Solution Manual For Essentials of Statistics for Business and Economics, 9th Edition

Chapter 2 Descriptive Statistics: Tabular and Graphical Displays Learning Objectives 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. © 2019 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Solutions: 1. 2. 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 a. 1 – (.22 + .18 + .40) = .20 b. .20(200) = 40 c/d. 3. 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 200 100 a. 360° × 58/120 = 174° b. 360° × 42/120 = 126° © 2019 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. c. No Opinion 16.7% No 35.0% Yes 48.3% d. 70 60 Frequency 50 40 30 20 10 0 Yes No No Opinion Response 4. a. These data are categorical. b. Website Frequency % Frequency FB 8 16 GOOG 14 28 © 2019 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. WIKI 9 18 YAH 13 26 YT 6 12 Total 50 100 c. The most frequently visited website is google.com (GOOG); the second is yahoo.com (YAH). 5. a. Name Frequency Relative Frequency Percent Frequency Brown 7 0.14 14 Johnson 10 0.20 20 Jones 7 0.14 14 Garcia 6 0.12 12 Smith 12 0.24 24 Williams 8 0.16 16 Total: 50 1 100 b. © 2019 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Common U.S. Last Names 14 Frequency 12 10 8 6 4 2 0 Brown Johnson Jones Garcia Smith Williams Last Name c. Common U.S. Last Names 14 Frequency 12 10 8 6 4 2 0 Smith Johnson Williams Brown Jones Garcia Last Name d. © 2019 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Common U.S. Last Names Brown 14% Williams 16% Johnson 20% Smith 24% Jones 14% Garcia 12% e. The three most common last names are Smith (24%), Johnson (20%), Williams (16%5). This is easily apparent from the sorted bar chart in c. Without the labeling of percentages, it is difficult to determine the most common names from the pie chart. 6. a. Network Relative Frequency % Frequency ABC 6 24 CBS 9 36 FOX 1 4 NBC 9 36 Total: 25 100 © 2019 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 10 9 8 Frequency 7 6 5 4 3 2 1 0 ABC CBS FOX NBC Network 10 Frequency 8 6 4 2 0 ABC CBS FOX Network NBC b. For these data, NBC and CBS tie for the number of top-rated shows. Each has nine (36%) of the top 25. ABC is third with six (24%) and the much younger FOX network has 1(4%). 7. a. Rating Frequency Percent Frequency Excellent 20 40 Very Good 23 46 Good 4 8 © 2019 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Percent Frequency Fair 1 2 Poor 2 4 50 100 50 45 40 35 30 25 20 15 10 5 0 Poor Fair Good Very Good Customer Rating Excellent Management should be very pleased with the survey results: 40% + 46% = 86% of the ratings are very good to excellent, and 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. Although 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 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. © 2019 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 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%) © 2019 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 9. a. Bachelor’s (%) Master’s (%) B 21 27 CSE 9 9 E 6 24 H 16 8 NSM 8 2 SBS 16 6 O 24 24 Total 100 100 b. 30% 25% Percent 20% 15% 10% 5% 0% B CSE E H NSM O SBS Bachelor’s Degree Field of Study © 2019 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 30% Percentage 25% 20% 15% 10% 5% 0% B CSE E H NSM O SBS Master’s Degree Field of Study c. The lowest percentage for a bachelor’s is education (6%) and for master’s in natural sciences and mathematics (2%). d. The highest percentage for a bachelor’s is other (24%) and for a master’s in business (27%). e. Bachelor’s (%) Master’s (%) Difference (%) B 21 27 6 CSE 9 9 0 E 6 24 18 H 16 8 8 NSM 8 2 6 SBS 16 6 10 O 24 24 0 Education has the largest increase in percent: 18%. © 2019 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 10. a. Rating Frequency Excellent 187 Very good 252 Average 107 Poor 62 Terrible 41 Total 649 b. Rating Percent Frequency Excellent 29 Very good 39 Average 16 Poor 10 Terrible 6 Total 100 c. © 2019 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 45 Percent Frequency 40 35 30 25 20 15 10 5 0 Excellent Very Good Average Poor Terrible Rating d. At the Lakeview Lodge, 29% + 39% = 68% of the guests rated the hotel as excellent or very good, but 10% + 6% = 16% of the guests rated the hotel as poor or terrible. e. The percent frequency distribution for the Timber Hotel follows: Rating Percent Frequency Excellent 48 Very good 31 Average 12 Poor 6 Terrible 3 Total 100 At the Lakeview Lodge, 48% + 31% = 79% of the guests 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 the Timber Hotel provided somewhat better ratings than guests at the Lakeview Lodge. © 2019 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 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 Cumulative Relative Frequency 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 © 2019 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 13. 18 16 14 Frequency 12 10 8 6 4 2 0 10-19 14. 20-29 30-39 40-49 50-59 a. b/c. Class Frequency Percent Frequency 6.0–7.9 4 20 8.0–9.9 2 10 10.0–11.9 8 40 12.0–13.9 3 15 14.0–15.9 3 15 20 100 © 2019 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 15. 16. 