bid 14,000. What is the probability that your bid will be accepted? c. What amount should you bid to maximize the probability that you get the property? d. Suppose you know someone who is willing to pay you 16,000 for the property. Would you consider bidding less than the amount in part (c)? Why or why not?28PThe Siler Construction Company is about to bid on a new industrial construction project. To formulate their bid, the company needs to estimate the time required for the project. Based on past experience, management expects that the project will require at least 24 months, and could take as long as 48 months if there are complications. The most likely scenario is that the project will require 30 months. a. Assume that the actual time for the project can be approximated using a triangular probability distribution. What is the probability that the project will take less than 30 months? b. What is the probability that the project will take between 28 and 32 months? c. To submit a competitive bid, the company believes that if the project takes more than 36 months, then the company will lose money on the project. Management does not want to bid on the project if there is greater than a 25% chance that they will lose money on this project. Should the company bid on this project?Suppose that the return for a particular large-cap stock fund is normally distributed with a mean of 14.4% and standard deviation of 4.4%. a. What is the probability that the large-cap stock fund has a return of at least 20%? b. What is the probability that the large-cap stock fund has a return of 10% or less?A person must score in the upper 2% of the population on an IQ test to qualify for membership in Mensa, the international high IQ society. If IQ scores are normally distributed with a mean of 100 and a standard deviation of 15, what score must a person have to qualify for Mensa?Assume that the traffic to the web site of Smileys People, Inc., which sells customized T-shirts, follows a normal distribution, with a mean of 4.5 million visitors per day and a standard deviation of 820,000 visitors per day. a. What is the probability that the web site has fewer than 5 million visitors in a single day? b. What is the probability that the web site has 3 million or more visitors in a single day? c. What is the probability that the web site has between 3 million and 4 million visitors in a single day? d. Assume that 85% of the time, the Smileys People web servers can handle the daily web traffic volume without purchasing additional server capacity. What is the amount of web traffic that will require Smileys People to purchase additional server capacity?Suppose that Motorola uses the normal distribution to determine the probability of defects and the number of defects in a particular production process. Assume that the production process manufactures items with a mean weight of 10 ounces. Calculate the probability of a defect and the suspected number of defects for a 1,000-unit production run in the following situations. a. The process standard deviation is 0.15, and the process control is set at plus or minus one standard deviation. Units with weights less than 9.85 or greater than 10.15 ounces will be classified as defects. b. Through process design improvements, the process standard deviation can be reduced to 0.05. Assume that the process control remains the same, with weights less than 9.85 or greater than 10.15 ounces being classified as defects. c. What is the advantage of reducing process variation, thereby causing process control limits to be at a greater number of standard deviations from the mean?34P35PSuppose that the time spent by players in a single session on the World of Warcraft multiplayer online role-playing game follows an exponential distribution with a mean of 38.3 minutes. a. Write the exponential probability distribution function for the time spent by players on a single session of World of Warcraft. b. What is the probability that a player will spend between 20 and 40 minutes on a single session of World of Warcraft? c. What is the probability that a player will spend more than one hour on a single session of World of Warcraft?The American League consists of 15 baseball teams. Suppose a sample of 5 teams is to be selected to conduct player interviews. The following table lists the 15 teams and the random numbers assigned by Excels RAND function. Use these random numbers to select a sample of size 5.The U.S. Golf Association is considering a ban on long and belly putters. This has caused a great deal of controversy among both amateur golfers and members of the Professional Golf Association (PGA). Shown below are the names of the top 10 finishers in the recent PGA Tour McGladrey Classic golf tournament.
Tommy Gainey
David Toms
Jim Furyk
Brendon de Jonge
D. J. Trahan
Davis Love III
Chad Campbell
Greg Owens
Charles Howell III
Arjun Atwal
Select a simple random sample of 3 of these players to assess their opinions on the use of long and belly putters.
A simple random sample of 5 months of sales data provided the following information: a. Develop a point estimate of the population mean number of units sold per month. b. Develop a point estimate of the population standard deviation.Morningstar publishes ratings data on 1,208 company stocks. A sample of 40 of these stocks is contained in the file named Morningstar. Use the Morningstar data set to answer the following questions. a. Develop a point estimate of the proportion of the stocks that receive Morningstars highest rating of 5 Stars. b. Develop a point estimate of the proportion of the Morningstar stocks that are rated Above Average with respect to business risk. c. Develop a point estimate of the proportion of the Morningstar stocks that are rated 2 Stars or less.One of the questions in the Pew Internet & American Life Project asked adults if they used the Internet at least occasionally. The results showed that 454 out of 478 adults aged 18–29 answered Yes; 741 out of 833 adults aged 30–49 answered Yes; and 1,058 out of 1,644 adults aged 50 and over answered Yes.
Develop a point estimate of the proportion of adults aged 18–29 who use the Internet.
Develop a point estimate of the proportion of adults aged 30–49 who use the Internet.
Develop a point estimate of the proportion of adults aged 50 and over who use the Internet.
Comment on any apparent relationship between age and Internet use.
Suppose your target population of interest is that of all adults (18 years of age and over). Develop an estimate of the proportion of that population who use the Internet.
In this chapter we showed how a simple random sample of 30 EAI employees can be used to develop point estimates of the population mean annual salary, the population standard deviation for annual salary, and the population proportion having completed the management training program. a. Use Excel to select a simple random sample of 50 EAI employees. b. Develop a point estimate of the mean annual salary. c. Develop a point estimate of the population standard deviation for annual salary. d. Develop a point estimate of the population proportion having completed the management training program.The College Board reported the following mean scores for the three parts of the SAT:
Assume that the population standard deviation on each part of the test is = 100.
For a random sample of 30 test takers, what is the sampling distribution of for scores on the Critical Reading part of the test?
For a random sample of 60 test takers, what is the sampling distribution of for scores on the Mathematics part of the test?
For a random sample of 90 test takers, what is the sampling distribution of for scores on the Writing part of the test?
8PThe Economic Policy Institute periodically issues reports on wages of entry-level workers. The institute reported that entry-level wages for male college graduates were $21.68 per hour and for female college graduates were $18.80 per hour in 2011. Assume that the standard deviation for male graduates is $2.30, and for female graduates it is $2.05.
What is the sampling distribution of for a random sample of 50 male college graduates?
What is the sampling distribution of for a random sample of 50 female college graduates?
In which of the preceding two cases, part (a) or part (b), is the standard error of smaller? Why?
The state of California has a mean annual rainfall of 22 inches, whereas the state of New York has a mean annual rainfall of 42 inches. Assume that the standard deviation for both states is 4 inches. A sample of 30 years of rainfall for California and a sample of 45 years of rainfall for New York has been taken. a. Show the sampling distribution of the sample mean annual rainfall for California. b. Show the sampling distribution of the sample mean annual rainfall for New York. c. In which of the preceding two cases, part (a) or part (b), is the standard error of x smaller? Why?The president of Doerman Distributors, Inc., believes that 30% of the firm’s orders come from first-time customers. A random sample of 100 orders will be used to estimate the proportion of first-time customers. Assume that the president is correct and p = 0.30. What is the sampling distribution of for this study?
12PPeople end up tossing 12% of what they buy at the grocery store. Assume this is the true population proportion and that you plan to take a sample survey of 540 grocery shoppers to further investigate their behavior. Show the sampling distribution of , the proportion of groceries thrown out by your sample respondents.
14PThe International Air Transport Association surveys business travelers to develop quality ratings for transatlantic gateway airports. The maximum possible rating is 10. Suppose a simple random sample of 50 business travelers is selected and each traveler is asked to provide a rating for the Miami International Airport. The ratings obtained from the sample of 50 business travelers follow. Develop a 95% confidence interval estimate of the population mean rating for Miami.16PHealth insurers are beginning to offer telemedicine services online that replace the common office visit. Wellpoint provides a video service that allows subscribers to connect with a physician online and receive prescribed treatments. Wellpoint claims that users of its LiveHealth Online service saved a significant amount of money on a typical visit. The data shown below (), for a sample of 20 online doctor visits, are consistent with the savings per visit reported by Wellpoint. Assuming that the population is roughly symmetric, construct a 95% confidence interval for the mean savings for a televisit to the doctor as opposed to an office visit.The average annual premium for automobile insurance in the United States is 1,503. The following annual premiums () are representative of the web sites findings for the state of Michigan. Assume the population is approximately normal. a. Provide a point estimate of the mean annual automobile insurance premium in Michigan. b. Develop a 95% confidence interval for the mean annual automobile insurance premium in Michigan. c. Does the 95% confidence interval for the annual automobile insurance premium in Michigan include the national average for the United States? What is your interpretation of the relationship between auto insurance premiums in Michigan and the national average?19PAccording to Thomson Financial, last year the majority of companies reporting profits had beaten estimates. A sample of 162 companies showed that 104 beat estimates, 29 matched estimates, and 29 fell short.
What is the point estimate of the proportion that fell short of estimates?
Determine the margin of error and provide a 95% confidence interval for the proportion that beat estimates.
How large a sample is needed if the desired margin of error is 0.05?
The Pew Research Center Internet Project conducted a survey of 857 Internet users. This survey provided a variety of statistics on them.
The sample survey showed that 90% of respondents said the Internet has been a good thing for them personally. Develop a 95% confidence interval for the proportion of respondents who say the Internet has been a good thing for them personally.
The sample survey showed that 67% of Internet users said the Internet has generally strengthened their relationship with family and friends. Develop a 95% confidence interval for the proportion of respondents who say the Internet has strengthened their relationship with family and friends.
Fifty-six percent of Internet users have seen an online group come together to help a person or community solve a problem, whereas only 25% have left an online group because of unpleasant interaction. Develop a 95% confidence interval for the proportion of Internet users who say online groups have helped solve a problem.
Compare the margin of error for the interval estimates in parts (a), (b), and (c). How is the margin of error related to the sample proportion?
For many years businesses have struggled with the rising cost of health care. But recently, the increases have slowed due to less inflation in health care prices and employees paying for a larger portion of health care benefits. A recent Mercer survey showed that 52% of U.S. employers were likely to require higher employee contributions for health care coverage. Suppose the survey was based on a sample of 800 companies. Compute the margin of error and a 95% confidence interval for the proportion of companies likely to require higher employee contributions for health care coverage.
The manager of the Danvers-Hilton Resort Hotel stated that the mean guest bill for a weekend is 600 or less. A member of the hotels accounting staff noticed that the total charges for guest bills have been increasing in recent months. The accountant will use a sample of future weekend guest bills to test the managers claim. a. Which form of the hypotheses should be used to test the managers claim? Explain. H0:600H0:600H0:=600Ha:600Ha:600Ha:600 b. What conclusion is appropriate when H0 cannot be rejected? c. What conclusion is appropriate when H0 can be rejected?The manager of an automobile dealership is considering a new bonus plan designed to increase sales volume. Currently, the mean sales volume is 14 automobiles per month. The manager wants to conduct a research study to see whether the new bonus plan increases sales volume. To collect data on the plan, a sample of sales personnel will be allowed to sell under the new bonus plan for a one-month period. a. Develop the null and alternative hypotheses most appropriate for this situation. b. Comment on the conclusion when H0 cannot be rejected. c. Comment on the conclusion when H0 can be rejected.A production line operation is designed to fill cartons with laundry detergent to a mean weight of 32 ounces. A sample of cartons is periodically selected and weighed to determine whether underfilling or overfilling is occurring. If the sample data lead to a conclusion of underfilling or overfilling, the production line will be shut down and adjusted to obtain proper filling. a. Formulate the null and alternative hypotheses that will help in deciding whether to shut down and adjust the production line. b. Comment on the conclusion and the decision when H0 cannot be rejected. c. Comment on the conclusion and the decision when H0 can be rejected.Because of high production-changeover time and costs, a director of manufacturing must convince management that a proposed manufacturing method reduces costs before the new method can be implemented. The current production method operates with a mean cost of $220 per hour. A research study will measure the cost of the new method over a sample production period.
Develop the null and alternative hypotheses most appropriate for this study.
Comment on the conclusion when H0 cannot be rejected.
Comment on the conclusion when H0 can be rejected.
Duke Energy reported that the cost of electricity for an efficient home in a particular neighborhood of Cincinnati, Ohio, was $104 per month. A researcher believes that the cost of electricity for a comparable neighborhood in Chicago, Illinois, is higher. A sample of homes in this Chicago neighborhood will be taken and the sample mean monthly cost of electricity will be used to test the following null and alternative hypotheses.
Assume the sample data lead to rejection of the null hypothesis. What would be your conclusion about the cost of electricity in the Chicago neighborhood?
