Essentials of Modern Business Statistics with Microsoft® Excel® – Solution Manual – Anderson, Sweeney, Williams – Complete Answers by Chapter

This document is the complete solution manual for Essentials of Modern Business Statistics with Microsoft® Excel® by David R. Anderson, Dennis J. Sweeney, and Thomas A. Williams. It provides fully worked-out answers to all exercises from each chapter, including both conceptual and computational problems. Topics covered include descriptive statistics, probability distributions, sampling, hypothesis testing, regression analysis, and Excel-based statistical techniques. The solutions are designed to support both student practice and instructor use, aligning closely with the textbook structure and learning outcomes. Full Chapters Include;....Chapter 1 Data and Statistics Learning Objectives 1. Obtain an appreciation for the breadth of statistical applications in business and economics. 2. Understand the meaning of the terms elements, variables, and observations as they are used in statistics. 3. Obtain an understanding of the difference between categorical, quantitative, crossectional and time series data. 4. Learn about the sources of data for statistical analysis both internal and external to the firm. 5. Be aware of how errors can arise in data. 6. Know the meaning of descriptive statistics and statistical inference. 7. Be able to distinguish between a population and a sample. 8. Understand the role a sample plays in making statistical inferences about the population. 9. Know the meaning of the term data mining. 10. Be aware of ethical guidelines for statistical practice. Solutions: 1. Statistics can be referred to as numerical facts. In a broader sense, statistics is the field of study dealing with the collection, analysis, presentation and interpretation of data. 2. a. The ten elements are the ten tablet computers b. 5 variables: Cost ($), Operating System, Display Size (inches), Battery Life (hours), CPU Manufacturer c. Categorical variables: Operating System and CPU Manufacturer Quantitative variables: Cost ($), Display Size (inches), and Battery Life (hours) d. Variable Measurement Scale Cost ($) Ratio Operating System Nominal Display Size (inches) Ratio Battery Life (hours) Ratio CPU Manufacturer Nominal 3. a. Average cost = 5829/10 = $582.90 b. Average cost with a Windows operating system = 3616/5 = $723.20 Average cost with an Android operating system = 1714/4 = $428.5 The average cost with a Windows operating system is much higher. c. 2 of 10 or 20% use a CPU manufactured by TI OMAP d. 4 of 10 or 40% use an Android operating system 4. a. There are eight elements in this data set; each element corresponds to one of the eight models of cordless telephones b. Categorical variables: Voice Quality and Handset on Base Quantitative variables: Price, Overall Score, and Talk Time c. Price – ratio measurement Overall Score – interval measurement Voice Quality – ordinal measurement Handset on Base – nominal measurement Talk Time – ratio measurement 5. a. Average Price = 545/8 = $68.13 b. Average Talk Time = 71/8 = 8.875 hours c. Percentage rated Excellent: 2 of 8 2/8 = .25, or 25% d. Percentage with Handset on Base: 4 of 8 4/8 = .50, or 50% 6. a. Categorical b. Quantitative c. Categorical d. Quantitative e. Quantitative 7. a. Each question has a yes or no categorical response. b. Yes and no are the labels for the customer responses. A nominal scale is being used. 8. a. 762 b. Categorical c. Percentages d. .67(762) = 510.54 510 or 511 respondents said they want the amendment to pass. 9. a. Categorical b. 30 of 71; 42.3% 10. a. Categorical b. Percentages c. 44 of 1080 respondents or approximately 4% strongly agree with allowing drivers of motor vehicles to talk on a hand-held cell phone while driving. d. 165 of the 1080 respondents or 15% of said they somewhat disagree and 741 or 69% said they strongly disagree. Thus, there does not appear to be general support for allowing drivers of motor vehicles to talk on a hand-held cell phone while driving. 11. a. Quantitative; ratio b. Categorical; nominal c. Categorical; ordinal d. Quantitative; ratio e. Categorical; ordinal. The response to this question was recorded as a numerical value from 1 to 10. While the data are numerical, they are not quantitative. The numerical values from 1 to 10 represent categories that order the overall rating somewhere between unacceptable and truly exceptional. The data may be ordered by response category with a higher number category indicating a higher overall rating. While we prefer the categorical; ordinal answer above, at times statisticians may make the assumption that the numerical responses are equal-interval measures on a quantitative scale from 1 to 10. When this assumption is made, the data may be considered quantitative with an interval scale of measurement. In this case, additional statistical computations such as the average overall rating become helpful in summarizing the data. 12. a. The population is all visitors coming to the state of Hawaii. b. Since airline flights carry the vast majority of visitors to the state, the use of questionnaires for passengers during incoming flights is a good way to reach this population. The questionnaire actually appears on the back of a mandatory plants and animals declaration form that passengers must complete during the incoming flight. A large percentage of passengers complete the visitor information questionnaire. c. Questions 1 and 4 provide quantitative data indicating the number of visits and the number of days in Hawaii. Questions 2 and 3 provide categorical data indicating the categories of reason for the trip and where the visitor plans to stay. 13. a. Federal spending measured in trillions of dollars b. Quantitative c. Time series d. Federal spending has increased over time 14. a. The graph of the time series follows: 100 50 0 Year Hertz Dollar Avis b. In 2007 and 2008 Hertz was the clear market share leader. In 2009 and 2010 Hertz and Avis have approximately the same market share. The market share for Dollar appears to be declining. c. The bar chart for 2010 is shown below. Cars in Service (1000s) 100 50 0 Hertz Dollar Avis Company This chart is based on cross-sectional data. 15. a. Quantitative – number of new drugs approved b. Time series c. July; 1100 d. 2.9%; Yes, because most recreational boating takes place during the summer months. e. The bar graph follows the shape of a bell curve. 16. The answer to this exercise depends on updating the time series of the average price per gallon of conventional regular gasoline as shown in Figure 1.1. Contact the website to obtain the most recent time series data. The answer should focus on the most recent changes or trend in the average price per gallon. 17. Internal data on salaries of other employees can be obtained from the personnel department. External data might be obtained from the Department of Labor or industry associations. Cars in Service (1000s) 18. a. 684/1021; or approximately 67% b. 612 c. Categorical 19. a. All subscribers of Business Week in North America at the time the survey was conducted. b. Quantitative c. Categorical (yes or no) d. Crossectional - all the data relate to the same time. e. Using the sample results, we could infer or estimate 59% of the population of subscribers have an annual income of $75,000 or more and 50% of the population of subscribers have an American Express credit card. 20. a. 43% of managers were bullish or very bullish. 