17. Leaf unit = .1 6 3 7 5 5 7 8 1 3 4 9 3 6 10 0 4 11 3 8 5 Leaf unit = 10 11 6 12 0 2 13 0 6 7 14 2 2 7 15 5 16 0 2 8 17 0 2 3 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 © 2019 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 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 10–12 1 12–14 3 14–16 7 16–18 19 18–20 9 20–22 4 22–24 2 24–26 0 © 2019 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 26–28 3 28–30 2 Total 50 b. PPG Relative Frequency 10–12 0.02 12–14 0.06 14–16 0.14 16–18 0.38 18–20 0.18 20–22 0.08 22–24 0.04 24–26 0.00 26–28 0.06 28–30 0.04 Total 1.00 c. PPG Cumulative Percent Frequency Less than 12 2 © 2019 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 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. 20 18 16 Frequency 14 12 10 8 6 4 2 0 10-12 12-14 14-16 16-18 18-20 20-22 22-24 24-26 26-28 28-30 PPG e. There is skewness to the right. f. (11/50)(100) = 22% 19. a. The busiest airport is Hartsfield-Jackson Atlanta (ATL) with 104.2 million total passengers. The least busy airport is Detroit Metropolitan (DTW) with 34.4 million total passengers. © 2019 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. b. Total Passengers (Millions) Frequency 30–39.9 4 40–49.9 9 50–59.9 3 60–69.9 1 70–79.9 1 80–89.9 1 90–99.9 0 100–109.9 1 c. 30-39.9 40-49.9 50-59.9 60-69.9 70-79.9 80-89.9 90-99.9 100-109.9 Most of the top 20 busiest North American airports service fewer than 60 million passengers. Only four of the 20 airports have more than 60 million passengers. © 2019 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 20. a. Lowest = 12, Highest = 23 b. Hours in Meetings per Week Frequency Percent Frequency (%) 11–12 1 4 13–14 2 8 15–16 6 24 17–18 3 12 19–20 5 20 21–22 4 16 23–24 4 16 25 100 c. 7 6 Fequency 5 4 3 2 1 0 11-12 13-14 15-16 17-18 19-20 21-22 23-24 Hours per Week in Meetings The distribution is slightly skewed to the left. © 2019 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 21. a/b/c/d. Endowment Amount ($ Billions) Frequency Relative Cumulative Cumulative Relative Frequency Frequency Frequency 0–1.9 10 0.17 10 0.17 2.0–3.9 24 0.40 34 0.57 4.0–5.9 7 0.12 41 0.68 6.0–7.9 5 0.08 46 0.77 8.0–9.9 3 0.05 49 0.82 10.0–11.9 4 0.07 53 0.88 12.0–13.9 1 0.02 54 0.90 14.0–15.9 1 0.02 55 0.92 16.0–17.9 0 0.00 55 0.92 18.0–19.9 0 0.00 55 0.92 20.0–21.9 0 0.00 55 0.92 © 2019 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 22.0–23.9 1 0.02 56 0.93 24.0–25.9 1 0.02 57 0.95 26.0–27.9 2 0.03 59 0.98 28.0–29.9 0 0.00 59 0.98 30.0–31.9 0 0.00 59 0.98 32.0–33.9 0 0.00 59 0.98 34.0–35.9 0 0.00 59 0.98 36.0–37.9 1 0.02 60 1.00 Total 60 1.00 e. Most universities (55) have endowments of less than $16 billion. Only five have endowments larger than $16 billion. We see that .92, or 92%, of the universities have endowments of less than $16 billion, and only .08, or 8%, of the universities have endowments larger than $16 billion. © 2019 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 36.0-37.9 34.0-35.9 32.0-33.9 30.0-31.9 28.0-29.9 26.0-27.9 24.0-25.9 22.0-23.9 20.0-21.9 18.0-19.9 16.0-17.9 14.0-15.9 12.0-13.9 10.0-11.9 8.0-9.9 6.0-7.9 4.0-5.9 2.0-3.9 0-1.9 f. The histogram shows the distribution is skewed to the right with five university endowments in the $22 billion to $38 billion range. g. Harvard University has the largest endowment at $16 36 billion. All other universities have endowments less than $28 billion. Most (92%) have endowments less than $16 billion. 22. a. No. U.S. Locations Frequency Percent Frequency 0–4,999 10 50 5,000–9,999 3 15 10,000–14,999 2 10 15,000–19,999 1 5 20,000–24,999 0 0 25,000–29,999 1 5 © 2019 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 30,000–34,999 2 10 35,000–39,999 1 5 Total: 20 100 Frequency b. 12 10 8 6 4 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 7Eleven 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 –20–15 1 –15–10 1 –10–5 3 © 2019 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. –5–0 3 0–5 4 5–10 5 10–15 8 15–20 3 20–25 1 30–35 1 c. 9 8 7 Frequency 6 5 4 3 2 1 0 -20–15 -15–10 -10–5 -5-0 0-5 5-10 10-15 15-20 20-25 30-35 YTD % Change The general shape of the distribution is skewed to the left. 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. © 2019 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 24. Starting Median Salary 4 6 8 5 1 2 3 3 5 6 6 0 1 1 1 2 2 7 1 2 5 8 8 Mid-Career Median Salary 8 0 0 4 9 3 3 5 10 5 6 6 11 0 1 4 12 2 3 6 6 7 4 4 There is a wider spread in the mid-career median salaries than in the starting median salaries. Also, as expected, the mid-career median salaries are higher that the starting median salaries. The mid-career median salaries were mostly in the $93,000 to $114,000 range while the starting median salaries were mostly in the $51,000 to $62,000 range. © 2019 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 25. a. 8 7 Frequency 6 5 4 3 2 1 0 40-50 50-60 60-70 70-80 80-90 90-100 100-110 110-120 % Increase b. The histogram is skewed to the right. c. 4 3 5 6 1 3 7 9 7 1 3 4 5 8 2 4 7 9 0 3 6 10 0 11 3 7 7 9 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 stem-and-leaf display may appear to offer the same information as a histogram, it has two primary © 2019 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. advantages: the stem-and-leaf display is easier to construct by hand, and