2.2 51 Summarizing Data for a Quantitative Variable Exercises Methods 11. Consider the following data. WEB 14 19 24 19 16 20 24 20 file Frequency a. b. SELF test 21 22 24 18 17 23 26 22 23 25 25 19 18 16 15 24 21 16 19 21 23 20 22 22 16 16 16 12 25 19 24 20 Develop a frequency distribution using classes of 12-14, 15-17, 18-20, 21-23, and 24-26. Develop a relative frequency distribution and a percent frequency distribution using the classes in part (a). 12. Consider the following frequency distribution. Class Frequency 10-19 20-29 30-39 40-49 50-59 10 14 17 7 2 Construct a cumulative frequency distribution and a cumulative relative frequency distribution. 13. Construct a histogram for the data in exercise 12. 14. Consider the following data. 8.9 6.8 10.2 9.5 a. b. c. SELF test 11.5 11.5 7.8 11.2 10.0 14.9 12.2 7.5 13.5 10.0 14.1 6.0 10.0 15.8 12.2 11.5 Construct a dot plot. Construct a frequency distribution. Construct a percent frequency distribution. 15. Construct a stem-and-leaf display for the following data. 11.3 9.3 9.6 8.1 10.4 7.7 7.5 7.5 8.3 8.4 10.5 6.3 10.0 8.8 16. Construct a stem-and-leaf display for the following data. Use a leaf unit of 10. 1161 1221 1206 1378 1478 1623 1300 1426 1604 1557 1725 1730 1361 1706 1422 1689 Applications SELF test 17. A doctor's office staff studied the waiting times for patients who arrive at the office with a request for emergency service. The following data with waiting times in minutes were collected over a one-month period. 2 5 10 12 4 4 5 17 11 8 9 8 Use classes of 0-4, 5-9, and so on in the following: a. Show the frequency distribution. b. Show the relative frequency distribution. 12 21 6 8 7 13 18 3 52 Chapter 2 c. d. e. Descriptive Statistics: Tabular and Graphical Displays Show the cumulative frequency distribution. Show the cumulative relative frequency distribution. What proportion of patients needing emergency service wait 9 minutes or less? 18. A shortage of candidates has required school districts to pay higher salaries and offer extras to attract and retain school district superintendents. The following data show the annual base salary ($1000s) for superintendents in 20 districts in the greater Rochester, New York, area (The Rochester Democrat and Chronicle, February 10, 2008). 187 175 165 162 172 184 172 208 172 175 174 202 215 182 170 185 197 164 156 183 Use classes of 150-159, 160-169, and so on in the following. a. Show the frequency distribution. b. Show the percent frequency distribution. c. Show the cumulative percent frequency distribution. d. Develop a histogram for the annual base salary. e. Do the data appear to be skewed? Explain. f. What percentage of the superintendents make more than $200,000? 19. The Dow Jones Industrial Average (DJIA) underwent one of its infrequent reshufflings of companies when General Motors and Citigroup were replaced by Cisco Systems and Travelers (The Wall Street Journal, June 8, 2009). At the time, the prices per share for the 30 companies in the DJIA were as follows: Company WEB file DJIAPrices 3M Alcoa American Express AT&T Bank of America Boeing Caterpillar Chevron Cisco Systems Coca-Cola DuPont ExxonMobil General Electric Hewlett-Packard Home Depot a. b. c. d. $/Share 61 11 25 24 12 52 38 69 20 49 27 72 14 37 24 Company IBM Intel J.P. Morgan Chase Johnson & Johnson Kraft Foods McDonald's Merck Microsoft Pfizer Procter & Gamble Travelers United Technologies Verizon Wal-Mart Stores Walt Disney $/Share 107 16 35 56 27 59 26 22 14 53 43 56 29 51 25 What is the highest price per share? What is the lowest price per share? Using a class width of 10, develop a frequency distribution for the data. Prepare a histogram. Interpret the histogram, including a discussion of the general shape of the histogram, the midprice range, and the most frequent price range. Use the The Wall Street Journal or another newspaper to find the current price per share for these companies. Prepare a histogram of the data and discuss any changes since June 2009. What company has had the largest increase in the price per share? What company has had the largest decrease in the price per share? 