21% of managers expected health care to be the leading industry over the next 12 months. b. We estimate the average 12-month return estimate for the population of investment managers to be 11.2%. c. We estimate the average over the population of investment managers to be 2.5 years. 21. a. The two populations are the population of women whose mothers took the drug DES during pregnancy and the population of women whose mothers did not take the drug DES during pregnancy. b. It was a survey. c. 63 / 3.980 = 15.8 women out of each 1000 developed tissue abnormalities. d. The article reported “twice” as many abnormalities in the women whose mothers had taken DES during pregnancy. Thus, a rough estimate would be 15.8/2 = 7.9 abnormalities per 1000 women whose mothers had not taken DES during pregnancy. e. In many situations, disease occurrences are rare and affect only a small portion of the population. Large samples are needed to collect data on a reasonable number of cases where the disease exists. 22. a. The population consists of all clients that currently have a home listed for sale with the agency or have hired the agency to help them locate a new home. b. Some of the ways that could be used to collect the data are as follows: • A survey could be mailed to each of the agency’s clients. • Each client could be sent an email with a survey attached. • The next time one of the firms agents meets with a client they could conduct a personal interview to obtain the data. 23. a. This finding is applicable to the population of all American adults. b. This finding is applicable to the population of American adults that own a cellphone and/or a tablet computer. c. They conducted a sample survey. It would be way too costly to survey all American adults or all American adults who own cellphones and/or tablet computers. As we will see later in the text, very good results can be obtained using a sample survey. d. These results should be quite interesting to restaurant owners. It suggests that it would be worthwhile for them to have a website and to consider advertising through an internet search company, such as Google. 24. a. This is a statistically correct descriptive statistic for the sample. b. An incorrect generalization since the data was not collected for the entire population. c. An acceptable statistical inference based on the use of the word “estimate.” d. While this statement is true for the sample, it is not a justifiable conclusion for the entire population. e. This statement is not statistically supportable. While it is true for the particular sample observed, it is entirely possible and even very likely that at least some students will be outside the 65 to 90 range of grades. 25. a. There are five variables: Exchange, Ticker Symbol, Market Cap, Price/Earnings Ratio and Gross Profit Margin. b. Categorical variables: Exchange and Ticker Symbol Quantitative variables: Market Cap, Price/Earnings Ratio, Gross Profit Margin c. Exchange variable: Exchange Frequency Percent Frequency AMEX 5 (5/25) 20% NYSE 3 (3/25) 12% OTC 17 (17/25) 68% AMEX NYSE OTC Exchange d. Gross Profit Margin variable: Gross Profit Margin Frequency Percent Frequency 0.0 – 14.9 2 15.0 – 29.9 6 30.0 – 44.9 8 45.0 – 59.9 6 60.0 – 74. 1 0 0 e. Sum the Price/Earnings Ratio data for all 25 companies. Sum = 505.4 Average Price/Earnings Ratio = Sum/25 = 505.4/25 = 20.2 Chapter 2 Descriptive Statistics: Tabular and Graphical Displays Learning Objectives Frequency 1. Learn how to construct and interpret summarization procedures for qualitative data such as: frequency and relative frequency distributions, bar graphs and pie charts. 2. Learn how to construct and interpret tabular summarization procedures for quantitative data such as: frequency and relative frequency distributions, cumulative frequency and cumulative relative frequency distributions. 3. Learn how to construct a dot plot and a histogram as graphical summaries of quantitative data. 4. Learn how the shape of a data distribution is revealed by a histogram. Learn how to recognize when a data distribution is negatively skewed, symmetric, and positively skewed. 5. Be able to use and interpret the exploratory data analysis technique of a stem-and-leaf display. 6. Learn how to construct and interpret cross tabulations, scatter diagrams, side-by-side and stacked bar charts. 7. Learn best practices for creating effective graphical displays and for choosing the appropriate type of display. Solutions: 1. Class Frequency Relative Frequency A 60 60/120 = 0.50 B 24 24/120 = 0.20 C 36 36/120 = 0.30 120 1.00 2. a. 1 – (.22 + .18 + .40) = .20 b. .20(200) = 40 c/d. Class Frequency Percent Frequency A .22(200) = 44 22 B .18(200) = 36 18 C .40(200) = 80 40 D .20(200) = 40 20 Total . a. 360° x 58/120 = 174° b. 360° x 42/120 = 126° c. No Opinion 16.7% No 35.0% Yes 48.3% d. 70 60 50 Yes No No Opinion Response 4. a. These data are categorical. b. Relative % Show Frequency Frequency Jep 10 20 JJ 8 16 OWS 7 14 THM 12 24 WoF 13 26 c. Total 50 100 Frequency 4 2 0 Jep JJ OWS THM WoF Syndicated Television Show Syndicated Television Shows WoF 26% THM 24% Jep 20% JJ 16% OWS 14% d. The largest viewing audience is for Wheel of Fortune and the second largest is for Two and a Half Men. 5. a. Name Frequency Relative Frequency Percent Frequency Brown 7 0.14 14% Johnson 10 0.20 20% Jones 7 0.14 14% Miller 6 0.12 12% Smith 12 0.24 24% Frequency Williams 8 0.16 16% Total: 50 1 100% b. Common U.S. Last Names 4 2 0 Brown Johnson Jones Miller Smith Williams Name c. Common U.S. Last Names Williams 16% Brown 14% Johnson 20% Smith 24% Miller 12% Jones 14% Frequency d. The three most common last names are Smith (24%), Johnson (20%), Williams (16%5) 6. a. Relative % Network Frequency Frequency ABC 6 24 CBS 9 36 FOX 1 4 NBC 9 36 Total: 2 1 0 ABC CBS FOX NBC Network b. For these data, NBC and CBS tie for the number of top-rated shows. Each has 9 (36%) of the top 25. ABC is third with 6 (24%) and the much younger FOX network has 1(4%).Frequency 7. a. Rating Frequency Percent Frequency Excellent 20 40 Very Good 23 46 Good 4 8 Fair 1 2 Poor Poor Fair Good Very Good Excellent Customer Rating Management should be very pleased with the survey results. 40% + 46% = 86% of the ratings are very good to excellent. 94% of the ratings are good or better. This does not look to be a Delta flight where significant changes are needed to improve the overall customer satisfaction ratings. b. While the overall ratings look fine, note that one customer (2%) rated the overall experience with the flight as Fair and two customers (4%) rated the overall experience with the flight as Poor. It might be insightful for the manager to review explanations from these customers as to how the flight failed to meet expectations. Perhaps, it was an experience Percent Frequency with other passengers that Delta could do little to correct or perhaps it was an isolated incident that Delta could take steps to correct in the future. 