it provides more information than the histogram because the stem-and-leaf shows the actual data. 26. a. 2 1 4 2 6 7 3 0 1 1 2 3 1 3 5 6 7 7 4 0 0 3 3 3 3 3 4 4 4 6 6 7 9 5 0 0 0 2 7 9 2 5 5 6 6 1 4 6 6 7 2 b. Most frequent age group: 40-44 with 9 runners c. 43 was the most frequent age with 5 runners 27. a. © 2019 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. y 1 2 Total A 5 0 5 B 11 2 13 C 2 10 12 Total 18 12 30 x b. y x 1 2 Total A 100.0 0.0 100.0 B 84.6 15.4 100.0 C 16.7 83.3 100.0 c. y x 1 2 A 27.8 0.0 B 61.1 16.7 C 11.1 83.3 Total 100.0 100.0 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. © 2019 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. © 2019 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 28. a. y 20–39 40–59 10–29 x 30–49 2 50–69 1 70–90 4 Grand Total 7 3 60–79 80–100 Grand Total 1 4 5 4 6 1 5 4 3 6 4 20 60–79 80–100 Grand Total 20.0 80.0 100 b. y 20–39 40–59 10–29 x 30–49 33.3 50–69 20.0 70–90 100.0 60.0 66.7 100 20.0 100 100 © 2019 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. c. y x 20–39 40–59 60–79 80–100 10–29 0.0 0.0 16.7 100.0 30–49 28.6 0.0 66.7 0.0 50–69 14.3 100.0 16.7 0.0 70–90 57.1 0.0 0.0 0.0 Grand Total 100 100 100 100 d. Higher values of x are associated with lower values of y and vice versa. © 2019 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 29. a. Average Miles per Hour Make 130–139.9 140–149.9 150–159.9 160–169.9 170–179.9 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 © 2019 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. c. Average Miles per Hour Make 130–139.9 140–149.9 150–159.9 160–169.9 170–179.9 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% © 2019 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 30. a. Year Average Speed 1988–1992 1993–1997 1998–2002 2003–2007 2008–2012 Total 130–139.9 16.7 0.0 0.0 33.3 50.0 100 140–149.9 25.0 25.0 12.5 25.0 12.5 100 150–159.9 0.0 50.0 16.7 16.7 16.7 100 160–169.9 50.0 0.0 50.0 0.0 0.0 100 170–179.9 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 the result of new regulations that take into account driver safety, fan safety, the environmental impact, and fuel consumption during races. © 2019 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 31. a. The cross-tabulation of condition of the greens by gender follows. Green Condition Gender Too Fast Fine Total Male 35 65 100 Female 40 60 100 Total 75 125 200 The female golfers have the highest percentage who say the greens are too fast: 40/100 = 40%. Of male golfers, 35/100 = 35% say 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. When the cross-tabulations are aggregated, however, 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. © 2019 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 32. a. Row percentages follow. Region Under $15,000 to $25,000 to $35,000 to $50,000 to $75,000 to $100,000 and $15,000 $24,999 $34,999 $49,999 $74,999 $99,999 Higher Northeast 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 © 2019 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Total 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. Northeast Percent Frequency 25.00 20.00 15.00 10.00 5.00 0.00 Under $15,000 $25,000 $35,000 $50,000 $75,000 $100,000 $15,000 to to to to to and over $24,999 $34,999 $49,999 $74,999 $99,999 Income Level © 2019 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Midwest Percent Frequency 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 Percent Frequency 25.00 20.00 15.00 10.00 5.00 0.00 Under $15,000 $15,000 to $25,000 to $35,000 to $50,000 to $75,000 to $100,000 $24,999 $34,999 $49,999 $74,999 $99,999 and over Income Level West Percent Frequency 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 © 2019 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. The largest difference appears to be a higher percentage of household incomes of $100,000 and higher for the Northeast and West regions. d. Column percentages follow. © 2019 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Region Under $15,000 to $25,000 to $35,000 to $50,000 to $75,000 to $100,000 and $15,000 $24,999 $34,999 $49,999 $74,999 $99,999 Higher 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 $49,999 the percent frequency distribution for the region variable is as follows: © 2019 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Region Percent Frequency Northeast 16.90 Midwest 22.68 South 39.00 West 21.42 Total 100.00 © 2019 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 33. a. Brand Value ($ billions) Industry 0–10 10–20 20–30 Automotive and luxury 10 4 1 Consumer packaged goods 7 5 12 Financial services 11 3 14 Other 14 10 2 Technology 7 4 1 1 2 15 Total 49 26 3 1 2 82 1 30–40 40–50 50–60 Total 15 26 © 2019 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. b. Industry Total Automotive and luxury 15 Consumer Packaged Goods 12 Financial Services 14 Other 26 Technology 15 Total 82 c. Brand Value ($ Billions) Frequency 0–10 49 10–20 26 20–30 1 30–40 3 40–50 1 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 four of the 15 technology brands (approximately 27%) had a brand value of $30 billion or higher. © 2019 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 34. a. Brand Revenue ($ billions) Industry 0–25 25–50 50–75 