20. The London School of Economics and the Harvard Business School conducted a study of how chief executive officers (CEOs) spend their day. The study found that CEOs spend on average about 18 hours per week in meetings, not including conference calls, business 2.2 53 Summarizing Data for a Quantitative Variable meals, and public events (The Wall Street Journal, February 14, 2012). Shown below is the time spent per week in meetings (hours) for a sample of 25 CEOs. 14 19 23 16 19 file WEB CEOTime a. b. c. 15 20 21 15 22 18 13 15 18 23 23 15 20 18 21 15 23 21 19 12 What is the least amount of time spent per week on meetings? The highest? Use a class width of two hours to prepare a frequency distribution and a percent frequency distribution for the data. Prepare a histogram and comment on the shape of the distribution. 21. Fortune provides a list of America's largest corporations based on annual revenue. Shown below are the 50 largest corporations, with annual revenue expressed in billions of dollars (CNN Money website, January 15, 2010). Corporation WEB file LargeCorp Amerisource Bergen Archer Daniels Midland AT&T Bank of America Berkshire Hathaway Boeing Cardinal Health Caterpillar Chevron Citigroup ConocoPhillips Costco Wholesale CVS Caremark Dell Dow Chemical ExxonMobil Ford Motors General Electric Goldman Sachs Hewlett-Packard Home Depot IBM JPMorgan Chase Johnson & Johnson Kroger Revenue 71 70 124 113 108 61 91 51 263 112 231 72 87 61 58 443 146 149 54 118 71 104 101 64 76 Corporation Lowe's Marathon Oil McKesson Medco Health MetLife Microsoft Morgan Stanley Pepsico Pfizer Procter & Gamble Safeway Sears Holdings State Farm Insurance Sunoco Target Time Warner United Parcel Service United Technologies UnitedHealth Group Valero Energy Verizon Walgreen Walmart WellPoint Wells Fargo Revenue 48 74 102 51 55 60 62 43 48 84 44 47 61 52 65 47 51 59 81 118 97 59 406 61 52 Summarize the data by constructing the following: a. A frequency distribution (classes 0-49, 50-99, 100-149, and so on). b. A relative frequency distribution. c. A cumulative frequency distribution. d. A cumulative relative frequency distribution. e. What do these distributions tell you about the annual revenue of the largest corporations in America? f. Show a histogram. Comment on the shape of the distribution. g. What is the largest corporation in America and what is its annual revenue? 22. Entrepreneur magazine ranks franchises using performance measures such as growth rate, number of locations, startup costs, and financial stability. The number of locations for the top 20 U.S. franchises follow (The World Almanac, 2012). 54 Chapter 2 Descriptive Statistics: Tabular and Graphical Displays No. U.S. Locations Franchise WEB file Franchise Hampton Inns ampm McDonald's 7-Eleven Inc. Supercuts Days Inn Vanguard Cleaning Systems Servpro Subway Denny's Inc. 1864 3183 32,805 37,496 2130 1877 2155 1572 34,871 1668 Franchise No. U.S. Locations Jan-Pro Franchising Intl. Inc. Hardee's Pizza Hut Inc. Kumon Math & Reading Centers Dunkin' Donuts KFC Corp. Jazzercise Inc. Anytime Fitness Matco Tools Stratus Building Solutions 12,394 1901 13,281 25,199 9947 16,224 7683 1618 1431 5018 Use classes 0-4999, 5000-9999, 10,000-14,999 and so forth to answer the following questions. a. Construct a frequency distribution and a percent frequency distribution of the number of U.S. locations for these top-ranked franchises. b. Construct a histogram of these data. c. Comment on the shape of the distribution. 23. The Nielsen Home Technology Report provided information about home technology and its usage. The following data are the hours of personal computer usage during one week for a sample of 50 persons. WEB file Computer 4.1 3.1 4.1 10.8 7.2 1.5 4.8 4.1 2.8 6.1 10.4 2.0 8.8 9.5 5.7 5.9 14.8 5.6 12.9 5.9 3.4 5.4 4.3 12.1 4.7 5.7 4.2 3.3 0.7 3.9 1.6 3.9 7.1 4.0 3.7 6.1 4.1 10.3 9.2 3.1 3.0 11.1 6.2 4.4 6.1 3.7 3.5 7.6 5.7 3.1 Summarize the data by constructing the following: a. A frequency distribution (use a class width of three hours) b. A relative frequency distribution c. A histogram d. Comment on what the data indicate about personal computer usage at home. 24. Money magazine listed top career opportunities for work that is enjoyable, pays well, and will still be around 10 years from now (Money, November 2009). Shown below are 20 top career opportunities, with the median pay and top pay for workers with two to seven years of experience in the field. Data are shown in thousands of dollars. Career WEB file Careers Account Executive Certified Public Accountant Computer Security Consultant