What is the Type I error in this situation? What are the consequences of making this error?
What is the Type II error in this situation? What are the consequences of making this error?
28PCarpetland salespersons average 8,000 per week in sales. Steve Contois, the firms vice president, proposes a compensation plan with new selling incentives. Steve hopes that the results of a trial selling period will enable him to conclude that the compensation plan increases the average sales per salesperson. a. Develop the appropriate null and alternative hypotheses. b. What is the Type I error in this situation? What are the consequences of making this error? c. What is the Type II error in this situation? What are the consequences of making this error?Suppose a new production method will be implemented if a hypothesis test supports the conclusion that the new method reduces the mean operating cost per hour.
State the appropriate null and alternative hypotheses if the mean cost for the current production method is $220 per hour.
What is the Type I error in this situation? What are the consequences of making this error?
What is the Type II error in this situation? What are the consequences of making this error?
31PA shareholders’ group, in lodging a protest, claimed that the mean tenure for a chief executive officer (CEO) was at least nine years. A survey of companies reported in The Wall Street Journal found a sample mean tenure of years for CEOs with a standard deviation of s = 6.38 years.
Formulate hypotheses that can be used to challenge the validity of the claim made by the shareholders’ group.
Assume that 85 companies were included in the sample. What is the p value for your hypothesis test?
At = 0.01, what is your conclusion?
33P34P35PAccording to the National Automobile Dealers Association, the mean price for used cars is 10,192. A manager of a Kansas City used car dealership reviewed a sample of 50 recent used car sales at the dealership in an attempt to determine whether the population mean price for used cars at this particular dealership differed from the national mean. The prices for the sample of 50 cars are shown in the tile named UsedCars. a. Formulate the hypotheses that can be used to determine whether a difference exists in the mean price for used cars at the dealership. b. What is the p value? c. At = 0.05, what is your conclusion?What percentage of the population live in their state of birth? According to the U.S. Census Bureaus American Community Survey, the figure ranges from 25% in Nevada to 78.7% in Louisiana. The average percentage across all states and the District of Columbia is 57.7%. The data in the file Homestate are consistent with the findings in the American Community Survey. The data are for a random sample of 120 Arkansas residents and for a random sample of 180 Virginia residents. a. Formulate hypotheses that can be used to determine whether the percentage of stay-at-home residents in the two states differs from the overall average of 57.7%. b. Estimate the proportion of stay-at-home residents in Arkansas. Does this proportion differ significantly from the mean proportion for all states? Use = 0.05. c. Estimate the proportion of stay-at-home residents in Virginia. Does this proportion differ significantly from the mean proportion for all states? Use = 0.05. d. Would you expect the proportion of stay-at-home residents to be higher in Virginia than in Arkansas? Support your conclusion with the results obtained in parts (b) and (c).38PTen years ago 53% of American families owned stocks or stock funds. Sample data collected by the Investment Company Institute indicate that the percentage is now 46%. a. Develop appropriate hypotheses such that rejection of H0 will support the conclusion that a smaller proportion of American families own stocks or stock funds this year than 10 years ago. b. Assume the Investment Company Institute sampled 300 American families to estimate that the percent owning stocks or stock funds is 46% this year. What is the p value for your hypothesis test? c. At = 0.01, what is your conclusion?40P41P42PThe Port Authority sells a wide variety of cables and adapters for electronic equipment online. Last year the mean value of orders placed with the Port Authority was 47.28, and management wants to assess whether the mean value of orders placed to date this year is the same as last year. The values of a sample of 49,896 orders placed this year are collected and recorded in the tile PortAuthority. a. Formulate hypotheses that can be used to test whether the mean value of orders placed this year differs from the mean value of orders placed last year. b. Use the data in the file PortAuthority to conduct your hypothesis test. What is the p value for your hypothesis test? At = 0.01, what is your conclusion?The Port Authority also wants to determine if the gender profile of its customers has changed since last year, when 59.4% of its orders placed were placed by males. The genders for a sample of 49,896 orders placed this year are collected and recorded in the file PortAuthority. a. Formulate hypotheses that can be used to test whether the proportion of orders placed by male customers this year differs from the proportion of orders placed by male customers placed last year. b. Use the data in the file PortAuthority to conduct your hypothesis test. What is the p value for your hypothesis test? At = 0.05, what is your conclusion?1CBicycling World, a magazine devoted to cycling, reviews hundreds of bicycles throughout the year. Its Road-Race category contains reviews of bicycles used by riders primarily interested in racing. One of the most important factors in selecting a bicycle for racing is its weight. The following data show the weight (pounds) and price () for ten racing bicycles reviewed by the magazine: a. Develop a scatter chart with weight as the independent variable. What does the scatter chart indicate about the relationship between the weight and price of these bicycles? b. Use the data to develop an estimated regression equation that could be used to estimate the price for a bicycle, given its weight. What is the estimated regression model? c. Test whether each of the regression parameters 0 and 1 is equal to zero at a 0.05 level of significance. What are the correct interpretations of the estimated regression parameters? Are these interpretations reasonable? d. How much of the variation in the prices of the bicycles in the sample does the regression model you estimated in part (b) explain? e. The manufacturers of the DOnofrio Pro plan to introduce the 15-lb DOnofrio Elite bicycle later this year. Use the regression model you estimated in part (a) to predict the price of the DOnonfrio Elite.In a manufacturing process the assembly line speed (feet per minute) was thought to affect the number of defective parts found during the inspection process. To test this theory, managers devised a situation in which the same batch of parts was inspected visually at a variety of line speeds. They collected the following data:
Develop a scatter chart with line speed as the independent variable. What does the scatter chart indicate about the relationship between line speed and the number of defective parts found?
Use the data to develop an estimated regression equation that could be used to predict the number of defective parts found, given the line speed. What is the estimated regression model?
Test whether each of the regression parameters β0 and β1 is equal to zero at a 0.01 level of significance. What are the correct interpretations of the estimated regression parameters? Are these interpretations reasonable?
How much of the variation in the number of defective parts found for the sample data does the model you estimated in part (b) explain?
Jensen Tire Auto is deciding whether to purchase a maintenance contract for its new computer wheel alignment and balancing machine. Managers feel that maintenance expense should be related to usage, and they collected the following information on weekly usage (hours) and annual maintenance expense (in hundreds of dollars). a. Develop a scatter chart with weekly usage hours as the independent variable. What does the scatter chart indicate about the relationship between weekly usage and annual maintenance expense? b. Use the data to develop an estimated regression equation that could be used to predict the annual maintenance expense for a given number of hours of weekly usage. What is the estimated regression model? c. Test whether each of the regression parameters 0 and 1 is equal to zero at a 0.05 level of significance. What are the correct interpretations of the estimated regression parameters? Are these interpretations reasonable? d. How much of the variation in the sample values of annual maintenance expense does the model you estimated in part (b) explain? e. If the maintenance contract costs 3,000 per year, would you recommend purchasing it? Why or why not?A sociologist was hired by a large city hospital to investigate the relationship between the number of unauthorized days that employees are absent per year and the distance (miles) between home and work for the employees. A sample of 10 employees was chosen, and the following data were collected. a. Develop a scatter chart for these data. Does a linear relationship appear reasonable? Explain. b. Use the data to develop an estimated regression equation that could be used to predict the number of days absent given the distance to work. What is the estimated regression model? c. What is the 99% confidence interval for the regression parameter 1? Based on this interval, what conclusion can you make about the hypotheses that the regression parameter 1 is equal to zero? d. What is the 99% confidence interval for the regression parameter 0? Based on this interval, what conclusion can you make about the hypotheses that the regression parameter 0 is equal to zero? e. How much of the variation in the sample values of number of days absent does the model you estimated in part (b) explain?The regional transit authority for a major metropolitan area wants to determine whether there is a relationship between the age of a bus and the annual maintenance cost. A sample of 10 buses resulted in the following data:
Develop a scatter chart for these data. What does the scatter chart indicate about the relationship between age of a bus and the annual maintenance cost?
Use the data to develop an estimated regression equation that could be used to predict the annual maintenance cost given the age of the bus. What is the estimated regression model?
Test whether each of the regression parameters β0 and β1 is equal to zero at a 0.05 level of significance. What are the correct interpretations of the estimated regression parameters? Are these interpretations reasonable?
How much of the variation in the sample values of annual maintenance cost does the model you estimated in part (b) explain?
What do you predict the annual maintenance cost to be for a 3.5-year-old bus?
A marketing professor at Givens College is interested in the relationship between hours spent studying and total points earned in a course. Data collected on 156 students who took the course last semester are provided in the tile MktHrsPts.
Develop a scatter chart for these data. What does the scatter chart indicate about the relationship between total points earned and hours spent studying?
Develop an estimated regression equation showing how total points earned is related to hours spent studying. What is the estimated regression model?
Test whether each of the regression parameters β0 and β1 is equal to zero at a 0.01 level of significance. What are the correct interpretations of the estimated regression parameters? Are these interpretations reasonable?
How much of the variation in the sample values of total point earned does the model you estimated in part (b) explain?
Mark Sweeney spent 95 hours studying. Use the regression model you estimated in part (b) to predict the total points Mark earned.
The Dow Jones Industrial Average (DJIA) and the Standard Poors 500 (SP 500) indexes are used as measures of overall movement in the stock market The DJIA is based on the price movements of 30 large companies: the SP 500 is an index composed of 500 stocks. Some say the SP 500 is a better measure of stock market performance because it is broader based. The closing price for the DJIA and the SP 500 for 15 weeks, beginning with January 6, 2012, follow (Barrons web site, April 17, 2012). a. Develop a scatter chart for these data with DJIA as the independent variable. What does the scatter chart indicate about the relationship between DJIA and SP 500? b. Develop an estimated regression equation showing how SP 500 is related to DJIA. What is the estimated regression model? c. What is the 95% confidence interval for the regression parameter 1? Based on this interval, what conclusion can you make about the hypotheses that the regression parameter 1 is equal to zero? d. What is the 95% confidence interval for the regression parameter 0? Based on this interval, what conclusion can you make about the hypotheses that the regression parameter 0 is equal to zero? e. How much of the variation in the sample values of SP 500 does the model estimated in part (b) explain? f. Suppose that the closing price for the DJIA is 13,500. Estimate the closing price for the SP 500. g. Should we be concerned that the DJIA value of 13,500 used to predict the SP 500 value in part (f) is beyond the range of the DJIA used to develop the estimated regression equation?The Toyota Camry is one of the best-selling cars in North America. The cost of a previously owned Camry depends on many factors, including the model year, mileage, and condition. To investigate the relationship between the car’s mileage and the sales price for Camrys, the following data show the mileage and sale price for 19 sales (PriceHub web site, February 24, 2012).
Develop a scatter chart for these data with miles as the independent variable. What does the scatter chart indicate about the relationship between price and miles?
Develop an estimated regression equation showing how price is related to miles. What is the estimated regression model?
Test whether each of the regression parameters β0 and β1 is equal to zero at a 0.01 level of significance. What are the correct interpretations of the estimated regression parameters? Are these interpretations reasonable?
How much of the variation in the sample values of price does the model estimated in part (b) explain?
For the model estimated in part (b), calculate the predicted price and residual for each automobile in the data. Identify the two automobiles that were the biggest bargains.
Suppose that you are considering purchasing a previously owned Camry that has been driven 60.000 miles. Use the estimated regression equation developed in part (b) to predict the price for this car. Is this the price you would offer the seller?