75–100 100–125 125–150 Total Automotive and luxury 10 1 1 1 2 15 Consumer packaged goods 12 Financial services 2 4 2 2 2 2 14 Other 13 5 3 2 2 1 26 Technology 4 4 4 1 2 Total 41 14 10 5 7 12 © 2019 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 15 5 82 b. Brand Revenue ($ Billion) Frequency 0–25 41 25–50 14 50–75 10 75–100 5 100–125 7 125–150 5 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. © 2019 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. d. One-Year Value Change (%) Industry –60–41 -40–21 –20–1 Automotive and luxury Consumer packaged goods Financial services 1 Other 0–19 20–39 11 4 40–60 Total 15 2 10 12 6 7 14 2 20 4 26 Technology 1 3 4 4 2 1 15 Total 1 4 14 52 10 1 82 © 2019 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. e. One-Year Value Change (%) Frequency –60–41 1 –40–21 4 –20–1 14 0–19 52 20–39 10 40–60 1 Total 82 f. The automotive & luxury brands all had a positive one-year value change (%). The technology brands had the greatest variability. © 2019 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 35. a. Hwy MPG Size 20–24 25–29 30–34 35–39 40–44 Total Compact 13 25 49 29 6 122 Large 10 31 19 11 1 72 Midsize 15 35 61 29 7 147 Total 9138 12991 69129 1469 9114 341 b. Midsize and compact seem to be more fuel efficient than large. c. City MPG Drive 10–14 15–19 20–24 25–29 A 3 43 57 5 8 48 82 F R 10 33 32 4 Total 13 84 137 91 30–34 Total 108 16 154 79 16 341 d. Higher fuel efficiencies are associated with front-wheel-drive cars. © 2019 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. e. City MPG Fuel Type 10–14 15–19 20–24 25–29 30–34 Total P 13 58 94 16 1 182 26 43 75 15 159 84 137 91 16 341 R Total 13 f. Higher fuel efficiencies are associated with cars that use regular gas. © 2019 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 36. a. 56 40 y 24 8 -8 -24 -40 -40 -30 -20 -10 0 x 10 20 30 40 b. There is a negative relationship between x and y; y decreases as x increases. 37. a. 1000 800 600 I 400 II 200 0 A B C D b. As X goes from A to D the frequency for I increases and the frequency of II decreases. y x Yes No Low 66.667 33.333 100 Medium 30.000 70.000 100 © 2019 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. High 38. 80.000 20.000 100 a. 100% 80% 60% No 40% Yes 20% 0% Low Medium x High 20 30 40 Driving Speed (MPH) b. a. Fuel Efficiency (MPG) 39. 40 35 30 25 20 15 10 5 0 0 10 50 60 70 b. For midsized cars, lower driving speeds seem to yield higher miles per gallon. 40. a. © 2019 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Avg. Snowfall (inches) 120 100 80 60 40 20 0 30 40 50 60 Avg. Low Temp 70 80 b. Colder average low temperature seems to lead to higher amounts of snowfall. c. Two cities have an average snowfall of nearly 100 inches of snowfall: Buffalo, New York, and Rochester, New York. Both are located near large lakes in the state. 41. a. 80.00% % with Hypertension 70.00% 60.00% 50.00% Male 40.00% Female 30.00% 20.00% 10.00% 0.00% 20-34 35-44 45-54 55-64 65-74 75+ Age b. The percentage of people with hypertension increases with age. c. For ages before 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 for © 2019 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. males. 42. a. 100% 90% 80% 70% 60% 50% No Cell Phone 40% Other Cell Phone 30% Smartphone 20% 10% 0% 18-24 25-34 35-44 45-54 55-64 65+ Age b. After increasing in ages 25–34, smartphone ownership decreases with increasing age. 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 necessity than luxury. 43. a. © 2019 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 100% 90% 80% 70% 60% Idle 50% Customers 40% Reports 30% Meetings 20% 10% 0% Bend Portland Seattle b. 0.6 0.5 0.4 Meetings Reports 0.3 Customers Idle 0.2 0.1 0 Bend Portland Seattle 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 800–999 1 © 2019 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 1000–1199 3 1200–1399 6 1400–1599 10 1600–1799 7 1800–1999 2 2000–2199 2 Total 30 12 Frequency 10 8 6 4 2 0 800-999 1000-1199 1200-1399 1400-1599 SAT Score 1600-1799 1800-1999 2000-2199 b. The distribution if nearly symmetrical. It could be approximated by a bell-shaped curve. c. Ten of 30, or 33%, of the scores are between 1400 and 1599. The average SAT score looks to be slightly more than 1500. Scores below 800 or above 2200 are unusual. 45. a. Median Household Income Frequency Percent Frequency 65.0–69.9 1 2 © 2019 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 70.0–74.9 6 12 75.0–79.9 17 34 80.0–84.9 6 12 85.0–89.9 7 14 90.0–94.9 5 10 95.0–99.9 4 8 100.0–104.9 0 0 105.0-109.9 3 6 110.0-114.9 1 2 50 100% b. 18 16 Frequency 14 12 10 8 6 4 2 0 Median Household Income – Two Earners c. The distribution is skewed to the right. There is a gap in the $100.0–$104.9 range. The most frequent range for the median household income is $75.0–$79.9 thousand. d. New Jersey $110.7 thousand e. Idaho $67.1 thousand © 2019 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 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 © 2019 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Frequency 16 14 12 10 8 6 4 2 0 Population Millions b. The distribution is skewed to the right. c. Fifteen states (30%) have a population less than 2.5 million. More than half of the states have populations of less than 5 million (28 states, or 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 (600.000). 