Director of Communications Financial Analyst Finance Director Financial Research Analyst Hotel General Manager Human Resources Manager Investment Banking IT Business Analyst IT Project Manager Median Pay 81 74 100 78 80 121 66 77 72 106 83 99 Top Pay 157 138 138 135 109 214 155 146 111 221 119 140 2.3 55 Summarizing Data for Two Variables Using Tables Career Median Pay Marketing Manager Quality-Assurance Manager Sales Representative Senior Internal Auditor Software Developer Software Program Manager Systems Engineer Technical Writer Top Pay 77 80 67 76 79 110 87 67 126 122 125 106 116 152 130 100 Develop a stem-and-leaf display for both the median pay and the top pay. Comment on what you learn about the pay for these careers. 25. A psychologist developed a new test of adult intelligence. The test was administered to 20 individuals, and the following data were obtained. 114 98 99 104 131 144 124 151 117 132 102 106 106 125 127 122 119 118 115 118 Construct a stem-and-leaf display for the data. 26. The 2011 Cincinnati Flying Pig Half-Marathon (13.1 miles) had 10,897 finishers (Cincinnati Flying Pig Marathon website). The following data show the ages for a sample of 40 half-marathoners. WEB 49 44 50 46 31 27 52 72 file Marathon a. b. c. 2.3 33 46 52 24 43 44 43 26 40 57 43 30 50 35 66 59 37 55 64 37 36 31 31 21 56 32 40 43 61 43 50 47 Construct a stretched stem-and-leaf display. What age group had the largest number of runners? What age occurred most frequently? Summarizing Data for Two Variables Using Tables Thus far in this chapter, we have focused on using tabular and graphical displays to summarize the data for a single categorical or quantitative variable. Often a manager or decision maker needs to summarize the data for two variables in order to reveal the relationshipif anybetween the variables. In this section, we show how to construct a tabular summary of the data for two variables. Crosstabulation A crosstabulation is a tabular summary of data for two variables. Although both variables can be either categorical or quantitative, crosstabulations in which one variable is categorical and the other variable is quantitative are just as common. We will illustrate this latter case by considering the following application based on data from Zagat's Restaurant Review. Data showing the quality rating and the typical meal price were collected for a sample of 68 Chapter 2 Descriptive Statistics: Tabular and Graphical Displays relationship between the variables. As we move from the low price category ($10-19) to the high price category ($40-49), the length of the blue bars decreases and the length of the green bars increases. NOTES AND COMMENTS 1. A time series is a sequence of observations on a variable measured at successive points in time or over successive periods of time. A scatter diagram in which the value of time is shown on the horizontal axis and the time series values are shown on the vertical axis is referred to in time series analysis as a time series plot. We will discuss time series plots and how to analyze time series data in Chapter 17. 2. A stacked bar chart can also be used to display frequencies rather than percentage frequencies. In this case, the different color segments of each bar represent the contribution to the total for that bar, rather than the percentage contribution. Exercises Methods SELF test 36. The following 20 observations are for two quantitative variables, x and y. Observation WEB Scatter a. b. y Observation x y 1 2 3 4 5 6 7 8 9 10 file x 22 33 2 29 13 21 13 23 14 3 22 49 8 16 10 28 27 35 5 3 11 12 13 14 15 16 17 18 19 20 37 34 9 33 20 3 15 12 20 7 48 29 18 31 16 14 18 17 11 22 Develop a scatter diagram for the relationship between x and y. What is the relationship, if any, between x and y? 37. Consider the following data on two categorical variables. The first variable, x, can take on values A, B, C, or D. The second variable, y, can take on values I or II. The following table gives the frequency with which each combination occurs. y x II A B C D a. b. I 143 200 321 420 857 800 679 580 Construct a side-by-side bar chart with x on the horizontal axis. Comment on the relationship between x and y. 38. The following crosstabulation summarizes the data for two categorical variables, x and y. The variable x can take on values low, medium, or high and the