Dixie Showtime Movie Theaters, Inc., owns and operates a chain of cinemas in several markets in the southern United States. The owners would like to estimate weekly gross revenue as a function of advertising expenditures. Data for a sample of eight markets for a recent week follow: a. Develop an estimated regression equation with the amount of television advertising as the independent variable. Test for a significant relationship between television advertising and weekly gross revenue at the 0.05 level of significance. What is the interpretation of this relationship? b. How much of the variation in the sample values of weekly gross revenue does the model in part (a) explain? c. Develop an estimated regression equation with both television advertising and newspaper advertising as the independent variables. Test whether each of the regression parameters 0, 1, and 2 is equal to zero at a 0.05 level of significance. What are the correct interpretations of the estimated regression parameters? Are these interpretations reasonable? d. How much of the variation in the sample values of weekly gross revenue does the model in part (c) explain? e. Given the results in parts (a) and (c), what should your next step be? Explain. f. What are the managerial implications of these results?10PThe American Association of Individual Investors (AAII) On-Line Discount Broker Survey polls members on their experiences with electronic trades handled by discount brokers. As part of the survey, members were asked to rate their satisfaction with the trade price and the speed of execution, as well as provide an overall satisfaction rating. Possible responses (scores) were no opinion (0), unsatisfied (1), somewhat satisfied (2), satisfied (3), and very satisfied (4). For each broker, summary scores were computed by computing a weighted average of the scores provided by each respondent. A portion the survey results follow (AAII web site. February 7, 2012). a. Develop an estimated regression equation using trade price and speed of execution to predict overall satisfaction with the broker. Interpret the coefficient of determination. b. Use the t test to determine the significance of each independent variable. What are your conclusions at the 0.05 level of significance? c. Interpret the estimated regression parameters. Are the relationships indicated by these estimates what you would expect? d. Finger Lakes Investments has developed a new electronic trading system and would like to predict overall customer satisfaction assuming they can provide satisfactory service levels (3) for both trade price and speed of execution. Use the estimated regression equation developed in part (a) to predict overall satisfaction level for Finger Lakes Investments if they can achieve these performance levels. e. What concerns (if any) do you have with regard to the possible responses the respondents could select on the survey.The National Football League (NFL) records a variety of performance data for individuals and teams. To investigate the importance of passing on the percentage of games won by a team, the following data show the conference (Conf), average number of passing yards per attempt (Yds/Att), the number of interceptions thrown per attempt (Int/Att), and the percentage of games won (Win%) for a random sample of 16 NFL teams for the 2011 season (NFL web site, February 12, 2012). a. Develop the estimated regression equation that could be used to predict the percentage of games won, given the average number of passing yards per attempt. What proportion of variation in the sample values of proportion of games won does this model explain? b. Develop the estimated regression equation that could be used to predict the percentage of games won, given the number of interceptions thrown per attempt. What proportion of variation in the sample values of proportion of games won does this model explain? c. Develop the estimated regression equation that could be used to predict the percentage of games won, given the average number of passing yards per attempt and the number of interceptions thrown per attempt. What proportion of variation in the sample values of proportion of games won does this model explain? d. The average number of passing yards per attempt for the Kansas City Chiefs during the 2011 season was 6.2, and the teams number of interceptions thrown per attempt was 0.036. Use the estimated regression equation developed in part (c) to predict the percentage of games won by the Kansas City Chiefs during the 2011 season. Compare your prediction to the actual percentage of games won by the Kansas City Chiefs. (Note: For the 2011 season, the Kansas City Chiefs record was 7 wins and 9 losses.) e. Did the estimated regression equation that uses only the average number of passing yards per attempt as the independent variable to predict the percentage of games won provide a good fit?Johnson Filtration. Inc., provides maintenance service for water filtration systems throughout southern Florida. Customers contact Johnson with requests for maintenance service on their water filtration systems. To estimate the service time and the service cost. Johnson’s managers want to predict the repair time necessary for each maintenance request. Hence, repair time in hours is the dependent variable. Repair time is believed to be related to three factors: the number of months since the last maintenance service, the type of repair problem (mechanical or electrical), and the repairperson who performs the repair (Donna Newton or Bob Jones). Data for a sample of 10 service calls are reported in the following table:
Develop the simple linear regression equation to predict repair time given the number of months since the last maintenance service, and use the results to test the hypothesis that no relationship exists between repair time and the number of months since the last maintenance service at the 0.05 level of significance. What is the interpretation of this relationship? What does the coefficient of determination tell you about this model?
Using the simple linear regression model developed in part (a), calculate the predicted repair time and residual for each of the 10 repairs in the data. Sort the data in ascending order by value of the residual. Do you see any pattern in the residuals for the two types of repair? Do you see any pattern in the residuals for the two repairpersons? Do these results suggest any potential modifications to your simple linear regression model? Now create a scatter chart with months since last service on the x-axis and repair time in hours on the y-axis for which the points representing electrical and mechanical repairs are shown in different shapes and/or colors. Create a similar scatter chart of months since last service and repair time in hours for which the points representing repairs by Bob Jones and Donna Newton are shown in different shapes and/or colors. Do these charts and the results of your residual analysis suggest the same potential modifications to your simple linear regression model?
Create a new dummy variable that is equal to zero if the type of repair is mechanical and one if the type of repair is electrical. Develop the multiple regression equation to predict repair time, given the number of months since the last maintenance service and the type of repair. What are the interpretations of the estimated regression parameters? What does the coefficient of determination tell you about this model?
Create a new dummy variable that is equal to zero if the repairperson is Bob Jones and one if the repairperson is Donna Newton. Develop the multiple regression equation to predict repair time, given the number of months since the last maintenance service and the repairperson. What are the interpretations of the estimated regression parameters? What does the coefficient of determination tell you about this model?
Develop the multiple regression equation to predict repair time, given the number of months since the last maintenance service, the type of repair, and the repairperson. What are the interpretations of the estimated regression parameters? What does the coefficient of determination tell you about this model?
Which of these models would you use? Why?
A study investigated the relationship between audit delay (the length of time from a company’s fiscal year-end to the date of the auditor’s report) and variables that describe the client and the auditor. Some of the independent variables that were included in this study follow:
Industry A dummy variable coded 1 if the firm was an industrial company or 0 if the firm was a bank, savings and loan, or insurance company.
Public A dummy variable coded 1 if the company was traded on an organized exchange or over the counter: otherwise coded 0.
Quality A measure of overall quality of internal controls, as judged by the auditor, on a 5-point scale ranging from “virtually none” (1) to “excellent” (5).
Finished A measure ranging from 1 to 4, as judged by the auditor, where 1 indicates “all work performed subsequent to year-end” and 4 indicates “most work performed prior to year-end.”
A sample of 40 companies provided the following data:
Develop the estimated regression equation using all of the independent variables included in the data.
How much of the variation in the sample values of delay does this estimated regression equation explain? What other independent variables could you include in this regression model to improve the fit?
Test the relationship between each independent variable and the dependent variable at the 0.05 level of significance, and interpret the relationship between each of the independent variables and the dependent variable.
On the basis of your observations about the relationships between the dependent variable Delay and the independent variables Quality and Finished, suggest an alternative model for the regression equation developed in part (a) to explain as much of the variability in Delay as possible.
The U.S. Department of Energys Fuel Economy Guide provides fuel efficiency data for cars and trucks. A portion of the data for 311 compact, midsized, and large cars follows. The Class column identifies the size of the car: Compact, Midsize, or Large. The Displacement column shows the engines displacement in liters. The FuelType column shows whether the car uses premium (P) or regular (R) fuel, and the HwyMPG column shows the fuel efficiency rating for highway driving in terms of miles per gallon. The complete data set is contained in the tile FuelData: a. Develop an estimated regression equation that can be used to predict the fuel efficiency for highway driving given the engines displacement. Test for significance using the 0.05 level of significance. How much of the variation in the sample values of HwyMPG does this estimated regression equation explain? b. Create a scatter chart with HwyMPG on the y-axis and displacement on the x-axis for which the points representing compact, midsize, and large automobiles are shown in different shapes and/or colors. What does this chart suggest about the relationship between the class of automobile (compact, midsize, and large) and HwyMPG? c. Now consider the addition of the dummy variables ClassMidsize and ClassLarge to the simple linear regression model in part (a). The value of ClassMidsize is 1 if the car is a midsize car and 0 otherwise; the value of ClassLarge is 1 if the car is a large car and 0 otherwise. Thus, for a compact car, the value of ClassMidsize and the value of ClassLarge are both 0. Develop the estimated regression equation that can be used to predict the fuel efficiency for highway driving, given the engines displacement and the dummy variables ClassMidsize and ClassLarge. How much of the variation in the sample values of HwyMPG is explained by this estimated regression equation? d. Use significance level of 0.05 to determine whether the dummy variables added to the model in part (c) are significant. e. Consider the addition of the dummy variable FuelPremium, where the value of FuelPremium is 1 if the car uses premium fuel and 0 if the car uses regular fuel. Develop the estimated regression equation that can be used to predict the fuel efficiency for highway driving given the engines displacement, the dummy variables ClassMidsize and ClassLarge, and the dummy variable FuelPremium. How much of the variation in the sample values of HwyMPG does this estimated regression equation explain? f. For the estimated regression equation developed in part (e), test for the significance of the relationship between each of the independent variables and the dependent variable using the 0.05 level of significance for each test.A highway department is studying the relationship between traffic flow and speed during rush hour on Highway 193. The data in the file TrafficFlow were collected on Highway 193 during 100 recent rush hours. a. Develop a scatter chart for these data. What does the scatter chart indicate about the relationship between vehicle speed and traffic flow? b. Develop an estimated simple linear regression equation for the data. How much variation in the sample values of traffic flow is explained by this regression model? Use a 0.05 level of significance to test the relationship between vehicle speed and traffic flow. What is the interpretation of this relationship? c. Develop an estimated quadratic regression equation for the data. How much variation in the sample values of traffic flow is explained by this regression model? Test the relationship between each of the independent variables and the dependent variable at a 0.05 level of significance. How would you interpret this model? Is this model superior to the model you developed in part (b)? d. As an alternative to fitting a second-order model, fit a model using a piecewise linear regression with a single knot. What value of vehicle speed appears to be a good point for the placement of the knot? Does the estimated piecewise linear regression provide a better fit than the estimated quadratic regression developed in part (c)? Explain. e. Separate the data into two sets such that one data set contains the observations of vehicle speed less than the value of the knot from part (d) and the other data set contains the observations of vehicle speed greater than or equal to the value of the knot from part (d). Then fit a simple linear regression equation to each data set. How does this pair of regression equations compare to the single piecewise linear regression with the single knot from part (d)? In particular, compare predicted values of traffic flow for values of the speed slightly above and slightly below the knot value from part (d). f. What other independent variables could you include in your regression model to explain more variation in traffic flow?A sample containing years to maturity and (percent) yield for 40 corporate bonds is contained in the file named CorporateBonds (Barron’s, April 2. 2012).
Develop a scatter chart of the data using years to maturity as the independent variable. Does a simple linear regression model appear to be appropriate?
Develop an estimated quadratic regression equation with years to maturity and squared values of years to maturity as the independent variables. How much variation in the sample values of yield is explained by this regression model? Test the relationship between each of the independent variables and the dependent variable at a 0.05 level of significance. How would you interpret this model?
Create a plot of the linear and quadratic regression lines overlaid on the scatter chart of years to maturity and yield. Does this helps you better understand the difference in how the quadratic regression model and a simple linear regression model fit the sample data? Which model does this chart suggest provides a superior fit to the sample data?
What other independent variables could you include in your regression model to explain more variation in yield?
In 2011, home prices and mortgage rates fell so far that in a number of cities the monthly cost of owning a home was less expensive than renting. The following data show the average asking rent for 10 markets and the monthly mortgage on the median priced home (including taxes and insurance) for 10 cites where the average monthly mortgage payment was less than the average asking rent (The Wall Street Journal, November 2627, 2011). a. Develop a scatter chart for these data, treating the average asking rent as the independent variable. Does a simple linear regression model appear to be appropriate? b. Use a simple linear regression model to develop an estimated regression equation to predict the monthly mortgage on the median priced home given the average asking rent. Construct a plot of the residuals against the independent variable rent. Based on this residual plot, does a simple linear regression model appear to be appropriate? c. Using a quadratic regression model, develop an estimated regression equation to predict the monthly mortgage on the median-priced home, given the average asking rent. d. Do you prefer the estimated regression equation developed in part (a) or part (c)? Create a plot of the linear and quadratic regression lines overlaid on the scatter chart of the monthly mortgage on the median-priced home and the average asking rent to help you assess the two regression equations. Explain your conclusions.A recent 10-year study conducted by a research team at the Great Falls Medical School was conducted to assess how age, systolic blood pressure, and smoking relate to the risk of strokes. Assume that the following data are from a portion of this study. Risk is interpreted as the probability (times 100) that the patient will have a stroke over the next 10-year period. For the smoking variable, define a dummy variable with 1 indicating a smoker and 0 indicating a nonsmoker.
Develop an estimated multiple regression equation that relates risk of a stroke to the person’s age, systolic blood pressure, and whether the person is a smoker.
Is smoking a significant factor in the risk of a stroke? Explain. Use a 0.05 level of significance.
What is the probability of a stroke over the next 10 years for Art Speen, a 68-year-old smoker who has a systolic blood pressure of 175? What action might the physician recommend for this patient?
What other factors could be included in the model as independent variables?
The Scholastic Aptitude Test (or SAT) is a standardized college entrance test that is used by colleges and universities as a means for making admission decisions. The critical reading and mathematics components of the SAT are reported on a scale from 200 to 800. Several universities believe these scores are strong predictors of an incoming student’s potential success, and they use these scores as important inputs when making admission decisions on potential freshman. The tile RugglesCollege contains freshman year GPA and the critical reading and mathematics SAT scores for a random sample of 200 students who recently completed their freshman year at Ruggles College.