47. a. © 2019 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 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% Percent Frequency 30% 25% 20% 15% 10% 5% 0% Bank c. Cable Car Industry Cell Collection 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 dealers and cable and satellite television companies also had about the same number of complaints. © 2019 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 49. a. Beta Frequency Percent Frequency 0.00–0.09 1 3.3 0.10–0.19 1 3.3 0.20–0.29 1 3.3 0.30–0.39 0 0.0 0.40–0.49 1 3.3 0.50–0.59 1 3.3 0.60–0.69 3 10.0 0.70–0.79 2 6.7 0.80–0.89 5 16.7 0.90–.99 4 13.3 1.00–1.09 0 0.0 1.10–1.19 2 6.7 1.20–1.29 5 16.7 1.30–1.39 2 6.7 1.40–1.49 0 0.0 1.50–1.59 0 0.0 1.60–1.69 0 0.0 © 2019 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 1.70–1.80 1 3.3 1.80–1.90 1 3.3 Total 30 100.0% b. 6 Frequency 5 4 3 2 1 0 Beta c. The distribution is somewhat skewed to the left. d. The stock with the highest beta is JP Morgan Chase & Company with a beta of 1.84. The stock with the lowest beta is Verizon Communications, Inc., with a beta of .04. 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. © 2019 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. b. Household Income Percent Frequency Less than $25,000 13,128/65,644(100) = 20.00 $25,000 to $49,999 15,499/65,644(100) = 23.61 $50,000 to $99,999 20,548/65,644(100) = 31.30 $100,000 and higher 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 to $50,000 to $100,000 and $25,000 $49,999 $99,999 Higher 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 High School graduate There is a large difference between the level of education for households with an income of less than $25,000 and households with an income of $100,000 or more. For © 2019 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. instance, 75.26% of households with an income of less than $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: Senior year: Allison Fealey 15/40 = .375 Emily Janson 70/200 = .350 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, she should receive the scholarship offer. b. The combined or aggregated two-year cross-tabulation is as follows: Combined Two-Year Batting Outcome A. Fealey E. Jansen Hit 90 105 No Hit 200 215 Total At Bats 290 320 Based on this cross-tabulation, the batting average for each player is as follows: Combined Junior–Senior Years © 2019 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 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, she 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 on separate or unaggregated cross-tabulations, 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 unaggregated or aggregated form of the crosstabulation is more helpful in identifying the desired conclusion. Note: The authors prefer the recommendation to offer the scholarship to Emily Janson because it is based on 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 unaggregated approach in part a. 52 a. Size of Company Job Growth (%) Small Midsized Large Total –10–0 4 6 2 12 0–10 18 13 29 60 10–20 7 2 4 13 20–30 3 3 2 8 © 2019 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 30–40 0 3 1 4 60–70 0 1 0 1 Total 32 28 38 98 b. Frequency distribution for growth rate. Job Growth (%) Total –10–0 12 0–10 60 10–20 13 20–30 8 30–40 4 60-70 1 Total 98 Frequency distribution for size of company. Size Total Small 32 Medium 28 Large 38 Total 98 c. Cross-tabulation showing column percentages. © 2019 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Size of Company Job Growth (%) Small Midsized Large –10–0 13 21 5 0–10 56 46 76 10–20 22 7 11 20–30 9 11 5 30–40 0 11 3 60–70 0 4 0 Total 100 100 100 d. Cross-tabulation showing row percentages. Size of Company Job Growth (%) Small Midsized Large Total –10–0 33 50 17 100 0–10 30 22 48 100 10–20 54 15 31 100 20–30 38 38 25 100 30–40 0 75 25 100 60–70 0 4 0 100 e. Twelve companies had negative job growth: 13% were small companies, 21% were © 2019 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 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 even though 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. © 2019 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 53. a. Tuition and Fees ($) Year Founded 1–5,000 10,001– 15,001– 20,001– 25,001– 30,001– 35,001– 40,001– 15,000 20,000 25,000 30,000 35,000 40,000 45,000 1600–1649 1 1700–1749 2 1750–1799 1800–1849 1850–1899 1 1900–1949 2 1 1950–2000 Total 1 1 Total 1 1 3 4 4 1 3 3 6 8 21 2 13 14 13 4 49 2 3 4 8 2 4 4 9 18 1 19 22 7 30 © 2019 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 17 103 b. Tuition and Fees ($) Year Founded 1–5,000 10,001– 15,001– 20,001– 25,001– 30,001– 35,001– 40,001 15,000 20,000 25,000 30,000 35,000 40,000 – Grand Total 45,00 0 1600–1649 100.00 1700–1749 66.67 1750–1799 1800–1849 1850–1899 1900–1949 1950–2000 2.04 4.08 5.56 28.57 100 33.33 100 100.00 100 4.76 14.29 14.29 28.57 38.10 100 4.08 26.53 28.57 26.53 8.16 100 11.11 16.67 22.22 44.44 57.14 14.29 c. Colleges in this sample founded before 1800 tend to be expensive in terms of tuition. © 2019 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 100 100 54. a. Percent Graduating Year 35– 40– 45– 50– 55– 60– 65– 70– 75– 80– 85– 90– 95– Grand Founded 40 45 50 55 60 65 70 75 80 85 90 95 100 Total 1600–1649 1 1 1700–1749 3 3 1 3 4 1750–1799 1800–1849 1850–1899 1 1900–1949 1 1950–2000 1 Grand Total 2 1 1 2 1 1 3 3 5 1 2 4 2 3 4 3 2 21 4 3 11 5 9 6 3 4 1 49 1 3 3 2 4 1 1 18 2 5 7 15 7 12 13 13 © 2019 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 8 9 10 103 b. c. Older colleges and universities tend to have higher graduation rates. a. Tuition & Fees ($) 55. 