variable y can take on values yes or no. 22 Chapter 1 Data and Statistics Categorical data Labels or names used to identify an attribute of each element. Categorical data use either the nominal or ordinal scale of measurement and may be nonnumeric or numeric. Quantitative data Numeric values that indicate how much or how many of something. Quantitative data are obtained using either the interval or ratio scale of measurement. Categorical variable A variable with categorical data. Quantitative variable A variable with quantitative data. Cross-sectional data Data collected at the same or approximately the same point in time. Time series data Data collected over several time periods. Descriptive statistics Tabular, graphical, and numerical summaries of data. Population The set of all elements of interest in a particular study. Sample A subset of the population. Census A survey to collect data on the entire population. Sample survey A survey to collect data on a sample. Statistical inference The process of using data obtained from a sample to make estimates or test hypotheses about the characteristics of a population. Data mining The process of using procedures from statistics and computer science to extract useful information from extremely large databases. Supplementary Exercises 1. Discuss the differences between statistics as numerical facts and statistics as a discipline or field of study. 2. The U.S. Department of Energy provides fuel economy information for a variety of motor vehicles. A sample of 10 automobiles is shown in Table 1.6 (Fuel Economy website, February 22, 2008). Data show the size of the automobile (compact, midsize, or large), the number of cylinders in the engine, the city driving miles per gallon, the highway driving miles per gallon, and the recommended fuel (diesel, premium, or regular). a. How many elements are in this data set? b. How many variables are in this data set? c. Which variables are categorical and which variables are quantitative? d. What type of measurement scale is used for each of the variables? SELF test 3. Refer to Table 1.6. a. What is the average miles per gallon for city driving? b. On average, how much higher is the miles per gallon for highway driving as compared to city driving? SELF test TABLE 1.6 FUEL ECONOMY INFORMATION FOR 10 AUTOMOBILES Car Audi A8 BMW 328Xi Cadillac CTS Chrysler 300 Ford Focus Hyundai Elantra Jeep Grand Cherokee Pontiac G6 Toyota Camry Volkswagen Jetta Size Cylinders City MPG Highway MPG Fuel Large Compact Midsize Large Compact Midsize Midsize Compact Midsize Compact 12 6 6 8 4 4 6 6 4 5 13 17 16 13 24 25 17 15 21 21 19 25 25 18 33 33 26 22 31 29 Premium Premium Regular Premium Regular Regular Diesel Regular Regular Regular 23 Supplementary Exercises TABLE 1.7 DATA FOR SEVEN COLLEGES AND UNIVERSITIES State School Amherst College Duke Harvard University Swarthmore College University of Pennsylvania Williams College Yale University c. d. Campus Setting Massachusetts North Carolina Massachusetts Pennsylvania Pennsylvania Massachusetts Connecticut Town: Fringe City: Midsize City: Midsize Suburb: Large City: Large Town: Fringe City: Midsize % Endowment Applicants NCAA ($ billions) Admitted Division 1.7 5.9 34.6 1.4 6.6 1.9 22.5 18 21 9 18 18 18 9 III I-A I-AA III I-AA III I-AA What percentage of the cars have four-cylinder engines? What percentage of the cars use regular fuel? 4. Table 1.7 shows data for seven colleges and universities. The endowment (in billions of dollars) and the percentage of applicants admitted are shown (USA Today, February 3, 2008). The state each school is located in, the campus setting, and the NCAA Division for varsity teams were obtained from the National Center of Education Statistics website, February 22, 2008. a. How many elements are in the data set? b. How many variables are in the data set? c. Which of the variables are categorical and which are quantitative? 5. Consider the data set in Table 1.7 a. Compute the average endowment for the sample. b. Compute the average percentage of applicants admitted. c. What percentage of the schools have NCAA Division III varsity teams? d. What percentage of the schools have a City: Midsize campus setting? 