8. a. Position Frequency Relative Frequency Pitcher 17 0.309 Catcher 4 0.073 1st Base 5 0.091 2nd Base 4 0.073 3rd Base 2 0.036 Shortstop 5 0.091 Left Field 6 0.109 Center Field 5 0.091 Right Field 7 0.127 55 1.000 b. Pitchers (Almost 31%) c. 3rd Base (3 – 4%) d. Right Field (Almost 13%) e. Infielders (16 or 29.1%) to Outfielders (18 or 32.7%) Living Area LiveNow Ideal Community City 32% 24% 9. Suburb Small 26% 25% a. Town 26% 30% Rural Area 16% 21% Total 100% 100% b. Where do you live now? 35% 30% 25% 20% 15% 10% 5% 0% City Suburb Small Town Rural Area Living Area What do you consider the ideal community? Percent 35% 30% 25% 20% 15% 10% 5% 0% City Suburb Small Town Rural Area Ideal Community c. Most adults are now living in a city (32%). d. Most adults consider the ideal community a small town (30%). e. Percent changes by living area: City –8%, Suburb –1%, Small Town +4%, and Rural Area +5%. Suburb living is steady, but the trend would be that living in the city would decline while living in small towns and rural areas would increase. 10. a. Rating Frequency Excellent 187 Very Good 252 Average 107 Poor 62 Percent Terrible 41 Total 649 b. Rating Percent Frequency Excellent 29 Very Good 39 Average 16 Poor 10 Terrible 6 Total 100 c. Excellent Very Good Average Poor Terrible Rating d. 29% + 39% = 68% of the guests at the Sheraton Anaheim Hotel rated the hotel as Excellent or Very Good. But, 10% + 6% = 16% of the guests rated the hotel as poor or terrible. Percent Frequency e. The percent frequency distribution for Disney’s Grand Californian follows: Rating Percent Frequency Excellent 48 Very Good 31 Average 12 Poor 6 Terrible 3 Total 100 48% + 31% = 79% of the guests at the Sheraton Anaheim Hotel rated the hotel as Excellent or Very Good. And, 6% + 3% = 9% of the guests rated the hotel as poor or terrible. Compared to ratings of other hotels in the same region, both of these hotels received very favorable ratings. But, in comparing the two hotels, guests at Disney’s Grand Californian provided somewhat better ratings than guests at the Sheraton Anaheim Hotel. 11. Class Frequency Relative Frequency Percent Frequency 12–14 2 0.050 5.0 15–17 8 0.200 20.0 18–20 11 0.275 27.5 21–23 10 0.250 25.0 24–26 9 0.225 22.5 Total 40 1.000 100.0 12. Class Cumulative Frequency Cumulative Relative Frequency less than or equal to 19 10 .20 less than or equal to 29 24 .48 less than or equal to 39 41 .82 less than or equal to 49 48 .96 less than or equal to 59 50 1.00 13. - 14. a. b/c. Class Frequency Percent Frequency 6.0 – 7.9 4 20 8.0 – 9..0 – 11.9 8 40 Frequency 12.0 – 13..0 – 15.. Leaf Unit = . . Leaf Unit = . a/b. Waiting Time Frequency Relative Frequency 0 – 4 4 0.20 5 – 9 8 0.40 10 – 14 5 0.25 15 – 19 2 0.10 20 – 24 1 0.05 Totals 20 1.00 c/d. Waiting Time Cumulative Frequency Cumulative Relative Frequency Less than or equal to 4 4 0.20 Less than or equal to 9 12 0.60 Less than or equal to 14 17 0.85 Less than or equal to 19 19 0.95 Less than or equal to 24 20 1.00 e. 12/20 = 0.60 18. a. PPG Frequency - - Total 50 b. PPG Relative Frequency 10-12 0..........04 Total 1.00 c. PPG Cumulative Percent Frequency less than 12 2 less than 14 8 less than 16 22 less than 18 60 less than 20 78 less than 22 86 less than 24 90 less than 26 90 less than 28 96 less than 30 100 d. 0 - - PPG e. There is skewness to the right. f. (11/50)(100) = 22% Frequency 19. a. The largest number of tons is 236.3 million (South Louisiana). The smallest number of tons is 30.2 million (Port Arthur). b. Millions Of Tons Frequency - -250 2 c. Histogram for 25 Busiest U.S Ports .9 50-74.9 75-99.......9 Millions of Tons Handled Most of the top 25 ports handle less than 75 million tons. Only five of the 25 ports handle above 75 million tons. 20. a. Lowest = 12, Highest = 23 b. Hours in Meetings per Week Frequency Percent Frequency % % % % % % % Frequency 25 100% c. 2 1 0 - Hours per Week in Meetings Fequency The distribution is slightly skewed to the left. 21. a/b/c/d. Revenue Frequency Relative Frequency Cumulative Frequency Cumulative Relative Frequency 0-49 6 .12 6 . .58 35 .70 .22 46 . .00 46 . .02 47 . .02 48 . .00 48 . .00 48 . .04 50 1.00 Total 50 1.00 e. The majority of the large corporations (40) have revenues in the $50 billion to $149 billion range. Only 4 corporations have revenues of over $200 billion and only 2 corporations have revenues over $400 billion. .70, or 70%, of the corporations have revenues under $100 billion. .30, or 30%, of the corporations have revenues of $100 billion or more. f. Revenue (Billion $) 400-449 The histogram shows the distribution is skewed to the right with four corporations in the $200 to $449 billion range. g. Exxon-Mobil is America’s largest corporation with an annual revenue of $443 billion. Walmart is the second largest corporation with an annual revenue of $406 billion. All other corporations have annual revenues less than $300 billion. Most (92%) have annual revenues less than $150 billion. Frequency 22. a. # U.S. Locations Frequency Percent Frequency Total: 20 100 b. 2 0 Number of U.S. Locations c. The distribution is skewed to the right. The majority of the franchises in this list have fewer than 20,000 locations (50% + 15% + 15% = 80%). McDonald's, Subway and 7- Eleven have the highest number of locations. 23. a. The highest positive YTD % Change for Japan’s Nikkei index with a YTD % Change of 31.4%. b. A class size of 10 results in 10 classes. YTD % Change Frequency - - -25 1 Frequency 30-35 1 c. --- - YTD % Change The general shape of the distribution is skewed to the right. Twenty two of the 30 indexes have a positive YTD % Change and 13 have a YTD % Change of 10% or more. Eight of the indexes had a negative YTD % Change. d. A variety of comparisons are possible depending upon when the study is done. 24. Median Pay Frequency 12 1 The median pay for these careers is generally in the $70 and $80 thousands. Only four careers have a median pay of $100 thousand or more. The highest median pay is $121 thousand for a finance director. Top Pay 4 22 1 The most frequent top pay is in the $130 thousand range. However, the top pay is rather evenly distributed between $100 and $160 thousand. Two unusually high top pay values occur at $214 thousand for a finance director and $221 thousand for an investment banker. Also, note that the top pay has more variability than the median pay. 25. a. % Increase b. The histogram is skewed to the right. c. 9 Frequency 11 3 d. Rotating the stem-and-leaf display counterclockwise onto its side provides a picture of the data that is similar to the histogram in shown in part (a). Although the stemand-leaf display may appear to offer the same information as a histogram, it has two primary advantages: the stem-and-leaf display is easier to construct by hand; and the stem-and-leaf display provides more information than the histogram because the stem-and-leaf shows the actual data. 26. a. 2 b. Most frequent age group: 40-44 with 9 runners c. 43 was the most frequent age with 5 runners 27. a. x b. y A B C Total 1 2 Total 12 30 y A x B C c. y x T d. Category A values for x are always associated with category 1 values for y. Category B values for x are usually associated with category 1 values for y. Category C values for x are usually associated with category 2 values for y. 28. a. y - Grand 1 2 Total 100.0 0.0 100.0 84.6 15.4 100.0 16.7 83.3 100.0 1 2 A 27.8 0.0 B 61.1 16.7 C 11.1 83.3 otal 100.0 100.0 Total x 50-69 1 3 1 Grand Total b. y - Grand Total 10-29 20.0 80.0 100 x 30-49 33.3 66..0 60.0 20.. 100 0 c. y - 10-29 0.0 0.0 16.7 100.0 x 30-49 28.6 0.0 66.7 0..3 100. 