Develop an estimated multiple regression equation that includes critical reading and mathematics SAT scores as independent variables. How much variation in freshman GPA is explained by this model? Test whether each of the regression parameters β0, β1, and β2, is equal to zero at a 0.05 level of significance. What are the correct interpretations of the estimated regression parameters? Are these interpretations reasonable?
Using the multiple linear regression model you developed in part (a), what is the predicted freshman GPA of Bobby Engle, a student who has been admitted to Ruggles College with a 660 SAT score on critical reading and at a 630 SAT score on mathematics?
The Ruggles College Director of Admissions believes that the relationship between a student’s scores on the critical reading component of the SAT and the student’s freshman GPA varies with the student’s score on the mathematics component of the SAT. Develop an estimated multiple regression equation that includes critical reading and mathematics SAT scores and their interaction as independent variables. How much variation in freshman GPA is explained by this model? Test whether each of the regression parameters β0, β1, β2, and β3 is equal to zero at a 0.05 level of significance. What are the correct interpretations of the estimated regression parameters? Do these results support the conjecture made by the Ruggles College Director of Admissions?
Do you prefer the estimated regression model developed in part (a) or part (c)? Explain.
What other factors could be included in the model as independent variables?
Consider again the example introduced in Section 7.5 of a credit card company that has a database of information provided by its customers when they apply for credit cards. An analyst has created a multiple regression model for which the dependent variable in the model is credit card charges accrued by a customer in the data set over the past year (y), and the independent variables are the customers annual household income (x1), number of members of the household (x2), and number of years of post-high school education (x3). Figure 7.23 provides Excel output for a multiple regression model estimated using a data set the company created. a. Estimate the corresponding simple linear regression with the customers annual household income as the independent variable and credit card charges accrued by a customer over the past year as the dependent variable. Interpret the estimated relationship between the customers annual household income and credit card charges accrued over the past year. How much variation in credit card charges accrued by a customer over the past year is explained by this simple linear regression model? b. Estimate the corresponding simple linear regression with the number of members in the customers household as the independent variable and credit card charges accrued by a customer over the past year as the dependent variable. Interpret the estimated relationship between the number of members in the customers household and credit card charges accrued over the past year. How much variation in credit card charges accrued by a customer over the past year is explained by this simple linear regression model? c. Estimate the corresponding simple linear regression with the customers number of years of posthigh school education as the independent variable and credit card charges accrued by a customer over the past year as the dependent variable. Interpret the estimated relationship between the customers number of years of posthigh school education and credit card charges accrued over the past year. How much variation in credit card charges accrued by a customer over the past year is explained by this simple linear regression model? d. Recall the multiple regression in Figure 7.23 with credit card charges accrued by a customer over the past year as the dependent variable and customers annual household income (x1), number of members of the household (x2), and number of years of post-high school education (x3) as the independent variables. Do the estimated slopes differ substantially from the corresponding slopes that were estimated using simple linear regression in parts (a), (b), and (c)? What does this tell you about multicollinearity in the multiple regression model in Figure 7.23? e. Add the coefficients of determination for the simple linear regression in parts (a), (b), and (c), and compare the result to the coefficient of determination for the multiple regression model in Figure 7.23. What does this tell you about multicollinearity in the multiple regression model in Figure 7.23? f. Add age, a dummy variable for sex, and a dummy variable for whether a customer has exceeded his or her credit limit in the past 12 months as independent variables to the multiple regression model in Figure 7.23. Code the dummy variable for sex as 1 if the customer is female and 0 if male, and code the dummy variable for whether a customer has exceeded his or her credit limit in the past 12 months as 1 if the customer has exceeded his or her credit limit in the past 12 months and 0 otherwise. Do these variables substantially improve the fit of your model?Alumni donations are an important source of revenue for colleges and universities. If administrators could determine the factors that could lead to increases in the percentage of alumni who make a donation, they might be able to implement policies that could lead to increased revenues. Research shows that students who are more satisfied with their contact with teachers are more likely to graduate. As a result, one might suspect that smaller class sizes and lower student/faculty ratios might lead to a higher percentage of satisfied graduates, which in turn might lead to increases in the percentage of alumni who make a donation. The following table shows data for 48 national universities. The Graduation Rate column is the percentage of students who initially enrolled at the university and graduated. The % of Classes Under 20 column shows the percentages of classes with fewer than 20 students that are offered. The Student/Faculty Ratio column is the number of students enrolled divided by the total number of faculty. Finally, the Alumni Giving Rate column is the percentage of alumni who made a donation to the university. Managerial Report 1. Use methods of descriptive statistics to summarize the data. 2. Develop an estimated simple linear regression model that can be used to predict the alumni giving rate, given the graduation rate. Discuss your findings. 3. Develop an estimated multiple linear regression model that could be used to predict the alumni giving rate using the Graduation Rate, % of Classes Under 20, and Student/Faculty Ratio as independent variables. Discuss your findings. 4. Based on the results in parts (2) and (3), do you believe another regression model may be more appropriate? Estimate this model, and discuss your results. 5. What conclusions and recommendations can you derive from your analysis? What universities are achieving a substantially higher alumni giving rate than would be expected, given their Graduation Rate, % of Classes Under 20, and Student/Faculty Ratio? What universities are achieving a substantially lower alumni giving rate than would be expected, given their Graduation Rate, % of Classes Under 20, and Student/Faculty Ratio? What other independent variables could be included in the model?Consider the following time series data:
Using the naĂŻve method (most recent value) as the forecast for the next week, compute the following measures of forecast accuracy:
Mean absolute error
Mean squared error
Mean absolute percentage error
What is the forecast for week 7?
Refer to the time series data in Problem 1. Using the average of all the historical data as a forecast for the next period, compute the following measures of forecast accuracy: a. Mean absolute error b. Mean squared error c. Mean absolute percentage error d. What is the forecast for week 7? 1. Consider the following time series data: Using the nave method (most recent value) as the forecast for the next week, compute the following measures of forecast accuracy: a. Mean absolute error b. Mean squared error c. Mean absolute percentage error d. What is the forecast for week 7?Problems 1 and 2 used different forecasting methods. Which method appears to provide the more accurate forecasts for the historical data? Explain.
Consider the following time series data:
Using the naĂŻve method (most recent value) as the forecast for the next week, compute the following measures of forecast accuracy:
Mean absolute error
Mean squared error
Mean absolute percentage error
What is the forecast for week 7?
Refer to the time series data in Problem 1. Using the average of all the historical data as a forecast for the next period, compute the following measures of forecast accuracy:
Mean absolute error
Mean squared error
Mean absolute percentage error
What is the forecast for week 7?
Consider the following time series data:
Compute MSE using the most recent value as the forecast for the next period. What is the forecast for month 8?
Compute MSE using the average of all the data available as the forecast for the next period. What is the forecast for month 8?
Which method appears to provide the better forecast?
Consider the following time series data:
Construct a time series plot. What type of pattern exists in the data?
Develop a three-week moving average for this time series. Compute MSE and a forecast for week 7.
Use α = 0.2 to compute the exponential smoothing values for the time series. Compute MSE and a forecast for week 7.
Compare the three-week moving average forecast with the exponential smoothing forecast using α = 0.2. Which appears to provide the better forecast based on MSE? Explain.
Use trial and error to find a value of the exponential smoothing coefficient α that results in a smaller MSE than what you calculated for α = 0.2.
Consider the following time series data:
Construct a time series plot. What type of pattern exists in the data?
Develop a three-week moving average for this time series. Compute MSE and a forecast for week 8.
Use α = 0.2 to compute the exponential smoothing values for the time series. Compute MSE and a forecast for week 8.
Compare the three-week moving average forecast with the exponential smoothing forecast using α = 0.2. Which appears to provide the better forecast based on MSE?
Use trial and error to find a value of the exponential smoothing coefficient α that results in a smaller MSE than what you calculated for α = 0.2.
Refer to the gasoline sales time series data in Table 8.1.
Compute four-week and five-week moving averages for the time series.
Compute the MSE for the four-week and five-week moving average forecasts.
What appears to be the best number of weeks of past data (three, four, or five) to use in the moving average computation? Recall that the MSE for the three-week moving average is 10.22.
8P9P10PFor the Hawkins Company, the monthly percentages of all shipments received on time over the past 12 months are 80, 82, 84, 83, 83, 84, 85, 84, 82, 83, 84, and 83.
Construct a time series plot. What type of pattern exists in the data?
Compare a three-month moving average forecast with an exponential smoothing forecast for α = 0.2. Which provides the better forecasts using MSE as the measure of model accuracy?
What is the forecast for next month?
Corporate triple A bond interest rates for 12 consecutive months are as follows:
Construct a time series plot. What type of pattern exists in the data?
Develop three-month and four-month moving averages for this time series. Does the three-month or the four-month moving average provide the better forecasts based on MSE? Explain.
What is the moving average forecast for the next month?
The values of Alabama building contracts (in millions of dollars) for a 12-month period are as follows:
Construct a time series plot. What type of pattern exists in the data?
Compare a three-month moving average forecast with an exponential smoothing forecast. Use α = 0.2. Which provides the better forecasts based on MSE?
What is the forecast for the next month using exponential smoothing with α = 0.2?
The following time series shows the sales of a particular product over the past 12 months.
Construct a time series plot. What type of pattern exists in the data?
Use α = 0.3 to compute the exponential smoothing values for the time series.
Use trial and error to find a value of the exponential smoothing coefficient α that results in a relatively small MSE.
15PThe following table reports the percentage of stocks in a portfolio for nine quarters: a. Construct a time series plot. What type of pattern exists in the data? b. Use trial and error to find a value of the exponential smoothing coefficient that results in a relatively small MSE. c. Using the exponential smoothing model you developed in part (b), what is the forecast of the percentage of stocks in a typical portfolio for the second quarter of year 3?Consider the following time series: a. Construct a time series plot. What type of pattern exists in the data? b. Use simple linear regression analysis to find the parameters for the line that minimizes MSE for this time series. c. What is the forecast for t = 6?Consider the following time series:
Construct a time series plot. What type of pattern exists in the data?
Use simple linear regression analysis to find the parameters for the line that minimizes MSE for this time series.
What is the forecast for t = 8?
Because of high tuition costs at state and private universities, enrollments at community colleges have increased dramatically in recent years. The following data show the enrollment for Jefferson Community College for the nine most recent years:
Construct a time series plot. What type of pattern exists in the data?
Use simple linear regression analysis to find the parameters for the line that minimizes MSE for this time series.
What is the forecast for year 10?
The Seneca Children’s Fund (SCF) is a local charity that runs a summer camp for disadvantaged children. The fund’s board of directors has been working very hard over recent years to decrease the amount of overhead expenses, a major factor in how charities are rated by independent agencies. The following data show the percentage of the money SCF has raised that was spent on administrative and fund-raising expenses over the past seven years:
Construct a time series plot. What type of pattern exists in the data?
Use simple linear regression analysis to find the parameters for the line that minimizes MSE for this time series.
Forecast the percentage of administrative expenses for year 8.
If SCF can maintain its current trend in reducing administrative expenses, how long will it take SCF to achieve a level of 5% or less?
The president of a small manufacturing firm is concerned about the continual increase in manufacturing costs over the past several years. The following figures provide a time series of the cost per unit for the firms leading product over the past eight years: a. Construct a time series plot. What type of pattern exists in the data? b. Use simple linear regression analysis to find the parameters for the line that minimizes MSE for this time series. c. What is the average cost increase that the firm has been realizing per year? d. Compute an estimate of the cost/unit for next year.Consider the following time series: a. Construct a time series plot. What type of pattern exists in the data? Is there an indication of a seasonal pattern? b. Use a multiple linear regression model with dummy variables as follows to develop an equation to account for seasonal effects in the data: Qtr1 = 1 if quarter 1, 0 otherwise; Qtr2 = 1 if quarter 2, 0 otherwise; Qtr3 = 1 if quarter 3, 0 otherwise. c. Compute the quarterly forecasts for next year.Consider the following time series data:
Construct a time series plot. What type of pattern exists in the data?
Use a multiple regression model with dummy variables as follows to develop an equation to account for seasonal effects in the data: Qtr1 = 1 if quarter 1, 0 otherwise; Qtr2 = 1 if quarter 2. 0 otherwise; Qtr3 = 1 if quarter 3, 0 otherwise.
Compute the quarterly forecasts for next year based on the model you developed in part (b).