50,000 45,000 40,000 35,000 30,000 25,000 20,000 15,000 10,000 5,000 0 1600 1700 1800 Year Founded 1900 2000 b. Older colleges and universities tend to be more expensive. 56. a. 120.00 % Graduate 100.00 80.00 60.00 40.00 20.00 0.00 0 10,000 20,000 30,000 40,000 Tuition & Fees ($) 50,000 © 2019 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. b. There appears to be a strong positive relationship between Tuition and Fees and Percent Graduating. 57. a. Electric Plug-in Vehicle Sales 250000 2013 Sales Amount 200000 2015 150000 100000 50000 0 Region China Western Europe United States Japan Canada Market Region b. Region 2013 2015 China 7.0% 37.9% Western Europe 33.4% 32.6% United States 45.6% 20.4% Japan 13.5% 8.2% Canada 0.4% 0.9% Total: 100.0% 100.0% © 2019 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Electric Vehicle Sales by Percentage 100.0% 90.0% 80.0% 70.0% 60.0% 50.0% 40.0% 30.0% 20.0% 10.0% 0.0% China Western Europe United States Japan Canada 2013 2013 2015 c. The graph ins part a is more insightful because is shows the change in vehicle sales over time for each market region. 58. a. Zoo attendance appears to be dropping over time. b. © 2019 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. c. General attendance is increasing, but not enough to offset the decrease in member attendance. School membership appears fairly stable. © 2019 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Case Solutions Chapter 2 Descriptive Statistics: Tabular and Graphical Presentations Case Problem 1 Pelican Stores 1. There were 70 promotional customers and 30 regular customers. Because there are 100 observations in the sample, the frequency and percent frequency distribution are the same. Percent frequency distributions for many of the variables are given. No. of Items 1 2 3 4 5 6 7 or more Total: Percent Frequency 29 27 10 10 9 7 8 100 Net Sales 0.00–24.99 25.00–49.99 50.00–74.99 75.00–99.99 100.00–124.99 125.00–149.99 150.00–174.99 175.00–199.99 200 or more Total: Percent Frequency 9 30 25 10 12 4 3 3 4 100 Method of Payment American Express Discover MasterCard Proprietary Card Visa Total: Gender Female Male Total: Martial Status Married Single Total: Age 20–29 30–39 Percent Frequency 2 4 14 70 10 100 Percent Frequency 93 7 100 Percent Frequency 84 16 100 Percent Frequency 10 30 © 2019 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 40–49 50–59 60–69 70–79 Total: 33 16 7 4 100 These percent frequency distributions provide a profile of Pelican’s customers. Many observations are possible, including: • A large majority of the customers use National Clothing’s proprietary credit card. • More than half of the customers purchase one or two items, but a few make numerous purchases. • The percent frequency distribution of net sales shows that 61% of the customers spent $50 or more. • Customers are distributed across all adult age groups. • The overwhelming majority of customers are female. • Most of the customers are married. 2. © 2019 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 3. A crosstabulation of type of customer versus net sales is shown. Customer Promotional Regular Total 0–25 7 2 9 25–50 17 13 30 50–75 17 8 25 75–100 8 2 10 100–125 9 3 12 Net Sales 125–175 175–200 3 2 1 1 4 3 200–225 3 225–250 250– 1 2 3 1 From the crosstabulation it appears that net sales are larger for promotional customers. © 2019 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 2 4. A scatter diagram of Net Sales versus Age is shown as follows. A trend line has been fitted to the data. From this, it appears that there is no relationship between net sales and age. 350.00 300.00 Net Sales 250.00 200.00 150.00 100.00 50.00 0.00 0 10 20 30 40 50 60 70 80 90 Age Age is not a factor in determining net sales. Case Problem 2 Movie Theater Releases This case provides the student with the opportunity to use tabular and graphical presentations to analyze data from the movie industry. Developing and interpreting frequency distributions, percent frequency distributions and scatter diagrams are emphasized. The interpretations and insights can be quite varied. We illustrate some below. Frequency Distribution and Percent Frequency Distribution The choice of the classes for frequency distributions or percent frequency distributions can be expected to vary. The frequency distributions we developed are as follows: Opening Gross Sales (Millions) $0–9.99 10–19.99 20–29.99 30–39.99 40–49.99 50–59.99 60–69.99 70–79.99 80–89.99 90–99.99 100–109.99 Frequency (or Percentage) 14 34 22 10 5 3 1 2 1 0 2 © 2019 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 110–119.99 120–129.99 130–139.99 140–149.99 150–159.99 160–169.99 170–179.99 Total Gross Sales (Millions) $0–49.99 