16.7 0..1 0.0 0.0 0.0 Grand Total d. Higher values of x are associated with lower values of y and vice versa 29. a. Average Miles per Hour - Make 139.9 149.9 159.9 169..9 Total Buick 100.00 0.00 0.00 0.00 0.00 100.00 Chevrolet 18.75 31.25 25.00 18.75 6.25 100.00 Dodge 0.00 100.00 0.00 0.00 0.00 100.00 Ford 33.33 16.67 33.33 16.67 0.00 100.00 b. 25.00 + 18.75 + 6.25 = 50 percent c. Average Miles per Hour - Make 139.9 149.9 159...9 Buick 16.67 0.00 0.00 0.00 0.00 Chevrolet 50.00 62.50 66.67 75.00 100.00 Dodge 0.00 25.00 0.00 0.00 0.00 Ford 33.33 12.50 33.33 25.00 0.00 Total 100.00 100.00 100.00 100.00 100.00 d. 75% 30. a. Year Average - Speed Total 130-139.9 16.7 0.0 0.0 33.3 50..9 25.0 25.0 12.5 25.0 12..9 0.0 50.0 16.7 16.7 16..9 50.0 0.0 50.0 0.0 0..9 0.0 0.0 100.0 0.0 0.0 100 b. It appears that most of the faster average winning times occur before 2003. This could be due to new regulations that take into account driver safety, fan safety, the environmental impact, and fuel consumption during races. 31. a. The crosstabulation of condition of the greens by gender is below. Green Condition Gender Too Fast Fine Total Male Female Total The female golfers have the highest percentage saying the greens are too fast: 40/100 = 40%. Male golfers have 35/100 = 35% saying the greens are too fast. b. Among low handicap golfers, 1/10 = 10% of the women think the greens are too fast and 10/50 = 20% of the men think the greens are too fast. So, for the low handicappers, the men show a higher percentage who think the greens are too fast. c. Among the higher handicap golfers, 39/51 = 43% of the woman think the greens are too fast and 25/50 = 50% of the men think the greens are too fast. So, for the higher handicap golfers, the men show a higher percentage who think the greens are too fast. d. This is an example of Simpson's Paradox. At each handicap level a smaller percentage of the women think the greens are too fast. But, when the crosstabulations are aggregated, the result is reversed and we find a higher percentage of women who think the greens are too fast. The hidden variable explaining the reversal is handicap level. Fewer people with low handicaps think the greens are too fast, and there are more men with low handicaps than women. 32. a. Row percentages are shown below. Region $15,000 $25,000 $35,000 $50,000 $75,000 Under to to to to to $15,000 $24,999 $34,999 $49,999 $74,999 $99,999 $100,00 0 and over Total Northeas t 12.72 10.45 10.54 13.07 17.22 11.57 24.42 100.00 Midwest 12.40 12.60 11.58 14.27 19.11 12.06 17.97 100.00 South 14.30 12.97 11.55 14.85 17.73 11.04 17.57 100.00 West 11.84 10.73 10.15 13.65 18.44 11.77 23.43 100.00 The percent frequency distributions for each region now appear in each row of the table. For example, the percent frequency distribution of the West region is as follows: Income Level Percent Frequency Under $15,000 11.84 $15,000 to $24,999 10.73 $25,000 to $34,999 10.15 $35,000 to $49,999 13.65 $50,000 to $74,999 18.44 $75,000 to $99,999 11.77 $100,000 and over 23.43 Total 100.00 b. West: 18.44 + 11.77 + 23.43 = 53.64% South: 17.73 + 11.04 + 17.57 = 46.34% c. 25.00 Northeast 20.00 15.00 10.00 5.00 0.00 Under $15,000 to$25,000 to$35,000 to$50,000 to$75,000 to $100,000 $15,000 $24,999 $34,999 $49,999 $74,999 $99,999 and over Income Level Percent Frequency Midwest 25.00 20.00 15.00 10.00 5.00 0.00 Under $15,000 to$25,000 to$35,000 to$50,000 to$75,000 to $100,000 $15,000 $24,999 $34,999 $49,999 $74,999 $99,999 and over Income Level South 25.00 20.00 15.00 10.00 5.00 0.00 Under $15,000 to$25,000 to$35,000 to$50,000 to$75,000 to $100,000 $15,000 $24,999 $34,999 $49,999 $74,999 $99,999 and over Income Level Percent Frequency Percent Frequency West 25.00 20.00 15.00 10.00 5.00 0.00 Under $15,000 to$25,000 to$35,000 to$50,000 to$75,000 to $100,000 $15,000 $24,999 $34,999 $49,999 $74,999 $99,999 and over Income Level The largest difference appears to be a higher percentage of household incomes of $100,000 and over for the Northeast and West regions. d. Column percentages are shown below. Region $15,000 $25,000 $35,000 $50,000 $75,000 Under to to to to to $15,000 $24,999 $34,999 $49,999 $74,999 $99,999 $100,000 and over Northeast 17.83 16.00 17.41 16.90 17.38 18.35 22.09 Midwest 21.35 23.72 23.50 22.68 23.71 23.49 19.96 South 40.68 40.34 38.75 39.00 36.33 35.53 32.25 West 20.13 19.94 20.34 21.42 22.58 22.63 25.70 Total 100.00 100.00 100.00 100.00 100.00 100.00 100.00 Each column is a percent frequency distribution of the region variable for one of the household income categories. For example, for an income level of $35,000 to $49,999 the percent frequency distribution for the region variable is as follows: Percent Region Frequency Northeast 16.90 Midwest 22.68 Percent Frequency South 39.00 West 21.42 Total 100.00 33. a. Brand Value ($ billions) Industry - 50-60 Total Automotive & Luxury Consumer Packaged Goods 7 5 12 Financial Services 11 3 14 Other Technology 15 Total 2 82 b. Industry Total Automotive & Luxury 15 Consumer Packaged Goods 12 Financial Services 14 Other 26 Technology 15 Total 82 c. Brand Value ($ billions) Frequency - 50-60 2 Total 82 d. The right margin shows the frequency distribution for the fund type variable and the bottom margin shows the frequency distribution for the brand value. e. Higher brand values are associated with the technology brands. For instance, the crosstabulation shows that 4 of the 15 technology brands (approximately 27%) had a brand value of $30 billion or higher. 34. a. b. Brand Revenue ($ billions) Industry - 125- 150 Tota l Automotive & Luxury 15 Consumer Packaged Goods 12 12 Financial Services 2 14 Other 1 26 Technology 15 Total 5 82 Brand Revenue ($ billions) Frequency - Total 82 c. Consumer packaged goods have the lowest brand revenues; each of the 12 consumer packaged goods brands in the sample data had a brand revenue of less than $25 billion. Approximately 57% of the financial services brands (8 out of 14) had a brand revenue of $50 billion or greater, and 47% of the technology brands (7 out of 15) had a brand revenue of at least $50 billion. d. 1-Yr Value Change (%) Industry --- 40- 60 Tota l Automotive & Luxury 11 4 15 Consumer Packaged Goods 2 10 12 Financial Services 1 6 7 14 Other Technology 1 15 Total 1 82 e. 1-Yr Value Change (%) Frequency - - 40-60 1 Total 82 f. The automotive & luxury brands all had a positive 1-year value change (%). The technology brands had the greatest variability. 35. a. Hwy MPG - Size 39 40-44 Total Compact 5 56 Large 24 Midsize 3 69 Total 8 149 b. Midsize and Compact seem to be more fuel efficient than Large. c. City MPG Drive 10- 25- Total A F 90 R Total 3 149 d. Higher fuel efficiencies are associated with front wheel drive cars. e. City MPG Fuel Type - 40-44 Total P R 93 Total 8 149 f. Higher fuel efficiencies are associated with cars that use regular gas. 36. a. -24 -40 - x b. There is a negative relationship between x and y; y decreases as x increases. 37. a. I II A B C D b. As X goes from A to D the frequency for I increases and the frequency of II decreases. 38. a. y Yes No y Low 66.667 33.333 100 x Medium 30.000 70.000 100 High 80.000 20.000 100 b. 100% 90% 80% 70% 60% 50% 40% 30% 20% 10% 0% Low Medium High x No Yes 39. a. 50 60 70 Driving Speed (MPH) b. For midsized cars, lower driving speeds seem to yield higher miles per gallon. Fuel Efficiency (MPG) 40. a. 20 0 80 Avg. Low Temp b. Colder average low temperature seems to lead to higher amounts of snowfall. Avg. Snowfall (inches) c. Two cities have an average snowfall of nearly 100 inches of snowfall: Buffalo, N.Y and Rochester, NY. Both are located near large lakes in New York. 41. a. 80.00% 70.00% 60.00% 50.00% 40.00% 30.00% 20.00% 10.00% 0.00% - 75+ Age Male Female b. The percentage of people with hypertension increases with age. c. For ages earlier than 65, the percentage of males with hypertension is higher than that for females. After age 65, the percentage of females with hypertension is higher than that for males. 