6. Foreign Affairs magazine conducted a survey to develop a profile of its subscribers (Foreign Affairs website, February 23, 2008). The following questions were asked. a. How many nights have you stayed in a hotel in the past 12 months? b. Where do you purchase books? Three options were listed: Bookstore, Internet, and Book Club. c. Do you own or lease a luxury vehicle? (Yes or No) d. What is your age? e. For foreign trips taken in the past three years, what was your destination? Seven international destinations were listed. Comment on whether each question provides categorical or quantitative data. 7. The Kroger Company is one of the largest grocery retailers in the United States with over 2000 grocery stores across the country. Kroger uses an online customer opinion questionnaire to obtain performance data about its products and services and learn about what motivates its customers (Kroger website, April 2012). In the survey, Kroger customers were asked if they would be willing to pay more for products that had each of the following four characteristics. The four questions were: Would you pay more for products that have a brand name? products that are environmentally friendly? products that are organic? products that have been recommended by others? For each question, the customers had the option of responding Yes if they would pay more or No if they would not pay more. 24 Chapter 1 a. b. Data and Statistics Are the data collected by Kroger in this example categorical or quantitative? What measurement scale is used? 8. The FinancialTimes/Harris Poll is a monthly online poll of adults from six countries in Europe and the United States. A January poll included 1015 adults in the United States. One of the questions asked was, \"How would you rate the Federal Bank in handling the credit problems in the financial markets?\" Possible responses were Excellent, Good, Fair, Bad, and Terrible (Harris Interactive website, January 2008). a. What was the sample size for this survey? b. Are the data categorical or quantitative? c. Would it make more sense to use averages or percentages as a summary of the data for this question? d. Of the respondents in the United States, 10% said the Federal Bank is doing a good job. How many individuals provided this response? 9. The Commerce Department reported receiving the following applications for the Malcolm Baldrige National Quality Award: 23 from large manufacturing firms, 18 from large service firms, and 30 from small businesses. a. Is type of business a categorical or quantitative variable? b. What percentage of the applications came from small businesses? 10. The Bureau of Transportation Statistics Omnibus Household Survey is conducted annually and serves as an information source for the U.S. Department of Transportation. In one part of the survey the person being interviewed was asked to respond to the following statement: \"Drivers of motor vehicles should be allowed to talk on a hand-held cell phone while driving.\" Possible responses were strongly agree, somewhat agree, somewhat disagree, and strongly disagree. Forty-four respondents said that they strongly agree with this statement, 130 said that they somewhat agree, 165 said they somewhat disagree, and 741 said they strongly disagree with this statement (Bureau of Transportation website, August 2010). a. Do the responses for this statement provide categorical or quantitative data? b. Would it make more sense to use averages or percentages as a summary of the responses for this statement? c. What percentage of respondents strongly agree with allowing drivers of motor vehicles to talk on a hand-held cell phone while driving? d. Do the results indicate general support for or against allowing drivers of motor vehicles to talk on a hand-held cell phone while driving? 11. J.D. Power and Associates conducts vehicle quality surveys to provide automobile manufacturers with consumer satisfaction information about their products (Vehicle Quality Survey, January 2010). Using a sample of vehicle owners from recent vehicle purchase records, the survey asks the owners a variety of questions about their new vehicles, such as those shown below. For each question, state whether the data collected are categorical or quantitative and indicate the measurement scale being used. a. What price did you pay for the vehicle? b. How did you pay for the vehicle? (Cash, Lease, or Finance) c. How likely would you be to recommend this vehicle to a friend? (Definitely Not, Probably