Use a multiple regression model to develop an equation to account for trend and seasonal effects in the data. Use the dummy variables you developed in part (b) to capture seasonal effects and create a variable t such that t = 1 for quarter 1 in year 1, t = 2 for quarter 2 in year 1, … t = 12 for quarter 4 in year 3.
Compute the quarterly forecasts for next year based on the model you developed in part (d).
Is the model you developed in part (b) or the model you developed in part (d) more effective? Justify your answer.
The quarterly sales data (number of copies sold) for a college textbook over the past three years are as follows: a. Construct a time series plot. What type of pattern exists in the data? b. Use a regression model with dummy variables as follows to develop an equation to account for seasonal effects in the data: Qtr1 = l if quarter l, 0 otherwise; Qtr2 = l if quarter 2, 0 otherwise; Qtr3 = 1 if quarter 3, 0 otherwise. c. Based on the model you developed in part (b), compute the quarterly forecasts for next year. d. Let t = 1 to refer to the observation in quarter 1 of year 1; t = 2 to refer to the observation in quarter 2 of year 1; ; and t = 12 to refer to the observation in quarter 4 of year 3. Using the dummy variables defined in part (b) and t, develop an equation to account for seasonal effects and any linear trend in the time series. e. Based upon the seasonal effects in the data and linear trend, compute the quarterly forecasts for next year. f. Is the model you developed in part (b) or the model you developed in part (d) more effective? Justify your answer.25PSouth Shore Construction builds permanent docks and seawalls along the southern shore of Long Island, New York. Although the firm has been in business only five years, revenue has increased from $308,000 in the first year of operation to $1,084,000 in the most recent year. The following data show the quarterly sales revenue in thousands of dollars:
Construct a time series plot. What type of pattern exists in the data?
Use a multiple regression model with dummy variables as follows to develop an equation to account for seasonal effects in the data: Qtr1 = 1 if quarter I, 0 otherwise; Qtr2 = 1 if quarter 2, 0 otherwise; Qtr3 = 1 if quarter 3, 0 otherwise.
Based on the model you developed in part (b), compute estimates of quarterly sales for year 6.
Let Period = 1 refer to the observation in quarter 1 of year 1; Period = 2 refer to the observation in quarter 2 of year 1; … and Period = 20 refer to the observation in quarter 4 of year 5. Using the dummy variables defined in part (b) and the variable Period, develop an equation to account for seasonal effects and any linear trend in the time series.
Based on the seasonal effects in the data and linear trend estimated in part (c), compute estimates of quarterly sales for year 6.
Is the model you developed in part (b) or the model you developed in part (d) more effective? Justify your answer.
Hogs & Dawgs is an ice cream parlor on the border of north-central Louisiana and southern Arkansas that serves 43 flavors of ice creams, sherbets, frozen yogurts, and sorbets. During the summer Hogs & Dawgs is open from 1:00 p.m. to 10:00 p.m. on Monday through Saturday, and the owner believes that sales change systematically from hour to hour throughout the day. She also believes her sales increase as the outdoor temperature increases. Hourly sales and the outside temperature at the start of each hour for the last week are provided in the file IceCreamSales.
Construct a time series plot of hourly sales and a scatter plot of outdoor temperature and hourly sales. What types of relationships exist in the data?
Use a simple regression model with outside temperature as the causal variable to develop an equation to account for the relationship between outside temperature and hourly sales in the data. Based on this model, compute an estimate of hourly sales for today from 2:00 p.m. to 3:00 p.m. if the temperature at 2:00 p.m. is 93°F.
Use a multiple linear regression model with the causal variable outside temperature and dummy variables as follows to develop an equation to account for both seasonal effects and the relationship between outside temperature and hourly sales in the data in the data:
Hour1 = 1 if the sales were recorded between 1:00 p.m. and 2:00 p.m., 0 otherwise;
Hour2 = 1 if the sales were recorded between 2:00 p.m. and 3:00 p.m., 0 otherwise;
Hour8 = 1 if the sales were recorded between 8:00 p.m. and 9:00 p.m., 0 otherwise.
Note that when the values of the 8 dummy variables are equal to 0, the observation corresponds to the 9:00-to-l0:00-p.m. hour.
Based on this model, compute an estimate of hourly sales for today from 2:00 p.m. to 3:00 p.m. if the temperature at 2:00 p.m. is 93°F.
Is the model you developed in part (b) or the model you developed in part (c) more effective? Justify your answer.
Donna Nickles manages a gasoline station on the corner of Bristol Avenue and Harpst Street in Arcata, California. Her station is a franchise, and the parent company calls her station every day at midnight to give her the prices for various grades of gasoline for the upcoming day. Over the past eight weeks Donna has recorded the price and sales (in gallons) of regular-grade gasoline at her station as well as the price of regular-grade gasoline charged by her competitor across the street. She is curious about the sensitivity of her sales to the price of regular gasoline she charges and the price of regular gasoline charged by her competitor across the street. She also wonders whether her sales differ systematically by day of the week and whether her station has experienced a trend in sales over the past eight weeks. The data collected by Donna for each day of the past eight weeks are provided in the tile GasStation.
Construct a time series plot of daily sales, a scatter plot of the price Donna charges for a gallon of regular gasoline and daily sales at Donna’s station, and a scatter plot of the price Donna’s competitor charges for a gallon of regular gasoline and daily sales at Donna’s station. What types of relationships exist in the data?
Use a multiple regression model with the price Donna charges for a gallon of regular gasoline and the price Donna’s competitor charges for a gallon of regular gasoline as causal variables to develop an equation to account for the relationships between these prices and Donna’s daily sales in the data. Based on this model, compute an estimate of sales for a day on which Donna is charging $3.50 for a gallon for regular gasoline and her competitor is charging $3.45 for a gallon of regular gasoline.
Use a multiple linear regression model with the trend and dummy variables as follows to develop an equation to account for both trend and seasonal effects in the data:
Monday = 1 if the sales were recorded on a Monday. 0 otherwise;
Tuesday = 1 if the sales were recorded on a Tuesday, 0 otherwise;
Saturday = 1 if the sales were recorded on a Saturday, 0 otherwise;
Note that when the values of the six dummy variables are equal to 0, the observation corresponds to Sunday.
Based on this model, compute an estimate of sales for Tuesday of the first week after Donna collected her data.
Use a multiple regression model with the price Donna charges for a gallon of regular gasoline and the price Donna’s competitor charges for a gallon of regular gasoline as causal variables and the trend and dummy variables from part (c) to create an equation to account for the relationships between these prices and daily sales as well as the trend and seasonal effects in the data. Based on this model, compute an estimate of sales for Tuesday of the first week after Donna collected her data a day if Donna is charging $3.50 for a gallon for regular gasoline and her competitor is charging $3.45 for a gallon of regular gasoline.
Which of the three models you developed in parts (b), (c), and (d) is most effective? Justify your answer.
The Vintage Restaurant, on Captiva Island near Fort Myers, Florida, is owned and operated by Karen Payne. The restaurant just completed its third year of operation. During those three years, Karen sought to establish a reputation for the restaurant as a high-quality dining establishment that specializes in fresh seafood. Through the efforts of Karen and her staff, her restaurant has become one of the best and fastest-growing restaurants on the island. To better plan for future growth of the restaurant, Karen needs to develop a system that will enable her to forecast food and beverage sales by month for up to one year in advance. The following table shows the value of food and beverage sales (1,000s) for the first three years of operation: Managerial Report Perform an analysis of the sales data for the Vintage Restaurant. Prepare a report for Karen that summarizes your findings, forecasts, and recommendations. Include the following: 1. A time series plot. Comment on the underlying pattern in the time series. 2. Using the dummy variable approach, forecast sales for January through December of the fourth year. How would you explain this model to Karen? Assume that January sales for the fourth year turn out to be 295,000. What was your forecast error? If this error is large, Karen may be puzzled about the difference between your forecast and the actual sales value. What can you do to resolve her uncertainty about the forecasting procedure?Consider again the scenario described in Problem 4.
The Center for Business Analytics is considering a refund policy for no-shows. No refund would be given for members who do not attend, but nonmembers who do not attend will be refunded 50% of the price. Extend the model you developed in Problem 4 for the Business Intelligence Symposium to account for the fact that, historically, 25% of members who registered do not show and 10% of registered nonmembers do not attend. The center pays the caterer for breakfast and lunch based on the number of registrants (not the number of attendees). However, the center pays for parking only for those who attend. What is the profit if each corporate member registers their full allotment of tickets and 127 nonmembers register?
Use a two-way data table to show how profit changes as a function of number of registered nonmembers and the no-show percentage of nonmembers. Vary the number of nonmember registrants from 80 to 160 in increments of 5 and the percentage of nonmember no-shows from 10 to 30% in increments of 2%.
4. The University of Cincinnati Center for Business Analytics is an outreach center that collaborates with industry partners on applied research and continuing education in business analytics. One of the programs offered by the center is a quarterly Business Intelligence Symposium. Each symposium features three speakers on the real-world use of analytics. Each corporate member of the center (there are currently 10) receives five free seats to each symposium. Nonmembers wishing to attend must pay $75 per person. Each attendee receives breakfast, lunch, and free parking. The following are the costs incurred for putting on this event:
Build a spreadsheet model that calculates a profit or loss based on the number of nonmember registrants.
Use Goal Seek to find the number of nonmember registrants that will make the event break even.
Kelson Sporting Equipment, Inc., makes two types of baseball gloves: a regular model and a catchers model. The firm has 900 hours of production time available in its cutting and sewing department, 300 hours available in its finishing department, and 100 hours available in its packaging and shipping department. The production time requirements and the profit contribution per glove are given in the following table: Assuming that the company is interested in maximizing the total profit contribution, answer the following: a. What is the linear programming model for this problem? b. Develop a spreadsheet model and find the optimal solution using Excel Solver. How many of each model should Kelson manufacture? c. What is the total profit contribution Kelson can earn with the optimal production quantities? d. How many hours of production time will be scheduled in each department? e. What is the slack time in each department?The Sea Wharf Restaurant would like to determine the best way to allocate a monthly advertising budget of 1,000 between newspaper advertising and radio advertising. Management decided that at least 25% of the budget must be spent on each type of media and that the amount of money spent on local newspaper advertising must be at least twice the amount spent on radio advertising. A marketing consultant developed an index that measures audience exposure per dollar of advertising on a scale from 0 to 100, with higher values implying greater audience exposure. If the value of the index for local newspaper advertising is 50 and the value of the index for spot radio advertising is 80, how should the restaurant allocate its advertising budget to maximize the value of total audience exposure? a. Formulate a linear programming model that can be used to determine how the restaurant should allocate its advertising budget in order to maximize the value of total audience exposure. b. Develop a spreadsheet model and solve the problem using Excel Solver.Blair Rosen. Inc. (BR) is a brokerage firm that specializes in investment portfolios designed to meet the specific risk tolerances of its clients. A client who contacted BR this past week has a maximum of 50,000 to invest. BRs investment advisor decides to recommend a portfolio consisting of two investment funds: an Internet fund and a Blue Chip fund. The Internet fund has a projected annual return of 12%, and the Blue Chip fund has a projected annual return of 9%. The investment advisor requires that at most 35,000 of the clients funds should be invested in the Internet fund. BR services include a risk rating for each investment alternative. The Internet fund, which is the more risky of the two investment alternatives, has a risk rating of 6 per 1,000 invested. The Blue Chip fund has a risk rating of 4 per 1,000 invested. For example, if 10,000 is invested in each of the two investment funds, BRs risk rating for the portfolio would be 6(10) + 4(10) = 100. Finally. BR developed a questionnaire to measure each clients risk tolerance. Based on the responses, each client is classified as a conservative, moderate, or aggressive investor. Suppose that the questionnaire results classified the current client as a moderate investor. BR recommends that a client who is a moderate investor limit his or her portfolio to a maximum risk rating of 240. a. Formulate a linear programming model to find the best investment strategy for this client. b. Build a spreadsheet model and solve the problem using Solver. What is the recommended investment portfolio for this client? What is the annual return for the portfolio? c. Suppose that a second client with 50,000 to invest has been classified as an aggressive investor. BR recommends that the maximum portfolio risk rating for an aggressive investor is 320. What is the recommended investment portfolio for this aggressive investor? d. Suppose that a third client with 50,000 to invest has been classified as a conservative investor. BR recommends that the maximum portfolio risk rating for a conservative investor is 160. Develop the recommended investment portfolio for the conservative investor.Round Tree Manor is a hotel that provides two types of rooms with three rental classes: Super Saver, Deluxe, and Business. The profit per night for each type of room and rental class is as follows: Type I rooms do not have high-speed wireless Internet access and are not available for the Business rental class. Round Trees management makes a forecast of the demand by rental class for each night in the future. A linear programming model developed to maximize profit is used to determine how many reservations to accept for each rental class. The demand forecast for a particular night is 130 rentals in the Super Saver class, 60 in the Deluxe class, and 50 in the Business class. Round Tree has 100 Type I rooms and 120 Type II rooms. a. Formulate and solve a linear program to determine how many reservations to accept in each rental class and how the reservations should be allocated to room types. b. For the solution in part (a), how many reservations can be accommodated in each rental class? Is the demand for any rental class not satisfied? c. With a little work, an unused office area could be converted to a rental room. If the conversion cost is the same for both types of rooms, would you recommend converting the office to a Type I or a Type II room? Why? d. Could the linear programming model be modified to plan for the allocation of rental demand for the next night? What information would be needed and how would the model change?Industrial Designs has been awarded a contract to design a label for a new wine produced by Lake View Winery. The company estimates that 150 hours will be required to complete the project. The firm’s three graphic designers available for assignment to this project are Lisa, a senior designer and team leader; David, a senior designer; and Sarah, a junior designer. Because Lisa has worked on several projects for Lake View Winery, management specified that Lisa must be assigned at least 40% of the total number of hours assigned to the two senior designers. To provide label designing experience for Sarah, the junior designer must be assigned at least 15% of the total project time. However, the number of hours assigned to Sarah must not exceed 25% of the total number of hours assigned to the two senior designers. Due to other project commitments, Lisa has a maximum of 50 hours available to work on this project. Hourly wage rates are $30 for Lisa, $25 for David, and $18 for Sarah.