42. a. % with Hypertension 100% 90% 80% 70% 60% 50% 40% 30% 20% 10% 0% - Age 65+ No Cell Phone Other Cell Phone Smartphone b. After an increase in age 25-34, smartphone ownership decreases as age increases. The percentage of people with no cell phone increases with age. There is less variation across age groups in the percentage who own other cell phones. c. Unless a newer device replaces the smartphone, we would expect smartphone ownership would become less sensitive to age. This would be true because current users will become older and because the device will become to be seen more as a necessity than a luxury. 43. a. 100% 90% 80% 70% 60% 50% 40% 30% 20% 10% 0% Bend Portland Seattle Idle Customers Reports Meetings b. 0.6 0.5 0.4 0.3 0.2 Meetings Reports Customers Idle 0.1 0 Bend Portland Seattle Frequency c. The stacked bar chart seems simpler than the side-by-side bar chart and more easily conveys the differences in store managers’ use of time. 44. a. Class Frequency Total 30 b. The distribution if nearly symmetrical. It could be approximated by a bell-shaped curve. c. 10 of 30 or 33% of the scores are between 1400 and 1599. The average SAT score looks to be a little over 1500. Scores below 800 or above 2200 are unusual. State Frequency Arizona 2 California 11 Florida 15 45. a. Georgia 2 Louisiana 8 Michigan 2 Minnesota 1 Texas 2 Total AZ CA FL GA LA MN MN TX State b. Florida has had the most Super Bowl with 15, or 15/43(100) = 35%. Florida and California have been the states with the most Super Bowls. A total of 15 + 11 = 26, or 26/43(100) = 60%. Only 3 Super Bowls, or 3/43(100) = 7%, have been played in the cold weather states of Michigan and Minnesota. Frequency c. 4 4 5 d. The most frequent winning points have been 0 to 4 points and 15 to 19 points. Both occurred in 10 Super Bowls. There were 10 close games with a margin of victory less than 5 points, 10/43(100) = 23% of the Super Bowls. There have also be 10 games, 23%, with a margin of victory more than 20 points. e. The closest games was the 25th Super Bowl with a 1 point margin. It was played in Florida. The largest margin of victory occurred one year earlier in the 24th Super Bowl. It had a 45 point margin and was played in Louisiana. More detailed information not available from the text information. 25th Super Bowl: 1991 New York Giants 20 Buffalo Bills 19, Tampa Stadium, Tampa, FL 24th Super Bowl: 1990 San Francisco 49ers 55 Denver Broncos 10, Superdome, New Orleans, LA Note: The data set SuperBowl contains a list of the teams and the final scores of the 43 Super Bowls. This data set can be used in Chapter 2 and Chapter 3 to provide interesting data summaries about the points scored by the winning team and the points scored by the losing team in the Super Bowl. For example, using the median scores, the median Super Bowl score was 28 to 13. 46. a. % Population in Millions Frequency Frequency 0.0 - 2.4 15 30.0% 2.5-4.9 13 26.0% 5.0-7.4 10 20.0% 7.5-9.9 5 10.0% 10.0-12.4 1 2.0% 12.5-14.9 2 4.0% 15.0-17.4 0 0.0% 17.5-19.9 2 4.0% 20.0-22.4 0 0.0% 22.5-24.9 0 0.0% 25.0-27.4 1 2.0% 27.5-29.9 0 0.0% 30.0-32.4 0 0.0% 32.5-34.9 0 0.0% 35.0-37.4 1 2.0% 37.5-39.9 0 0.0% More 0 0.0% Population Millions b. The distribution is skewed to the right. c. 15 states (30%) have a population less than 2.5 million. Over half of the states have population less than 5 million (28 states – 56%). Only seven states have a population greater than 10 million (California, Florida, Illinois, New York, Ohio, Pennsylvania and Texas). The largest state is California (37.3 million) and the smallest states are Vermont and Wyoming (600 thousand). 47. a. Frequency b. The majority of the start-up companies in this set have less than $90 million in venture capital. Only 6 of the 50 (12%) have more than $150 million. 48. a. % Industry Frequency Frequency Bank 26 13% Cable 44 22% Car 42 21% Cell 60 30% Collection 28 14% Total 200 100% b. 35% 30% 25% 20% 15% 10% 5% 0% Bank Cable Car Cell Collection Industry c. The cellular phone providers had the highest number of complaints. d. The percentage frequency distribution shows that the two financial industries (banks and collection agencies) had about the same number of complaints. Also, new car Percent Frequency dealers and cable and satellite television companies also had about the same number of complaints. 49. a. Yield% Frequency Percent Frequency 0.0-0.9 4 13.3 1.0-1.9 2 6.7 2.0-2.9 6 20.0 3.0-3.9 10 33.3 4.0-4.9 3 10.0 5.0-5.9 2 6.7 6.0-6.9 2 6.7 7.0-7.9 0 0.0 8.0-8.9 0 0.0 9.0-9.9 1 3.3 Total 30 100.0 b. 2 0 0.0-0.9 1.0-1.9 2.0-2.9 3.0-3.9 4.0-4.9 5.0-5.9 6.0-6.9 7.0-7.9 8.0-8.9 9.0-9.9 Dividend Yields Frequency c. The distribution is skewed to the right. d. Dividend yield ranges from 0% to over 9%. The most frequent range is 3.0% to 3.9%. Average dividend yields looks to be between 3% and 4%. Over 50% of the companies (16) pay from 2.0 % to 3.9%. Five companies (AT&T, DuPont, General Electric, Merck, and Verizon) pay 5.0% or more. Four companies (Bank of America, Cisco Systems, Hewlett-Packard, and J.P. Morgan Chase) pay less than 1%. e. General Electric had an unusually high dividend yield of 9.2%. 500 shares at $14 per share is an investment of 500($14) = $7,000. A 9.2% dividend yield provides .092(7,000) = $644 of dividend income per year. 50. a. Level of Education Percent Frequency High School graduate 32,773/65,644(100) = 49.93 Bachelor's degree 22,131/65,644(100) = 33.71 Master's degree 9003/65,644(100) = 13.71 Doctoral degree 1737/65,644(100) = 2.65 Total 100.00 13.71 + 2.65 = 16.36% of heads of households have a master’s or doctoral degree. b. Household Income Percent Frequency Under $25,000 $25,000 to $49,999 $50,000 to $99,999 $100,000 and over 13,128/65,644(100) = 20.00 15,499/65,644(100) = 23.61 20,548/65,644(100) = 31.30 16,469/65,644(100) = 25.09 Total 100.00 31.30 + 25.09 = 56.39% of households have an income of $50,000 or more. c. Household Income Level of Education Under $25,000 $25,000 to $49,999 $50,000 to $99,999 $100,000 and over High School graduate 75.26 64.33 45.95 21.14 Bachelor's degree 18.92 26.87 37.31 47.46 Master's degree 5.22 7.77 14.69 24.86 Doctoral degree 0.60 1.03 2.05 6.53 Total 100.00 100.00 100.00 100.00 There is a large difference between the level of education for households with an income of under $25,000 and households with an income of $100,000 or more. For instance, 75.26% of households with an income of under $25,000 are households in which the head of the household is a high school graduate. But, only 21.14% of households with an income level of $100,000 or more are households in which the head of the household is a high school graduate. It is interesting to note, however, that 45.95% of households with an income of $50,000 to $99,999 are households in which the head of the household his a high school graduate. 