Not, Probably Will, and Definitely Will) d. What is the current mileage? e. What is your overall rating of your new vehicle? A 10-point scale, ranging from 1 for unacceptable to 10 for truly exceptional, was used. 12. The Hawaii Visitors Bureau collects data on visitors to Hawaii. The following questions were among 16 asked in a questionnaire handed out to passengers during incoming airline flights. This trip to Hawaii is my: 1st, 2nd, 3rd, 4th, etc. The primary reason for this trip is: (10 categories, including vacation, convention, honeymoon) 25 Supplementary Exercises a. b. c. SELF test Where I plan to stay: (11 categories, including hotel, apartment, relatives, camping) Total days in Hawaii What is the population being studied? Is the use of a questionnaire a good way to reach the population of passengers on incoming airline flights? Comment on each of the four questions in terms of whether it will provide categorical or quantitative data. 13. Figure 1.8 provides a bar chart showing the amount of federal spending for the years 2004 to 2010 (Congressional Budget Office website, May 15, 2011). a. What is the variable of interest? b. Are the data categorical or quantitative? c. Are the data time series or cross-sectional? d. Comment on the trend in federal spending over time. 14. The following data show the number of rental cars in service for three rental car companies: Hertz, Avis, and Dollar. The data are for the years 2007-2010 and are in thousands of vehicles (Auto Rental News website, May 15, 2011). Company Hertz Dollar Avis a. b. c. 2010 290 108 270 Construct a time series graph for the years 2007 to 2010 showing the number of rental cars in service for each company. Show the time series for all three companies on the same graph. Comment on who appears to be the market share leader and how the market shares are changing over time. Construct a bar chart showing rental cars in service for 2010. Is this chart based on cross-sectional or time series data? FEDERAL SPENDING 4.0 3.5 Federal Spending ($ trillions) FIGURE 1.8 Cars in Service (1000s) 2008 2009 311 286 140 106 220 300 2007 327 167 204 3.0 2.5 2.0 1.5 1.0 0.5 0 2004 2005 2006 2007 Year 2008 2009 2010 Chapter 1 Data and Statistics NUMBER OF RECREATIONAL BOATING ACCIDENTS D ec O ct N ov p Se Ju l A ug Ju n M ay pr A ar M b Ja n 1200 1100 1000 900 800 700 600 500 400 300 200 100 0 Fe FIGURE 1.9 Number of Accidents 26 Month 15. Every year, the U.S. Coast Guard collects data and compiles statistics on reported recreational boating accidents. These statistics are derived from accident reports that are filed by the owners/operators of recreational vessels involved in accidents. In 2009, 4730 recreational boating accident reports were filed. Figure 1.9 provides a bar chart summarizing the number of accident reports that were filed each month (U.S. Coast Guard's Boating Safety Division website, August 2010). a. Are the data categorical or quantitative? b. Are the data time series or cross-sectional? c. In what month were the most accident reports filed? Approximately how many? d. There were 61 accident reports filed in January and 76 accident reports filed in December. What percentage of the total number of accident reports for the year was filed in these two months? Does this seem reasonable? e. Comment on the overall shape of the bar graph. 16. The Energy Information Administration of the U.S. Department of Energy provided time series data for the U.S. average price per gallon of conventional regular gasoline between January 2007 and March 2012 (Energy Information Administration website, April 2012). Use the Internet to obtain the average price per gallon of conventional regular gasoline since March 2012. a. Extend the graph of the time series shown in Figure 1.1. b. What interpretations can you make about the average price per gallon of conventional regular gasoline since March 2012? c. Does the time series continue to show a summer increase in the average price per gallon? Explain. 