Formulate a linear program that can be used to determine the number of hours each graphic designer should be assigned to the project to minimize total cost.
How many hours should be assigned to each graphic designer? What is the total cost?
Suppose Lisa could be assigned more than 50 hours. What effect would this have on the optimal solution? Explain.
If Sarah were not required to work a minimum number of hours on this project, would the optimal solution change? Explain.
Vollmer Manufacturing makes three components for sale to refrigeration companies. The components are processed on two machines: a shaper and a grinder. The times (in minutes) required on each machine are as follows: The shaper is available for 120 hours, and the grinder for 110 hours. No more than 200 units of component 3 can be sold, but up to 1,000 units of each of the other components can be sold. In fact, the company already has orders for 600 units of component 1 that must be satisfied. The profit contributions for components 1, 2, and 3 are 8, 6, and 9, respectively. a. Formulate and solve for the recommended production quantities. b. What are the objective coefficient ranges for the three components? Interpret these ranges for company management. c. What are the right-hand-side ranges? Interpret these ranges for company management. d. If more time could be made available on the grinder, how much would it be worth? e. If more units of component 3 can be sold by reducing the sales price by 4, should the company reduce the price?Photon Technologies, Inc., a manufacturer of batteries for mobile phones, signed a contract with a large electronics manufacturer to produce three models of lithium-ion battery packs for a new line of phones. The contract calls for the following:
Photon Technologies can manufacture the battery packs at manufacturing plants located in the Philippines and Mexico. The unit cost of the battery packs differs at the two plants because of differences in production equipment and wage rates. The unit costs for each battery pack at each manufacturing plant are as follows:
The PT-100 and PT-200 battery packs are produced using similar production equipment available at both plants. However, each plant has a limited capacity for the total number of PT-100 and PT-200 battery packs produced. The combined PT-100 and PT-200 production capacities are 175,000 units at the Philippines plant and 160,000 units at the Mexico plant. The PT-300 production capacities are 75,000 units at the Philippines plant and 100,000 units at the Mexico plant. The cost of shipping from the Philippines plant is $0.18 per unit, and the cost of shipping from the Mexico plant is $0.10 per unit.
Develop a linear program that Photon Technologies can use to determine how many units of each battery pack to produce at each plant to minimize the total production and shipping cost associated with the new contract.
Solve the linear program developed in part (a), to determine the optimal production plan.
Use sensitivity analysis to determine how much the production and/or shipping cost per unit would have to change to produce additional units of the PT-100 in the Philippines plant.
Use sensitivity analysis to determine how much the production and/or shipping cost per unit would have to change to produce additional units of the PT-200 in the Mexico plant.
The Westchester Chamber of Commerce periodically sponsors public service seminars and programs. Currently, promotional plans are under way for this year’s program. Advertising alternatives include television, radio, and online. Audience estimates, costs, and maximum media usage limitations are as shown:
To ensure a balanced use of advertising media, radio advertisements must not exceed 50% of the total number of advertisements authorized. In addition, television should account for at least 10% of the total number of advertisements authorized.
If the promotional budget is limited to $18,200, how many commercial messages should be run on each medium to maximize total audience contact? What is the allocation of the budget among the three media, and what is the total audience reached?
By how much would audience contact increase if an extra $100 were allocated to the promotional budget?
The management of Hartman Company is trying to determine the amount of each of two products to produce over the coming planning period. The following information concerns labor availability, labor utilization, and product profitability: a. Develop a linear programming model of the Hartman Company problem. Solve the model to determine the optimal production quantities of products 1 and 2. b. In computing the profit contribution per unit, management does not deduct labor costs because they are considered fixed for the upcoming planning period. However, suppose that overtime can be scheduled in some of the departments. Which departments would you recommend scheduling for overtime? How much would you be willing to pay per hour of overtime in each department? c. Suppose that 10, 6, and 8 hours of overtime may be scheduled in departments A, B, and C, respectively. The cost per hour of overtime is 18 in department A, 22.50 in department B, and 12 in department C. Formulate a linear programming model that can be used to determine the optimal production quantities if overtime is made available. What are the optimal production quantities, and what is the revised total contribution to profit? How much overtime do you recommend using in each department? What is the increase in the total contribution to profit if overtime is used?The employee credit union at State University is planning the allocation of funds for the coming year. The credit union makes four types of loans to its members. In addition, the credit union invests in risk-free securities to stabilize income. The various revenue-producing investments, together with annual rates of return, are as follows: The credit union will have 2 million available for investment during the coming year. State laws are credit union policies impose the following restrictions on the composition of the loans and investments: Risk-free securities may not exceed 30% of the total funds available for investment. Signature loans may not exceed 10% of the funds invested in all loans (automobile, furniture, other secured, and signature loans). Furniture loans plus other secured loans may not exceed the automobile loans. Other secured loans plus signature loans may not exceed the funds invested in risk-free securities. How should the 2 million be allocated to each of the loan/investment alternatives to maximize total annual return? What is the projected total annual return?The Atlantic Seafood Company (ASC) is a buyer and distributor of seafood products that are sold to restaurants and specialty seafood outlets throughout the Northeast. ASC has a frozen storage facility in New York City that serves as the primary distribution point for all products. One of the ASC products is frozen large black tiger shrimp, which are sized at 16 to 20 pieces per pound. Each Saturday, ASC can purchase more tiger shrimp or sell the tiger shrimp at the existing New York City warehouse market price. ASCs goal is to buy tiger shrimp at a low weekly price and sell it later at a higher price. ASC currently has 20,000 pounds of tiger shrimp in storage. Space is available to store a maximum of 100,000 pounds of tiger shrimp each week. In addition, ASC developed the following estimates of tiger shrimp prices for the next four weeks: ASC would like to determine the optimal buying/storing/selling strategy for the next four weeks. The cost to store a pound of shrimp for one week is 0.15, and to account for unforeseen changes in supply or demand, management also indicated that 25,000 pounds of tiger shrimp must be in storage at the end of week 4. Determine the optimal buying/storing/selling strategy for ASC. What is the projected four-week profit? (Hint: Define variables for buying, selling, and inventory held in each week. Then use a constraint to define the relationship between these: inventory from end of previous period + bought this period sold this period = inventory at end of this period. This type of constraint is referred to as an inventory balance constraint.)The Silver Star Bicycle Company will manufacture both mens and womens models for its Easy-Pedal bicycles during the next two months. Management wants to develop a production schedule indicating how many bicycles of each model should be produced in each month. Current demand forecasts call for 150 mens and 125 womens models to be shipped during the first month and 200 mens and 150 womens models to be shipped during the second month. Additional data are as follows: Last month, the company used a total of 1,000 hours of labor. The companys labor relations policy will not allow the combined total hours of labor (manufacturing plus assembly) to increase or decrease by more than 100 hours from month to month. In addition, the company charges monthly inventory at the rate of 2% of the production cost based on the inventory levels at the end of the month. The company would like to have at least 25 units of each model in inventory at the end of the two months. (Hint: Define variables for production and inventory held in each period for each product. Then use a constraint to define the relationship between these: inventory from end of previous period + produced this period demand this period = inventory at end of this period.) a. Establish a production schedule that minimizes production and inventory costs and satisfies the labor-smoothing, demand, and inventory requirements. What inventories will be maintained and what are the monthly labor requirements? b. If the company changed the constraints so that monthly labor increases and decreases could not exceed 50 hours, what would happen to the production schedule? How much will the cost increase? What would you recommend?The Clark County Sheriff’s Department schedules police officers for 8-hour shifts. The beginning times for the shifts are 8:00 a.m., noon, 4:00 p.m., 8:00 p.m., midnight, and 4:00 a.m. An officer beginning a shift at one of these times works for the next 8 hours. During normal weekday operations, the number of officers needed varies depending on the time of day. The department staffing guidelines require the following minimum number of officers on duty:
Determine the number of police officers that should be scheduled to begin the 8-hour shifts at each of the six times to minimize the total number of officers required. (Hint: Let x1 = the number of officers beginning work at 8:00 a.m., x2 = the number of officers beginning work at noon, and so on.)
Bay Oil produces two types of fuel (regular and super) by mixing three ingredients. The major distinguishing feature of the two products is the octane level required. Regular fuel must have a minimum octane level of 90, whereas super must have a level of at least 100. The cost per barrel, octane levels, and available amounts (in barrels) for the upcoming two-week period appear in the following table, along with the maximum demand for each end product and the revenue generated per barrel:
Develop and solve a linear programming model to maximize contribution to profit. What is the optimal contribution to profit?
Consider the following network representation of a transportation problem:
The supplies, demands, and transportation costs per unit are shown on the network. What is the optimal (cost minimizing) distribution plan?
Refer to the transportation problem described in Problem 16. Use the procedure described in Section 11.7 to try to find an alternative optimal solution.
16. Consider the following network representation of a transportation problem:
The supplies, demands, and transportation costs per unit are shown on the network. What is the optimal (cost minimizing) distribution plan?
Aggie Power Generation supplies electrical power to residential customers for many U.S. cities. Its main power generation plants are located in Los Angeles. Tulsa, and Seattle. The following table shows Aggie Power Generation’s major residential markets, the annual demand in each market (in Megawatts or MWs), and the cost to supply electricity to each market from each power generation plant (prices are in $/MW).
If there are no restrictions on the amount of power that can be supplied by any of the power plants, what is the optimal solution to this problem? Which cities should be supplied by which power plants? What is the total annual power distribution cost for this solution?
If at most 4,000 MWs of power can be supplied by any one of the power plants, what is the optimal solution? What is the annual increase in power distribution cost that results from adding these constraints to the original formulation?
The Calhoun Textile Mill is in the process of deciding on a production schedule. It wishes to know how to weave the various fabrics it will produce during the coming quarter. The sales department has continued orders for each of the 15 fabrics produced by Calhoun. These demands are given in the following table. Also given in this table is the variable cost for each fabric. The mill operates continuously during the quarter: 13 weeks, 7 days a week, and 24 hours a day. There are two types of looms: dobbie and regular. Dobbie looms can be used to make all fabrics and are the only looms that can weave certain fabrics, such as plaids. The rate of production for each fabric on each type of loom is also given in the table. Note that if the production rate is zero, the fabric cannot be woven on that type of loom. Also, if a fabric can be woven on each type of loom, then the production rates are equal. Calhoun has 90 regular looms and 15 dobbie looms. For this problem, assume that the time requirement to change over a loom from one fabric to another is negligible. Management would like to know how to allocate the looms to the fabrics and which fabrics to buy on the market so as to minimize the cost of meeting demand.Refer to the Calhoun Textile Mill production problem described in Problem 19. Use the procedure described in Section 11.7 to try to find an alternative optimal solution. If you are successful, discuss the differences in the solution you found versus that found in Problem 19.
19. The Calhoun Textile Mill is in the process of deciding on a production schedule. It wishes to know how to weave the various fabrics it will produce during the coming quarter. The sales department has confirmed orders for each of the 15 fabrics produced by Calhoun. These demands are given in the following table. Also given in this table is the variable cost for each fabric. The mill operates continuously during the quarter: 13 weeks, 7 days a week, and 24 hours a day.