51. a. The batting averages for the junior and senior years for each player are as follows: Junior year: Allison Fealey 15/40 = .375 Emily Janson 70/200 = .350 Senior year: Allison Fealey 75/250 = .300 Emily Janson 35/120 = .292 Because Allison Fealey had the higher batting average in both her junior year and senior year, Allison Fealey should receive the scholarship offer. b. The combined or aggregated two-year crosstabulation is as follows: Combined 2-Year Batting Outcome A. Fealey E. Jansen Hit 90 105 No Hit 200 215 Total At Bats 290 320 Based on this crosstabulation, the batting average for each player is as follows: Combined Junior/Senior Years Allison Fealey 90/290 = .310 Emily Janson 105/320 = .328 Because Emily Janson has the higher batting average over the combined junior and senior years, Emily Janson should receive the scholarship offer. c. The recommendations in parts (a) and (b) are not consistent. This is an example of Simpson’s Paradox. It shows that in interpreting the results based upon separate or un-aggregated crosstabulations, the conclusion can be reversed when the crosstabulations are grouped or aggregated. When Simpson’s Paradox is present, the decision maker will have to decide whether the un-aggregated or the aggregated form of the crosstabulation is the most helpful in identifying the desired conclusion. Note: The authors prefer the recommendation to offer the scholarship to Emily Janson because it is based upon the aggregated performance for both players over a larger number of at-bats. But this is a judgment or personal preference decision. Others may prefer the conclusion based on using the un-aggregated approach in part (a). 52 a. Size of Company Job Growth (%) Small Midsized Large Total - - 1 Total b. Frequency distribution for growth rate. Job Growth (%) Total - - 60-70 1 Total 98 Frequency distribution for size of company. Size Total Small 32 Medium 28 Large 38 Total 98 c. Crosstabulation showing column percentages. Size of Company Job Growth (%) Small Midsized Large - Total d. Crosstabulation showing row percentages. Size of Company Job Growth (%) Small Midsized Large Total - 100 e. 12 companies had a negative job growth: 13% were small companies; 21% were midsized companies; and 5% were large companies. So, in terms of avoiding negative job growth, large companies were better off than small and midsized companies. But, although 95% of the large companies had a positive job growth, the growth rate was below 10% for 76% of these companies. In terms of better job growth rates, midsized companies performed better than either small or large companies. For instance, 26% of the midsized companies had a job growth of at least 20% as compared to 9% for small companies and 8% for large companies. 53. a. Tution & Fees ($) Year Founded - - - Total - - 7 Total b. Tuition & Fees ($) Year - - Grand Founded Total .00 100 .67 33.33 100 .00 100 .76 14.29 14.29 28.57 38.10 100 .04 4.08 4.08 26.53 28.57 26.53 8.16 100 .56 11.11 16.67 22.22 44.44 100 .57 57.14 14.29 100 c. Colleges in this sample founded before 1800 tend to be expensive in terms of tuition. 54. a. % Graduate Year Founded - - - 100 Grand Total - 2 7 Grand Total b. c. Older colleges and universities tend to have higher graduation rates. 55. a. 50,000 45,000 40,000 35,000 30,000 25,000 20,000 15,000 10,000 5, Year Founded Tuition & Fees ($) b. Older colleges and universities tend to be more expensive. 56. a. 120.00 100.00 80.00 60.00 40.00 20.00 0.00 0 10,000 20,000 30,000 40,000 50,000 Tuition & Fees ($) b. There appears to be a strong positive relationship between Tuition & Fees and % Graduation. % Graduate 57. a. 140.0 120.0 100.0 80.0 60.0 40.0 20.0 0.0 2008 Year 2011 b. Internet 86.7% 57.8% Newspaper etc. 13.3% 9.7% Television 0.0% 32.5% Total 100.0% 100.0% Internet Newspaper etc. Television Advertising Spend $Millions 100% 90% 80% 70% 60% 50% 40% 30% 20% 10% 0% Year Television Newspaper etc. Internet c. The graph is part a is more insightful because is shows the allocation of the budget across media, but also dramatic increase in the size of the budget. 58. a. Year Zoo attendance appears to be dropping over time. b. Attendance Advertising Spend $Millions 180,000 160,000 140,000 120,000 100,000 80,000 60,000 40,000 20,000 0 Year General Member School c. General attendance is increasing, but not enough to offset the decrease in member attendance. School membership appears fairly stable. Chapter 3 Descriptive Statistics: Numerical Measures Learning Objectives 1. Understand the purpose of measures of location. 2. Be able to compute the mean, weighted mean, geometric mean, median, mode, quartiles, and various percentiles. Attendance 3. Understand the purpose of measures of variability. 4. Be able to compute the range, interquartile range, variance, standard deviation, and coefficient of variation. 5. Understand skewness as a measure of the shape of a data distribution. Learn how to recognize when a data distribution is negatively skewed, roughly symmetric, and positively skewed. 6. Understand how z scores are computed and how they are used as a measure of relative location of a data value. 7. Know how Chebyshev’s theorem and the empirical rule can be used to determine the percentage of the data within a specified number of standard deviations from the mean. 8. Learn how to construct a 5–number summary and a box plot. 9. Be able to compute and interpret covariance and correlation as measures of association between two variables. 10. Understand the role of summary measures in data dashboards. Solutions: 1. x = xi = 75 = 15 n 5 10, 12, 16, 17, 20 Median = 16 (middle value) 2. x = xi = 96 = 16 n 6 10, 12, 16, 17, 20, 21 Median = 16 +17 = 16.5 2 5 0.8925 3. a. x = wi xi = 6(3.2) + 3(2) + 2(2.5) + 8(5) = 70.2 = 3.69 wi 6 + 3+ 2 + 8 19 b. 3.2 + 2 + 2.5+5 = 12.7 = 3.. Period Rate of Return (%) 1 -6.0 2 -8.0 3 -4.0 4 2.0 5 5.4 The mean growth factor over the five periods is: xg = = = 0.9775 So the mean growth rate (0.9775 – 1)100% = –2.25%. 5. 15, 20, 25, 25, 27, 28, 30, 34 L = p (n +1) = 20 (8 +1) =1.8 n (x1 )(x2 ) (x5 ) = 5 (0.940)(0.920)(0.960)(1.020)(1.054) 20th percentile = 15 + .8(20 ̶15) = 19 L = p (n +1) = 25 (8+1) = 2.25 25th percentile = 20 + .25(25 ̶20) = 21.25 L = p (n +1) = 65 (8 +1) = 5.85 65th percentile = 27 + .85(28 ̶27) = 27.85 L = p (n +1) = 75 (8 +1) = 6.75 75th percentile = 28 + .75(30 ̶28) = 29.5 6. Mean = xi = 657 = 59.73 n 11 Median = 57 6th item Mode = 53 It appears 3 times 7. a. The mean commute time is 26.9 minutes. b. The median commute time is 25.95 minutes. c. The data are bimodal. The modes are 23.4 and 24.8. d. L = p (n +1) = 75 (48+1) = 36.75 75th percentile = 28.5 + .75(28.5 ̶28.5) = 28.5 8. a. Median = 80 or $80,000. The median salary for the sample of 15 middle-level managers working at firms in Atlanta is slightly lower than the median salary reported by the Wall Street Journal. b. x = xi = 1260 = 84 n 15 Mean salary is $84,000. The sample mean salary for the sample of 15 middle-level managers is greater than the median salary. This indicates that the distribution of salaries for middle-level managers working at firms in Atlanta is positively skewed. c. The sorted data are as follows: i = p (n +1) = 25 (16) = First quartile or 25th percentile is the value in position 4 or 67. i = p (n +1) = 75 (16) = Third quartile or 75th percentile is the value in position 12 or 106. 9. a. x = xi = 148 = 14.8 n 10 b. Order the data from low 6.7 to high 36.6 6.7, 7.2, 7.2, 7.6, 10.1, 16.1, 16.4, 17.2, 22.9, 36.6 L = p (n +1) = 50 (10 +1) = 5.5 Median or 50th percentile = 10.1 + . 