17. A manager of a large corporation recommends a $10,000 raise be given to keep a valued subordinate from moving to another company. What internal and external sources of data might be used to decide whether such a salary increase is appropriate? Supplementary Exercises 27 18. A random telephone survey of 1021 adults (aged 18 and older) was conducted by Opinion Research Corporation on behalf of CompleteTax, an online tax preparation and e-filing service. The survey results showed that 684 of those surveyed planned to file their taxes electronically (CompleteTax Tax Prep Survey 2010). a. Develop a descriptive statistic that can be used to estimate the percentage of all taxpayers who file electronically. b. The survey reported that the most frequently used method for preparing the tax return is to hire an accountant or professional tax preparer. If 60% of the people surveyed had their tax return prepared this way, how many people used an accountant or professional tax preparer? c. Other methods that the person filing the return often used include manual preparation, use of an online tax service, and use of a software tax program. Would the data for the method for preparing the tax return be considered categorical or quantitative? 19. A Bloomberg Businessweek North American subscriber study collected data from a sample of 2861 subscribers. Fifty-nine percent of the respondents indicated an annual income of $75,000 or more, and 50% reported having an American Express credit card. a. What is the population of interest in this study? b. Is annual income a categorical or quantitative variable? c. Is ownership of an American Express card a categorical or quantitative variable? d. Does this study involve cross-sectional or time series data? e. Describe any statistical inferences Bloomberg Businessweek might make on the basis of the survey. 20. A survey of 131 investment managers in Barron's Big Money poll revealed the following: 43% of managers classified themselves as bullish or very bullish on the stock market. The average expected return over the next 12 months for equities was 11.2%. 21% selected health care as the sector most likely to lead the market in the next 12 months. When asked to estimate how long it would take for technology and telecom stocks to resume sustainable growth, the managers' average response was 2.5 years. a. Cite two descriptive statistics. b. Make an inference about the population of all investment managers concerning the average return expected on equities over the next 12 months. c. Make an inference about the length of time it will take for technology and telecom stocks to resume sustainable growth. 21. A seven-year medical research study reported that women whose mothers took the drug DES during pregnancy were twice as likely to develop tissue abnormalities that might lead to cancer as were women whose mothers did not take the drug. a. This study compared two populations. What were the populations? b. Do you suppose the data were obtained in a survey or an experiment? c. For the population of women whose mothers took the drug DES during pregnancy, a sample of 3980 women showed that 63 developed tissue abnormalities that might lead to cancer. Provide a descriptive statistic that could be used to estimate the number of women out of 1000 in this population who have tissue abnormalities. d. For the population of women whose mothers did not take the drug DES during pregnancy, what is the estimate of the number of women out of 1000 who would be expected to have tissue abnormalities? e. Medical studies often use a relatively large sample (in this case, 3980). Why? 28 Chapter 1 Data and Statistics 22. The Nielsen Company surveyed consumers in 47 markets from Europe, Asia-Pacific, the Americas, and the Middle East to determine which factors are most important in determining where they buy groceries. Using a scale of 1 (low) to 5 (high), the highest rated factor was good value for money, with an average point score of 4.32. The second highest rated factor was better selection of high-quality brands and products, with an average point score of 3.78, and the lowest rated factor was uses recyclable bags and packaging, with an average point score of 2.71 (Nielsen website, February 24, 2008). Suppose that you have been hired by a grocery store chain to conduct a similar study to determine what factors customers at the chain's stores in Charlotte, North Carolina, think are most important in determining where they buy groceries. a. What is the population for the survey that you will be conducting? b. How would you collect the data for this study? 