There are two types of looms: dobbie and regular. Dobbie looms can be used to make all fabrics and are the only looms that can weave certain fabrics, such as plaids. The rate of production for each fabric on each type of loom is also given in the table. Note that if the production rate is zero, the fabric cannot be woven on that type of loom. Also, if a fabric can be woven on each type of loom, then the production rates are equal. Calhoun has 90 regular looms and 15 dobbie looms. For this problem, assume that the time requirement to change over a loom from one fabric to another is negligible.
Management would like to know how to allocate the looms to the fabrics and which fabrics to buy on the market so as to minimize the cost of meeting demand.
STAR Co. provides paper to smaller companies with volumes that are not large enough to warrant dealing directly with the paper mill. STAR receives 100-feet-wide paper rolls from the mill and cuts the rolls into smaller rolls of widths 12, 15, and 30 feet. The demands for these widths vary from week to week. The following cutting patterns have been established: Trim loss is the leftover paper from a pattern (e.g., for pattern 4, 2(12) + 1(15) + 2(30) = 99 feet used results in 100 99 = 1 foot of trim loss). Demands this week are 5,670 12-foot rolls, 1,680 15-foot rolls, and 3,350 30-foot rolls. Develop an all-integer model that will determine how many 100-foot rolls to cut into each of the live patterns in order to meet demand and minimize trim loss (leftover paper from a pattern).The following questions refer to a capital budgeting problem with six projects represented by binary variables x1, x2, x3, x4, x5, and x6. a. Write a constraint modeling a situation in which two of the projects 1, 3, 5, and 6 must be undertaken. b. Write a constraint modeling a situation in which, if project 3 or 5 is undertaken, they must both be undertaken. c. Write a constraint modeling a situation in which project 1 or 4 must be undertaken, but not both. d. Write constraints modeling a situation in which project 4 cannot be undertaken unless projects 1 and 3 also are undertaken. e. Revise the requirement in part (d) to accommodate the case in which, when projects 1 and 3 are undertaken, project 4 also must be undertaken.Spencer Enterprises is attempting to choose among a series of new investment alternatives. The potential investment alternatives, the net present value of the future stream of returns, the capital requirements, and the available capital funds over the next three years are summarized as follows:
Develop and solve an integer programming model for maximizing the net present value.
Assume that only one of the warehouse expansion projects can be implemented. Modify your model from part (a).
Suppose that if test marketing of the new product is carried out, the advertising campaign also must be conducted. Modify your formulation from part (b) to reflect this new situation.
Hawkins Manufacturing Company produces connecting rods for 4- and 6-cylindcr automobile engines using the same production line. The cost required to set up the production line to produce the 4-cylinder connecting rods is 2,000, and the cost required to set up the production line for the 6-cylinder connecting rods is 3,500. Manufacturing costs are 15 for each 4-cylinder connecting rod and 18 for each 6-cylinder connecting rod. There is no production on weekends, so on Friday the line is disassembled and cleaned. On Monday, the line must be set up to run whichever product will be produced that week. Once the line has been set up, the weekly production capacities are 6,000 6-cylinder connecting rods and 8,000 4-cylinder connecting rods. Let x4 = the number of 4-cylinder connecting rods produced next week x6 = the number of 6-cylinder connecting rods produced next week s4 = 1 if the production line is set up to produce the 4-cylinder connecting rods; 0 if otherwise s6 = 1 if the production line is set up to produce the 6-cylinder connecting rods; 0 if otherwise a. Using the decision variables x4 and s4, write a constraint that sets next weeks maximum production of the 4-cylinder connecting rods to either 0 or 8,000 units. b. Using the decision variables x6 and s6, write a constraint that sets next weeks maximum production of the 6-cylinder connecting rods to either 0 or 6,000 units. c. Write a constraint that requires that production be set up for exactly one of the two rods. d. Write the cost function to be minimized.Grave City is considering the relocation of several police substations to obtain better enforcement in high-crime areas. The locations under consideration together with the areas that can be covered from these locations are given in the following table: a. Formulate an integer programming model that could be used to find the minimum number of locations necessary to provide coverage to all areas. b. Solve the problem in part (a).Hart Manufacturing makes three products. Each product requires manufacturing operations in three departments: A, B, and C. The labor-hour requirements, by department, are as follows: During the next production period the labor-hours available are 450 in department A, 350 in department B, and 50 in department C. The profit contributions per unit are 25 for product 1, 28 for product 2, and 30 for product 3. a. Formulate a linear programming model for maximizing total profit contribution. b. Solve the linear program formulated in part (a). How much of each product should be produced, and what is the projected total profit contribution? c. After evaluating the solution obtained in part (b), one of the production supervisors noted that production setup costs had not been taken into account. She noted that setup costs are 400 for product 1, 550 for product 2, and 600 for product 3. If the solution developed in part (b) is to be used, what is the total profit contribution after taking into account the setup costs? d. Management realized that the optimal product mix, taking setup costs into account, might be different from the one recommended in part (b). Formulate a mixed-integer linear program that takes setup costs provided in part (c) into account. Management also stated that we should not consider making more than 175 units of product 1, 150 units of product 2, or 140 units of product 3. e. Solve the mixed-integer linear program formulated in part (d). How much of each product should be produced and what is the projected total profit contribution? Compare this profit contribution to that obtained in part (c).Galaxy Cloud Services operates several data centers across the United States containing servers that store and process the data on the Internet. Suppose that Galaxy Cloud Services currently has five outdated data centers: one each in Michigan, Ohio, and California and two in New York. Management is considering increasing the capacity of these data centers to keep up with increasing demand. Each data center contains servers that are dedicated to Secure data and to Super Secure data. The cost to update each data center and the resulting increase in server capacity for each type of server are as follows:
The projected needs are for a total increase in capacity of 90 Secure servers and 90 Super Secure servers. Management wants to determine which data centers to update to meet projected needs and, at the same time, minimize the total cost of the added capacity.
Formulate a binary integer programming model that could be used to determine the optimal solution to the capacity increase question facing management.
Solve the model formulated in part (a) to provide a recommendation for management.
East Coast Trucking provides service from Boston to Miami using regional offices located in Boston, New York, Philadelphia, Baltimore, Washington, Richmond, Raleigh, Florence, Savannah, Jacksonville, and Tampa. The number of miles between the regional offices is provided in the following table: The companys expansion plans involve constructing service facilities in some of the cities where regional offices are located. Each regional office must be within 400 miles of a service facility. For instance, if a service facility is constructed in Richmond, it can provide service to regional offices located in New York, Philadelphia, Baltimore, Washington, Richmond, Raleigh, and Florence. Management would like to determine the minimum number of service facilities needed and where they should be located. a. Formulate an integer linear program that can be used to determine the minimum number of service facilities needed and their locations. b. Solve the integer linear program formulated in part (a). How many service facilities are required, and where should they be located? c. Suppose that each service facility can provide service only to regional offices within 300 miles. Re-solve the integer linear program with the 300-mile requirement. How many service facilities are required and where should they be located?GreenLawns provides a lawn fertilizing and weed control service. The company is adding a special aeration treatment as a low-cost extra service option that it hopes will help attract new customers. Management is planning to promote this new service in two media: radio and direct-mail advertising. A media budget of 3,000 is available for this promotional campaign. Based on past experience in promoting its other services, GreenLawns has obtained the following estimate of the relationship between sales and the amount spent on promotion in these two media: S=2R210M28RM+18R+34M, where S = total sales in thousands of dollars R = thousands of dollars spent on radio advertising M = thousands of dollars spent on direct-mail advertising GreenLawns would like to develop a promotional strategy that will lead to maximum sales subject to the restriction provided by the media budget. a. What is the value of sales if 2,000 is spent on radio advertising and 1,000 is spent on direct-mail advertising? b. Formulate an optimization problem that can be solved to maximize sales subject to the media budget of spending no more than 3,000 on total advertising. c. Determine the optimal amount to spend on radio and direct-mail advertising. How much in sales will be generated?The Cobb-Douglas production function is a classic model from economics used to model output as a function of capital and labor. It has the form: f(L,C)=c0Lc1Cc2, where c0, c1, and c2 are constants. The variable L represents the units of input of labor, and the variable C represents the units of input of capital. a. In this example, assume c0 = 5, c1 = 0.25, and c2 = 0.75. Assume each unit of labor costs 25 and each unit of capital costs 75. With 75,000 available in the budget, develop an optimization model to determine how the budgeted amount should be allocated between capital and labor in order to maximize output. b. Find the optimal solution to the model you formulated in part (a). (Hint: When using Excel Solver, use the Multistart option with bounds 0 L 3,000 and 0 C 1,000.)Let S represent the amount of steel produced (in tons). Steel production is related to the amount of labor used (L) and the amount of capital used (C) by the following function: S=20L0.30C0.70. In this formula L represents the units of labor input and C the units of capital input. Each unit of labor costs 50, and each unit of capital costs 100. a. Formulate an optimization problem that will determine how much labor and capital are needed to produce 50,000 tons of steel at minimum cost. b. Solve the optimization problem you formulated in part (a). (Hint: When using Excel Solver, start with an initial L 0 and C 0.)The profit function for two products is: Profit3x12+42x13x22+48x2+700, where x1 represents units of production of product 1, and x2 represents units of production of product 2. Producing one unit of product 1 requires 4 labor-hours, and producing one unit of product 2 requires 6 labor-hours. Currently, 24 labor-hours are available. The cost of labor-hours is already factored into the profit function, but it is possible to schedule overtime at a premium of 5 per hour. a. Formulate an optimization problem that can be used to find the optimal production quantity of products 1 and 2 and the optimal number of overtime hours to schedule. b. Solve the optimization model you formulated in part (a). How much should be produced and how many overtime hours should be scheduled?Jims Camera shop sells two high-end cameras, the Sky Eagle and Horizon. The demand for these two cameras are as follows: Ds = demand for the Sky Eagle, Ps is the selling price of the Sky Eagle, DH is the demand for the Horizon, and PH is the selling price of the Horizon. DS=2220.60Ps+0.35PHDH270+0.10Ps0.64PH The store wishes to determine the selling price that maximizes revenue for these two products. Develop the revenue function for these two models, and find the prices that maximize revenue.Heller Manufacturing has two production facilities that manufacture baseball gloves. Production costs at the two facilities differ because of varying labor rates, local property taxes, type of equipment, capacity, and so on. The Dayton plant has weekly costs that can be expressed as a function of the number of gloves produced: TCD(X)=X2X+5, where X is the weekly production volume in thousands of units, and TCD(X) is the cost in thousands of dollars. The Hamilton plants weekly production costs are given by: TCH(Y)=Y2+2Y+3, where Y is the weekly production volume in thousands of units, and TCH(Y) is the cost in thousands of dollars. Heller Manufacturing would like to produce 8,000 gloves per week at the lowest possible cost. a. Formulate a mathematical model that can be used to determine the optimal number of gloves to produce each week at each facility. b. Solve the optimization model to determine the optimal number of gloves to produce at each facility.Andalus Furniture Company has two manufacturing plants, one at Aynor and another at Spartanburg. The cost in dollars of producing a kitchen chair at each of the two plants is given here. The cost of producing Q1 chairs at Aynor is: 75Q1+5Q12+100 and the cost of producing Q2 kitchen chairs at Spartanburg is: 25Q2+2.5Q22+150. Andalus needs to manufacture a total of 40 kitchen chairs to meet an order just received. How many chairs should be made at Aynor, and how many should be made at Spartanburg in order to minimize total production cost?Phillips Inc. produces two distinct products, A and B. The products do not compete with each other in the marketplace; that is, neither cost, price, nor demand for one product will impact the demand for the other. Phillips’ analysts have collected data on the effects of advertising on profits. These data suggest that, although higher advertising correlates with higher profits, the marginal increase in profits diminishes at higher advertising levels, particularly for product B. Analysts have estimated the following functions:
where XA and XB are the advertising amount allocated to products A and B, respectively, in thousands of dollars, profit is in millions of dollars, and LN is the natural logarithm function. The advertising budget is $500,000, and management has dictated that at least $50,000 must be allocated to each of the two products.
(Hint: To compute a natural logarithm for the value X in Excel, use the formula = LN(X). For Solver to find an answer, you also need to start with decision variable values greater than 0 in this problem.)
Build an optimization model that will prescribe how Phillips should allocate its marketing budget to maximize profit.
Solve the model you constructed in part (a) using Excel Solver.