5(16.1 ̶10.1) = 13.1 c. Mode = 7.2 (occurs 2 times) d. L = p (n +1) = 25 (10 +1) = 2.75 25th percentile = 7.2 + .75(7.2 ̶7.2) = 7.2 L = p (n +1) = 75 (10 +1) = 8.25 75th percentile = 17.2 + .25(22.9 ̶17.2) = 18.625 e. xi = $148 billion The percentage of total endowments held by these 2.3% of colleges and universities is (148/413)(100) = 35.8%. f. A decline of 23% would be a decline of .23(148) = $34 billion for these 10 colleges and universities. With this decline, administrators might consider budget cutting strategies such as • Hiring freezes for faculty and staff • Delaying or eliminating construction projects • Raising tuition • Increasing enrollments 10. a. x = xi = 1318 = 65.9 n 20 Order the data from the lowest rating (42) to the highest rating (83) Position Rating Position Rating 81 L = p (n +1) = 50 (20+1) =10.5 Median or 50th percentile = 66 + . 5(67 ̶66) = 66.5 Mode is 61. b. L = p (n +1) = 25 (20 +1) = 5.25 First quartile or 25th percentile = 61 L = p (n +1) = 75 (20 +1) =15.75 Third quartile or 75th percentile = 71 c. L = p (n +1) = 90 (20 +1) =18.9 90th percentile = 78 + .9(81 ̶78) = 80.7 90% of the ratings are 80.7 or less;10% of the ratings are 80.7 or greater. 11. a. The median number of hours worked per week for high school science teachers is 54. b. The median number of hours worked per week for high school English teachers is 47. c. The median number of hours worked per week for high school science teachers is greater than the median number of hours worked per week for high school English teachers; the difference is 54 – 47 = 7 hours. 12. a. The minimum number of viewers that watched a new episode is 13.3 million, and the maximum number is 16.5 million. b. The mean number of viewers that watched a new episode is 15.04 million or approximately 15.0 million; the median also 15.0 million. The data is multimodal (13.6, 14.0, 16.1, and 16.2 million); in such cases the mode is usually not reported. c. The data are first arranged in ascending order. L = p (n +1) = 25 (21+1) = 5.50 First quartile or 25th percentile = 14 + .50(14.1 ̶14) = 14.05 L = p (n +1) = 75 (21+1) =16.5 Third quartile or 75th percentile = 16 + . 5(16.1 ̶16) = 16.05 d. A graph showing the viewership data over the air dates follows. Period 1 corresponds to the first episode of the season, period 2 corresponds to the second episode, and so on. 18.0 16.0 14.0 12.0 10.0 8.0 6.0 4.0 2.0 0.0 25 Period This graph shows that viewership of The Big Bang Theory has been relatively stable over the 2011–2012 television season. Viewers (millions) 13. Using the mean we get xcity =15.58, xhighway = 18.92 For the samples we see that the mean mileage is better on the highway than in the city. City 13.2 14.4 15.2 15.3 15.3 15.3 15.9 16 16.1 16.2 16.2 16.7 16.8  Median Mode: 15.3 Highway 17.2 17.4 18.3 18.5 18.6 18.6 18.7 19.0 19.2 19.4 19.4 20.6 21.1  Median Mode: 18.6, 19.4 The median and modal mileages are also better on the highway than in the city. 14. For March 2011: L = p (n +1) = 25 (50 +1) =12.75 First quartile or 25th percentile = 6.8 + .75(6.8 ̶6.8) = 6.8 L = p (n +1) = 50 (50 +1) = 25.5 Second quartile or median = 8 + .5(8 ̶8) = 8 L = p (n +1) = 75 (50 +1) = 38.25 Third quartile or 75th percentile = 9.4 + . 25(9.6 ̶9.4) = 9.45 For March 2012: L = p (n +1) = 25 (50 +1) =12.75 First quartile or 25th percentile = 6.2 + .75(6.2 ̶6.2) = 6.2 L = p (n +1) = 50 (50 +1) = 25.5 Second quartile or median = 7.3 + .5(7.4 ̶7.3) = 7.35 L = p (n +1) = 75 (50 +1) = 38.25 Third quartile or 75th percentile = 8.6 + . 25(8.6 ̶8.6) = 8.6 It may be easier to compare these results if we place them in a table. March 2011 March 2012 First Quartile 6.80 6.20 Median 8.00 7.35 Third 9.45 8.60 Quarti le The results show that in March 2012 approximately 25% of the states had an unemployment rate of 6.2% or less, lower than in March 2011. And, the median of 7.35% and the third quartile of 8.6% in March 2012 are both less than the corresponding values in March 2011, indicating that unemployment rates across the states are decreasing. 15. To calculate the average sales price we must compute a weighted mean. The weighted mean is 501(34.99) +1425(38.99) + 294(36.00) + 882(33.59) + 715(40.99) +1088(38.59) +1644(39.59) + 819(37.99) 501+1425 + 294 + 882 + 715 +1088 +1644 + 819 = 38.11 Thus, the average sales price per case is $38.11. 16. a. Grade xi Weight wi 4 (A) 9 3 (B) 15 2 (C) 33 1 (D) 3 0 (F) 0 60 Credit Hours x = wi xi = 9(4) +15(3) + 33(2) + 3(1) = 150 = 2.50 wi 9 +15 + 33 + 3 60 b. Yes; satisfies the 2.5 grade point average requirement 17. a. x = wi xi = 9191(4.65) + 2621(18.15) +1419(11.36) + 2900(6.75) wi 9191+ 2621+1419 + 2900 = 126,004.14 = 7.81 16,131 The weighted average total return for the Morningstar funds is 7.81%. b. If the amount invested in each fund was available, it would be better to use those amounts as weights. The weighted return computed in part (a) will be a good approximation, if the amount invested in the various funds is approximately equal. c. Portfolio Return = 2000(4.65) + 4000(18.15) + 3000(11.36) +1000(6.75) 2000 + 4000 + 3000 +1000 = 122,730 =12.27 10,000 The portfolio return would be 12.27%. 18.18. 5 1.036275 wi 180 Recruiters: x = wi xi = 444 = 3.7 wi 120 19. To calculate the mean growth rate we must first compute the geometric mean of the five growth factors: xg = = =1.007152 The mean annual growth rate is (1.007152 – 1)100 = 0.7152%. 20. n (x1 )(x2 ) (x5 ) = 5 (1.055)(1.011)(0.965)(0.989)(1.018) Assessment Deans wixi Recruiters wixi Total 180 Deans: x = wi xi = 684 = 3.8 Year % Growth Growth Factor xi 2010 5.5 1..1 1..5 0..1 0..8 1.018 Year Stivers End of Year Value Growth Factor Trippi End of Year Value Growth Factor 2004 $11,000 1.100 $5,600 1.120 2005 $12,000 1.091 $6,300 1.125 2006 $13,000 1.083 $6,900 1.095 2007 $14,000 1.077 $7,600 1.101 2008 $15,000 1.071 $8,500 1.118 2009 $16,000 1.067 $9,200 1.082 2010 $17,000 1.063 $9,900 1.076 2011 $18,000 1.059 $10,600 1.071 For the Stivers mutual fund we have: 18000=10000  ( x1 )( x2 ) ( x8 ) , so  ( x1 )( x2 ) ( x8 ) =1.8 and xg = =1.07624 So the mean annual return for the Stivers mutual fund is (1.07624 – 1)100 = 7.624% n (x )(x ) (x ) = 1. For the Trippi mutual fund we have: 10600=5000  ( x1 )( x2 ) ( x8 ) , so  ( x1 )( x2 ) ( x8 ) =2.12 and xg = =1.09848 So the mean annual return for the Trippi mutual fund is (1.09848 – 1)100 = 9.848%. While the Stivers mutual fund has generated a nice annual return of 7.6%, the annual return of 9.8% earned by the Trippi mutual fund is far superior. 21. 5000=3500  ( x1 )( x2 ) ( x9 ) , so  ( x1 )( x2 ) ( x9 ) =1.428571, and so xg = =1.040426 So the mean annual growth rate is (1.040426 – 1)100 = 4.0404% 22. 25,000,000=10,000,000  ( x1 )( x2 ) ( x6 ) , so  ( x1 )( x2 ) ( x6 ) =2.50, and so xg = =1.165 So the mean annual growth rate is (1.165 – 1)100 = 16.5% 23. Range 20 - 10 = 10 n (x )(x ) (x ) = 2. n (x )(x ) (x ) = 1. 9 n (x )(x ) (x ) = 2. 16 10, 12, 16, 17, 20 L = p (n +1) = 25 (5+1) =1.5 First quartile or 25th percentile = 10 + . 5(12 ̶10) = 11 L = p (n +1) = 75 (5+1) = 4.5 Third quartile or 75th percentile = 17 + . 5(20 ̶17) = 18.5 IQR = Q3 – Q1 = 18.5 – 11 = 7.5 24. x = xi = 75 = 15 n 5 (x − x) 2 64 s 2 = i = = 16 n −1 4 s = = 4 25. 15, 20, 25, 25, 27, 28, 30, 34 Range = 34 – 15 = 19 34.57 L = p (n +1) = 25 (8+1) = 2.25 First quartile or 25th percentile = 20 + . 25(20 ̶15) = 21.25 L = p (n +1) = 75 (8+1) = 6.75 Third quartile or 75th percentile = 28 + . 