23. Pew Research Center is a nonpartisan polling organization that provides information about issues, attitudes, and trends shaping America. In a recent poll, Pew researchers found that 47% of American adult respondents reported getting at least some local news on their cell phone or tablet computer (Pew Research website, May 14, 2011). Further findings showed that 42% of respondents who own cell phones or tablet computers use those devices to check local weather reports and 37% use the devices to find local restaurants or other businesses. a. One statistic concerned using cell phones or tablet computers for local news. What population is that finding applicable to? b. Another statistic concerned using cell phones or tablet computers to check local weather reports and to find local restaurants. What population is this finding applicable to? c. Do you think the Pew researchers conducted a census or a sample survey to obtain their results? Why? d. If you were a restaurant owner, would you find these results interesting? Why? How could you take advantage of this information? 24. A sample of midterm grades for five students showed the following results: 72, 65, 82, 90, 76. Which of the following statements are correct, and which should be challenged as being too generalized? a. The average midterm grade for the sample of five students is 77. b. The average midterm grade for all students who took the exam is 77. c. An estimate of the average midterm grade for all students who took the exam is 77. d. More than half of the students who take this exam will score between 70 and 85. e. If five other students are included in the sample, their grades will be between 65 and 90. 25. Table 1.8 shows a data set containing information for 25 of the shadow stocks tracked by the American Association of Individual Investors. Shadow stocks are common stocks of smaller companies that are not closely followed by Wall Street analysts. The data set is also on the website that accompanies the text in the file named Shadow02. a. How many variables are in the data set? b. Which of the variables are categorical and which are quantitative? c. For the Exchange variable, show the frequency and the percent frequency for AMEX, NYSE, and OTC. Construct a bar graph similar to Figure 1.5 for the Exchange variable. d. Show the frequency distribution for the Gross Profit Margin using the five intervals: 0-14.9, 15-29.9, 30-44.9, 45-59.9, and 60-74.9. Construct a histogram similar to Figure 1.6. e. What is the average price/earnings ratio? 2.1 39 Summarizing Data for a Categorical Variable Exercises Methods 1. The response to a question has three alternatives: A, B, and C. A sample of 120 responses provides 60 A, 24 B, and 36 C. Show the frequency and relative frequency distributions. 2. A partial relative frequency distribution is given. Class a. b. c. d. SELF test WEB file Syndicated Relative Frequency A B C D .22 .18 .40 What is the relative frequency of class D? The total sample size is 200. What is the frequency of class D? Show the frequency distribution. Show the percent frequency distribution. 3. A questionnaire provides 58 Yes, 42 No, and 20 no-opinion answers. a. In the construction of a pie chart, how many degrees would be in the section of the pie showing the Yes answers? b. How many degrees would be in the section of the pie showing the No answers? c. Construct a pie chart. d. Construct a bar chart. Applications 4. For the 2010-2011 viewing season, the top five syndicated programs were Wheel of Fortune (WoF), Two and Half Men (THM), Jeopardy (Jep), Judge Judy (JJ), and the Oprah Winfrey Show (OWS) (Nielsen Media Research website, April 16, 2012). Data indicating the preferred shows for a sample of 50 viewers follow. WoF THM Jep WoF THM OWS JJ THM Jep WoF a. b. c. d. Jep WoF OWS THM JJ OWS WoF THM THM THM JJ OWS WoF OWS JJ JJ THM WoF WoF OWS Jep Jep WoF THM Jep JJ WoF JJ Jep OWS THM THM WoF WoF THM Jep WoF JJ Jep Jep Are these data categorical or quantitative? Provide frequency and percent frequency distributions. Construct a bar chart and a pie chart. On the basis of the sample, which television show has the largest viewing audience? Which one is second? 5. In alphabetical order, the six most common last names in the United States are Brown, Johnson, Jones, Miller, Smith, and Williams (The World Almanac, 2012). Assume that a sample of 50 individuals with one of these last names provided the following data