50–99.99 100–149.99 150–199.99 200–249.99 250–299.99 300–349.99 350–399.99 400–449.99 450–499.99 500–549.99 Total Number of Theaters 0–499 500–999 1,000–1,499 1,500–1,999 2,000–2,499 2,500–2,999 3,000–3,499 3,500–3,999 4,000–4,499 100 Number of Weeks in Release 0–4 5–9 10–14 15–19 20–24 25–29 30–34 35–39 40–44 45–49 100 0 0 3 0 1 1 1 Frequency (or Percentage) 34 36 11 6 3 1 3 3 1 1 1 100 Frequency (or Percentage) 0 0 1 4 6 17 37 21 14 Frequency (or Percentage) 0 15 43 23 14 4 0 0 0 1 Histograms © 2019 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 400–449.99 350–399.99 300–349.99 250–299.99 200–249.99 150–199.99 100–149.99 50–99.99 0–49.99 Frequency 500–549.99 450–499.99 400–449.99 350–399.99 300–349.99 250–299.99 200–249.99 150–199.99 100–149.99 50–99.99 0–49.99 Frequency 0–9.99 170–179.99 160–169.99 150–159.99 140–149.99 130–139.99 120–129.99 110–119.99 100–109.99 90–99.99 80–89.99 70–79.99 60–69.99 50–59.99 40–49.99 30–39.99 20–29.99 10–19.99 Frequency 40 35 30 25 20 15 10 5 0 Opening Gross Sales ($ millions) 40 35 30 25 20 15 10 5 0 Total Gross Sales ($ millions) 40 35 30 25 20 15 10 5 0 Number of Theaters The following histograms are based on the frequency distributions shown above. © 2019 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 50 45 40 Frequency 35 30 25 20 15 10 5 0 5-9 10-14 15-19 20-24 25-29 30-34 35-39 40-44 45-50 Number of Weeks in Release Interpretation Opening Weekend Gross Sales The distribution is skewed to the right. Numerous movies have somewhat low opening weekend gross sales, while a relatively few (8%) have an opening weekend gross sales of $100 million or more. Only 3% had opening weekend gross sales of $150 million or more. Eighty percent of the movies had opening weekend gross sales less than $40 million, and 92% of the movies had opening weekend gross sales less than $100 million. Total Gross Sales This distribution is also skewed to the right. Again, the majority of the movies have relatively low total gross sales with 70% of movies having gross sales less than $100 million and 91% less than $300 million. Highly successful blockbuster movies are rare. Total gross sales of more than $400 million occurred only 3% of the time, and gross sales of more than $500 million occurred only 1% of the time. Unless there is something unusually attractive about the movie, a total gross sales less than $100 million appears typical. Number of Theaters This distribution is skewed to the left. The number of theaters range from slightly more than 1,000 to almost 4,500. Eighty-nine percent of the movies had large market exposure, playing in 2,500 or more theaters. No movies were in fewer than 1,000 theaters, and only 11% were in fewer than 2,500 theaters. Most top movies in 2016 appeared to receive large market exposure in 2,500 or more theaters. Number of Weeks in Release This distribution is skewed to the right, but not as much as the distributions on sales. Almost all movies in 2016 spent at least 10 weeks in release. Only 15% of movies in 2016 spent fewer than 10 weeks in release. One movie (Hidden Figures) spent much longer in release than any other movie at 46 weeks. General Observations The data show there are relatively few high-end, highly successful movies. The financial rewards are there for the pictures that make the blockbuster level. But the majority of movies will have relatively low opening weekend gross sales and low total gross sales. Movies being shown in more than 2500 theaters and movies that spend at least 10 weeks in release are common. © 2019 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Scatter Diagrams Three scatter diagrams are suggested to show how Total Gross Sales is related to each of the other three variables. Total Gross Sales ($ millions) 600.00 500.00 400.00 300.00 200.00 100.00 0.00 0.00 50.00 100.00 150.00 200.00 Opening Gross Sales ($ millions) Total Gross Sales ($ millions) 600.00 500.00 400.00 300.00 200.00 100.00 0.00 0 1,000 2,000 3,000 4,000 5,000 40 50 Number of Theaters Total Gross Sales ($ millions) 600.00 500.00 400.00 300.00 200.00 100.00 0.00 0 10 20 30 Number of Weeks in Release Interpretation Opening Weekend Gross Sales The scatter plot of total gross sales and opening weekend gross sales shows a strong positive relationship. Movies with the highest total gross sales were © 2019 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. those with the highest opening gross sales. How a movie does during its opening weekend should be a strong predictor of how the movie will do in terms of total gross sales. Note in the scatter diagram that the majority of the movies show a low opening weekend gross sales and a low total gross sales. Number of Theaters The scatter plot of the total gross sales and number of theaters also shows a positive relationship. For movies playing in fewer than 3,500 theaters, the total gross sales were significantly less than those movies playing in more than 3,500 theaters. If the movie is shown in more theaters, higher total gross sales are anticipated. For movies playing in more than 3,500 theaters, the positive relationship is especially strong. This scatter chart also appears to show a nonlinear relationship because movies playing in the most theaters