1PSouthland Corporation’s decision to produce a new line of recreational products resulted in the need to construct either a small plant or a large plant. The best selection of plant size depends on how the marketplace reacts to the new product line. To conduct an analysis, marketing management has decided to view the possible long-run demand as low, medium, or high. The following payoff table shows the projected profit in millions of dollars:
What is the decision to be made, and what is the chance event for Southland’s problem?
Construct a decision tree.
Recommend a decision based on the use of the optimistic, conservative, and minimax regret approaches.
Amy Lloyd is interested in leasing a new Honda and has contacted three automobile dealers for pricing information. Each dealer offered Amy a closed-end 36-month lease with no down payment due at the time of signing. Each lease includes a monthly charge and a mileage allowance. Additional miles receive a surcharge on a per-mile basis. The monthly lease cost, the mileage allowance, and the cost for additional miles follow: Amy decided to choose the lease option that will minimize her total 36-month cost. The difficulty is that Amy is not sure how many miles she will drive over the next three years. For purposes of this decision, she believes it is reasonable to assume that she will drive 12,000 miles per year, 15,000 miles per year, or 18,000 miles per year. With this assumption Amy estimated her total costs for the three lease options. For example, she figures that the Hepburn Honda lease will cost her 36(299) + 0.15(36,000 36,000) = 10,764 if she drives 12,000 miles per year, 36(299) + 0.15(45,000 36,000) = 12,114 if she drives 15,000 miles per year, or 36(299) + 0.15(54.000 36,000) = 13,464 if she drives 18,000 miles per year. a. What is the decision, and what is the chance event? b. Construct a payoff table for Amys problem. c. If Amy has no idea which of the three mileage assumptions is most appropriate, what is the recommended decision (leasing option) using the optimistic, conservative, and minimax regret approaches? d. Suppose that the probabilities that Amy drives 12,000, 15,000, and 18,000 miles per year are 0.5, 0.4, and 0.1, respectively. What option should Amy choose using the expected value approach? e. Develop a risk profile for the decision selected in part (d). What is the most likely cost, and what is its probability? f. Suppose that, after further consideration, Amy concludes that the probabilities that she will drive 12,000, 15,000, and 18,000 miles per year are 0.3, 0.4, and 0.3, respectively. What decision should Amy make using the expected value approach?Investment advisors estimated the stock market returns for four market segments: computers, financial, manufacturing, and pharmaceuticals. Annual return projections vary depending on whether the general economic conditions are improving, stable, or declining. The anticipated annual return percentages for each market segment under each economic condition are as follows:
Assume that an individual investor wants to select one market segment for a new investment. A forecast shows improving to declining economic conditions with the following probabilities: improving (0.2), stable (0.5), and declining (0.3). What is the preferred market segment for the investor, and what is the expected return percentage?
At a later date, a revised forecast shows a potential for an improvement in economic conditions. New probabilities are as follows: improving (0.4), stable (0.4), and declining (0.2). What is the preferred market segment for the investor based on these new probabilities? What is the expected return percentage?
Hudson Corporation is considering three options for managing its data warehouse: continuing with its own staff, hiring an outside vendor to do the managing, or using a combination of its own staff and an outside vendor. The cost of the operation depends on future demand. The annual cost of each option (in thousands of dollars) depends on demand as follows:
If the demand probabilities are 0.2, 0.5, and 0.3, which decision alternative will minimize the expected cost of the data warehouse? What is the expected annual cost associated with that recommendation?
Construct a risk profile for the optimal decision in part (a). What is the probability of the cost exceeding $700,000?
6PMyrtle Air Express decided to offer direct service from Cleveland to Myrtle Beach. Management must decide between a full-price service using the companys new fleet of jet aircraft and a discount service using smaller-capacity commuter planes. It is clear that the best choice depends on the market reaction to the service Myrtle Air offers. Management developed estimates of the contribution to profit for each type of service based on two possible levels of demand for service to Myrtle Beach: strong and weak. The following table shows the estimated quarterly profits (in thousands of dollars): a. What is the decision to be made, what is the chance event, and what is the consequence for this problem? How many decision alternatives are there? How many outcomes are there for the chance event? b. If nothing is known about the probabilities of the chance outcomes, what is the recommended decision using the optimistic, conservative, and minimax regret approaches? c. Suppose that management of Myrtle Air Express believes that the probability of strong demand is 0.7 and the probability of weak demand is 0.3. Use the expected value approach to determine an optimal decision. d. Suppose that the probability of strong demand is 0.8 and the probability of weak demand is 0.2. What is the optimal decision using the expected value approach? e. Use sensitivity analysis to determine the range of demand probabilities for which each of the decision alternatives has the largest expected value.Video Tech is considering marketing one of two new video games for the coming holiday season: Battle Pacific or Space Pirates. Battle Pacific is a unique game and appears to have no competition. Estimated profits (in thousands of dollars) under high, medium, and low demand are as follows:
Video Tech is optimistic about its Space Pirates game. However, the concern is that profitability will be affected by a competitor’s introduction of a video game viewed as similar to Space Pirates. Estimated profits (in thousands of dollars) with and without competition are as follows:
Develop a decision tree for the Video Tech problem.
For planning purposes, Video Tech believes there is a 0.6 probability that its competitor will produce a new game similar to Space Pirates. Given this probability of competition, the director of planning recommends marketing the Battle Pacific video game. Using expected value, what is your recommended decision?
Show a risk profile for your recommended decision.
Use sensitivity analysis to determine what the probability of competition for Space Pirates would have to be for you to change your recommended decision alternative.
Seneca Hill Winery recently purchased land for the purpose of establishing a new vineyard. Management is considering two varieties of white grapes for the new vineyard: Chardonnay and Riesling. The Chardonnay grapes would be used to produce a dry Chardonnay wine, and the Riesling grapes would be used to produce a semidry Riesling wine. It takes approximately four years from the time of planting before new grapes can be harvested. This length of time creates a great deal of uncertainty concerning future demand and makes the decision about the type of grapes to plant difficult. Three possibilities are being considered: Chardonnay grapes only; Riesling grapes only; and both Chardonnay and Riesling grapes. Seneca management decided that for planning purposes it would be adequate to consider only two demand possibilities for each type of wine: strong or weak. With two possibilities for each type of wine, it was necessary to assess four probabilities. With the help of some forecasts in industry publications, management made the following probability assessments: Revenue projections show an annual contribution to profit of 20,000 if Seneca Hill plants only Chardonnay grapes and demand is weak for Chardonnay wine, and 70,000 if Seneca plants only Chardonnay grapes and demand is strong for Chardonnay wine. If Seneca plants only Riesling grapes, the annual profit projection is 25,000 if demand is weak for Riesling grapes and 45,000 if demand is strong for Riesling grapes. If Seneca plants both types of grapes, the annual profit projections are shown in the following table: a. What is the decision to be made, what is the chance event, and what is the consequence? Identify the alternatives for the decisions and the possible outcomes for the chance events. b. Develop a decision tree. c. Use the expected value approach to recommend which alternative Seneca Hill Winery should follow in order to maximize expected annual profit. d. Suppose management is concerned about the probability assessments when demand for Chardonnay wine is strong. Some believe it is likely for Riesling demand to also be strong in this case. Suppose that the probability of strong demand for Chardonnay and weak demand for Riesling is 0.05 and that the probability of strong demand for Chardonnay and strong demand for Riesling is 0.40. How does this change the recommended decision? Assume that the probabilities when Chardonnay demand is weak are still 0.05 and 0.50. e. Other members of the management team expect the Chardonnay market to become saturated at some point in the future, causing a fall in prices. Suppose that the annual profit projections fall to 50,000 when demand for Chardonnay is strong and only Chardonnay grapes are planted. Using the original probability assessments, determine how this change would affect the optimal decision.Hemmingway, Inc. is considering a $5 million research and development (R&D) project. Profit projections appear promising, but Hemmingway’s president is concerned because the probability that the R&D project will be successful is only 0.50. Furthermore, the president knows that even if the project is successful, it will require that the company build a new production facility at a cost of $20 million in order to manufacture the product. If the facility is built, uncertainty remains about the demand and thus uncertainty about the profit that will be realized. Another option is that if the R&D project is successful, the company could sell the rights to the product for an estimated $25 million. Under this option, the company would not build the $20 million production facility.
The decision tree follows. The profit projection for each outcome is shown at the end of the branches. For example, the revenue projection for the high demand outcome is $59 million. However, the cost of the R&D project ($5 million) and the cost of the production facility ($20 million) show the profit of this outcome to be $59 – $5 – $20 = $34 million. Branch probabilities are also shown for the chance events.
Analyze the decision tree to determine whether the company should undertake the R&D project. If it does, and if the R&D project is successful, what should the company do? What is the expected value of your strategy?
What must the selling price be for the company to consider selling the rights to the product?
Develop a risk profile for the optimal strategy.
The following profit payoff table was presented in Problem 1:
The probabilities for the states of nature are P(s1) = 0.65, P(s2) = 0.15, and P(s3) = 0.20.
What is the optimal decision strategy if perfect information were available?
What is the expected value for the decision strategy developed in part (a)?
Using the expected value approach, what is the recommended decision without perfect information? What is its expected value?
What is the expected value of perfect information?
Suppose that you are given a decision situation with three possible states of nature: s1, s2, and s3. The prior probabilities are P(s1) = 0.2, P(s2) = 0.5, and P(s3) = 0.3. With sample information I, P(I|s1) = 0.1, P(I|s2) = 0.05, and P(I|s3) = 0.2. Compute the revised (or posterior) probabilities: P(s1|I), P(s2|I), and P(s3|I).
A firm has three investment alternatives. Payoffs are in thousands of dollars. a. Using the expected value approach, which decision is preferred? b. For the lottery having a payoff of 100,000 with probability p and 0 with probability (1 p), two decision makers expressed the following indifference probabilities. Find the most preferred decision for each decision maker using the expected utility approach. c. Why dont decision makers A and B select the same decision alternative?Alexander Industries is considering purchasing an insurance policy for its new office building in St. Louis, Missouri. The policy has an annual cost of $10,000. If Alexander Industries doesn’t purchase the insurance and minor fire damage occurs, a cost of $100,000 is anticipated; the cost if major or total destruction occurs is $200,000. The costs, including the state-of-nature probabilities, are as follows:
Using the expected value approach, what decision do you recommend?
What lottery would you use to assess utilities? (Note: Because the data are costs, the best payoff is $0.)
Assume that you found the following indifference probabilities for the lottery defined in part (b). What decision would you recommend?
Do you favor using expected value or expected utility for this decision problem? Why?
In a certain state lottery, a lottery ticket costs 2. In terms of the decision to purchase or not to purchase a lottery ticket, suppose that the following payoff table applies: a. A realistic estimate of the chances of winning is 1 in 250,000. Use the expected value approach to recommend a decision. b. If a particular decision maker assigns an indifference probability of 0.000001 to the 0 payoff, would this individual purchase a lottery ticket? Use expected utility to justify your answer.Three decision makers have assessed utilities for the following decision problem (payoff in dollars):
The indifference probabilities are as follows:
Plot the utility function for money for each decision maker.
Classify each decision maker as a risk avoider, a risk taker, or risk-neutral.
For the payoff of 20, what is the premium that the risk avoider will pay to avoid risk? What is the premium that the risk taker will pay to have the opportunity of the high payoff?
In Problem 22, if P(s1) = 0.25, P(s2) = 0.50, and P(s3) = 0.25, find a recommended decision for each of the three decision makers. (Note: For the same decision problem, different utilities can lead to different decisions.) 22. Three decision makers have assessed utilities for the following decision problem (payoff in dollars): The indifference probabilities are as follows: a. Plot the utility function for money for each decision maker. b. Classify each decision maker as a risk avoider, a risk taker, or risk-neutral. c. For the payoff of 20, what is the premium that the risk avoider will pay to avoid risk? What is the premium that the risk taker will pay to have the opportunity of the high payoff?Translate the following monetary payoffs into utilities for a decision maker whose utility function is described by an exponential function with R = 250: –$200, –$100, $0, $100, $200, $300, $400, $500.
Consider a decision maker who is comfortable with an investment decision that has a 50% chance of earning $25,000 and a 50% chance of losing $12,500, but not with any larger investments that have the same relative payoffs.
Write the equation for the exponential function that approximates this decision maker’s utility function.
Plot the exponential utility function for this decision maker for x values between –20,000 and 35,000. Is this decision maker risk-seeking, risk-neutral, or risk-averse?
Suppose the decision maker decides that she would actually be willing to make an investment that has a 50% chance of earning $30,000 and a 50% chance of losing $15,000. Plot the exponential function that approximates this utility function and compare it to the utility function from part (b). Is the decision maker becoming more risk-seeking or more risk-averse?
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