75(30 ̶28) = 29.5 IQR = Q3 – Q1 = 29.5 – 21.25 = 8.25 x = xi = 204 = 25.5 n 8 (x − x) 2 242 s 2 = i = = 34.57 n −1 7 s = = 5.88 26. Excel’s Descriptive Statistics tool provides the following values: Mean 3.72 Standard Error 0.0659 Median 3.605 Mode 3.59 Standard Deviation 0.2948 Sample Variance 0.0869 Kurtosis 9.4208 Skewness 2.9402 Range 1.24 Minimum 3.55 Maximum 4.79 Sum 74.4 Count 20 a. x = 3.72 b. s = .2948 c. The average price for a gallon of unleaded gasoline in San Francisco is much higher than the national average. This indicates that the cost of living in San Francisco is higher than it would be for cities that have an average price close to the national average. 27. a. The mean price for a round–trip flight into Atlanta is $356.73, and the mean price for a round–trip flight into Salt Lake City is $400.95. Flights into Atlanta are less expensive than flights into Salt Lake City. This possibly could be explained by the locations of these two cities relative to the 14 departure cities; Atlanta is generally closer than Salt Lake City to the departure cities. b. For flights into Atlanta, the range is $290.0, the variance is 5517.41, and the standard Deviation is $74.28. For flights into Salt Lake City, the range is $458.8, the variance is 18933.32, and the standard deviation is $137.60. The prices for round–trip flights into Atlanta are less variable than prices for round– trip flights into Salt Lake City. This could also be explained by Atlanta’s relative nearness to the 14 departure cities. 92.75 4.1 9 i 28. a. The mean serve speed is 180.95, the variance is 21.42, and the standard deviation is 4.63. b. Although the mean serve speed for the twenty Women's Singles serve speed leaders for the 2011 Wimbledon tournament is slightly higher, the difference is very small. Furthermore, given the variation in the twenty Women's Singles serve speed leaders from the 2012 Australian Open and the twenty Women's Singles serve speed leaders from the 2011 Wimbledon tournament, the difference in the mean serve speeds is most likely due to random variation in the players’ performances. 29. a. Range = 60 – 28 = 32 IQR = Q3 – Q1 = 56.5 – 43.5 = 13 b. x = 435 = 48.33 9 (x − x ) 2 = 742 (x − x) 2 742 s 2 = i = = 92.75 n −1 8 s = = 9.63 c. The average air quality is about the same. But, the variability is greater in Anaheim. 30. Dawson Supply: Range = 11 – 9 = 2 s = = 0.67 60.1 9 (x − x ) 2 i (n −1) 4, 407,720.95 19 (x − x) 2 i (n −1) 456804.55 19 J.C. Clark: Range = 15 – 7 = 8 s = = 2.58 31. a. mean median standard deviation b. The 45+ group appears to spend less on coffee than the other two groups, and the 18– 34 and 35–44 groups spend similar amounts of coffee. 32. a. Automotive : x = xi = 39201 = 1960.05 n 20 Department store: x = xi = 13857 = 692.85 n 20 b. Automotive : s = = = 481.65 Department store: s = = = 155.06 18–34 35–44 45+ 1368.0 1330.1 1070.4 1423.0 1382.5 1163.5 540.8 431.7 334.5 c. Automotive: 2901 – 598 = 2303 Department Store: 1011 – 448 = 563 d. Order the data for each variable from the lowest to highest. Automotive Department Store (x − x ) 2 i n −1 30 7 i = p (n +1) = 25 (21) = 5. Automotive: First quartile or 25th percentile = 1714 + .25(1720 – 1714) = 1715.5 Department Store: First quartile or 25th percentile = 589 + .25(597 – 589) = 591 i = p (n +1) = 75 (21) =15. Automotive: Third quartile or 75th percentile = 2202 + .75(2254 – 2202) = 2241 Department Store: First quartile or 75th percentile = 782 + .75(824 – 782) = 813.5 e. Automotive spends more on average, has a larger standard deviation, larger max and min, and larger range than Department Store. Autos have all new model years and may spend more heavily on advertising. 33. a. For 2011 x = xi = 608 = 76 n 8 s = = = 2.07 For 2012 S - The Marketplace to Buy and Sell your Study Material Downloaded by: Nurstuvia | Distribution of this document is illegal Want to earn $1.236 extra per year? (x − x ) 2 i n −1 194 7 x = xi = 608 = 76 n 8 s = = = 5.26 b. The mean score is 76 for both years, but there is an increase in the standard deviation for the scores in 2012. The golfer is not as consistent in 2012 and shows a sizeable increase in the variation with golf scores ranging from 71 to 85. The increase in variation might be explained by the golfer trying to change or modify the golf swing. In general, a loss of consistency and an increase in the standard deviation could be viewed as a poorer performance in 2012. The optimism in 2012 is that three of the eight scores were better than any score reported for 2011. If the golfer can work for consistency, eliminate the high score rounds, and reduce the standard deviation, golf scores should show improvement. 34. Quarter milers s = 0.0564 Coefficient of Variation = (s/ x )100% = (0.0564/0.966)100% = 5.8% Milers s = 0.1295 Coefficient of Variation = (s/ x )100% = (0.1295/4.534)100% = 2.9% Yes; the coefficient of variation shows that as a percentage of the mean the quarter milers’ times show more variability. S - The Marketplace to Buy and Sell your Study Material Downloaded by: Nurstuvia | Distribution of this document is illegal Want to earn $1.236 extra per year? i n − 1 ( x − x ) . x = xi = 75 = 15 n 5 s 2 = = = 4 10 z = 10 −15 = −1.25 4 20 z = 20 −15 = +1.25 4 12 z = 12 −15 = −.75 4 17 z = 17 −15 = +.50 4 16 z = 16 −15 = +.25 4 36. z = 520 − 500 = +.20 100 z = 650 − 500 = +1.50 100 z = 500 − 500 = 0.00 100 z = 450 − 500 = −.50 100 S - The Marketplace to Buy and Sell your Study Material Downloaded by: Nurstuvia | Distribution of this document is illegal Want to earn $1.236 extra per year? z = 280 − 500 = −2.. a. z = 20 −30 = −2, z = 40 −30 = 2 1− 1 = .75 At least 75% 5 5 2 2 b. z = 15−30 = −3, z = 45 −30 = 3 1− 1 = .89 At least 89% 5 5 3 2 c. z = 22 −30 = −1.6, z = 38 −30 =1.6 1− 1 = .61At least 61% 5 5 1.62 d. z = 18 −30 = −2.4, z = 42 −30 = 2.4 1− 1 = .83 At least 83% 5 5 2.42 e. z = 12 −30 = −3.6, z = 48 − 30 = 3.6 1− 1 = .92 At least 92% 5 5 3.62 38. a. Approximately 95% b. Almost all 39. c. a. Approximately 68% This is from 2 standard deviations below the mean to 2 standard deviations above the mean. S - The Marketplace to Buy and Sell your Study Material Downloaded by: Nurstuvia | Distribution of this document is illegal Want to earn $1.236 extra per year? With z = 2, Chebyshev’s theorem gives: 1− 1 = 1− 1 = 1− 1 = 3 z Therefore, at least 75% of adults sleep between 4.5 and 9.3 hours per day. b. This is from 2.5 standard deviations below the mean to 2.5 standard deviations above the mean. With z = 2.5, Chebyshev’s theorem gives: 1− 1 z 2 =1− 1 2.52 = 1− 1 6.25 = .84 Therefore, at least 84% of adults sleep between 3.9 and 9.9 hours per day. c. With z = 2, the empirical rule suggests that 95% of adults sleep between 4.5and 9.3 hours per day. The percentage obtained using the empirical rule is greater than the percentage obtained using Chebyshev’s theorem. 40. a. $3.33 is one standard deviation below the mean and $3.53 is one standard deviation above the mean. The empirical rule says that approximately 68% of gasoline sales are in this price range. b. Part (a) shows that approximately 68% of the gasoline sales are between $3.33 and $3.53. Since the bell-shaped distribution is symmetric, approximately half of 68%, or 34%, of the gasoline sales should be between $3.33 and the mean price of $3.43. $3.63 is two standard deviations above the mean price of $3.43. The empirical rule says that approximately 95% of