increase in total gross sales rapidly compared to those playing in fewer theaters. Number of Weeks in Release The scatter plot of the total gross sales and number of weeks in release shows a positive relationship, but this relationship appears to be the weakest of the three relationships studied. Generally, the more successful movies with higher gross sales are in release for more weeks. However, this is not always the case. The longest released movie (Hidden Figures) had less in total gross sales than many movies that had shorter release times. And many movies that were in release for more than 20 weeks had less total gross revenue than those with fewer than 20 weeks in release. This suggests that in some cases blockbuster movies with high gross sales may run their course quickly and not have an excessively long run in release. At the same time, perhaps quality movies with a limited audience may not generate the high total gross sales but may still show a run of 20 or more weeks. The number of weeks in release does not appear to the best predictor of total gross sales. Case Problem 3 Queen City This case provides the student with the opportunity to use basic tabular and graphical presentations to describe data from the annual expenditures for the city of Cincinnati, Ohio. The data set is large relative to others in the text. It contains 5,427 records of expenditures. As such, one point of this case is to expose students to a larger data set and help them understand that the pivot tables and charts can be used on a larger data set. In some cases, the student will have to copy, paste, and aggregate data to create the desired tables and charts. Style of presentation may vary by student (for example, vertical versus horizontal bar charts may be used). We illustrate with results and comments below. Expenditures by Category The pivot table shows expenditures and percentage of total expenditures by category. The bar chart shows percentage of total expenditures by category (both the table and the bar chart are sorted in descending order). Capital expenditures and payroll account for more than 50% of all expenditures. Total expenditures are more than $660 million. Debt Service seems somewhat high with more than 10% of total expenditures. Category Capital Total Expenditures ($) 198,365,854 % of Total Expenditures 29.98 © 2019 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Payroll Debt Service Contractual Services Fringe Benefits Fixed Costs Materials and Supplies Inventory Payables Grand Total 145,017,555 86,913,978 85,043,249 66,053,340 53,732,177 19,934,710 6,393,394 180,435 661,634,693 21.92 13.14 12.85 9.98 8.12 3.01 0.97 0.03 100.0 Expenditures by Department Capital Payroll Debt Service Category Contractual Services Fringe Benefits Fixed Costs Materials and Supplies Inventory Payables 0.00% 5.00% 10.00% 15.00% 20.00% 25.00% 30.00% % of Total Expenditures The following table and bar chart show the percentages of total expenditures incurred by department. Note that we have combined all departments that individually incurred less than 1% of the total expenditures. Of all 119 departments, 96 each account for less than 1% of the total expenditures. As shown as follows, only six individual departments incur 5% or more of the total expenditures. These include Police, Sewers, Transportation Engineering (Engineering). Fire, Sewer Debt Service, and Finance and Risk Management. Debt service on sewers as a percentage of total expenditures appears to be especially high. © 2019 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Department Department of Police Department of Sewers Transportation and Engineering (Engineering) Department of Fire Sewer Debt Service Finance, Risk Management SORTA Operations Water Works, Debt Service Department of Water Works Finance, Treasury Economic Development Division of Parking Services Community Development, Housing Enterprise Technology Solutions Public Services, Fleet Services Finance, Accounts and Audits Transportation and Engineering, Planning Public Services, Neighborhood Operations Sewers, Millcreek Health, Primary Health Care Centers Water Works, Water Supply Public Services, Facilities Management Sewers, Wastewater Administration Other Depts. (< 1% each) Total % of Total Expenditures 9.7 8.8 8.7 7.2 6.6 5.4 3.6 3.2 3.1 2.8 2.1 1.9 1.7 1.7 1.7 1.7 1.6 1.4 1.3 1.2 1.2 1.1 1.0 21.2% 100.0% © 2019 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Expenditures by Fund The following table and bar chart show the percentages of total expenditures charged by the fund used to pay. Note that we have combined those funds that each cover less than 1% of the total expenditures. Of 129 funds in the data base, 117 each account for less than 1% of total expenditures. © 2019 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Fund 050–GENERAL FUND 980–CAPITAL PROJECTS 701–METROPOLITAN SEWER DISTRICT OF GREATER CINCINNATI 704–METROPOLITAN SEWER DISTRICT CAPITAL IMPROVEMENTS 101–WATER WORKS 711–RISK MANAGEMENT 759–INCOME TAX–TRANSIT 151–BOND RETIREMENT–CITY 202–FLEET SERVICES 898–WATER WORKS IMPROVEMENT 12 897–WATER WORKS IMPROVEMENT 11 302–INCOME TAX–INFRASTRUCTURE Other (< 1 % each). Total % of Total Expenditures Covered 25.5 16.0 12.7 8.8 7.9 4.9 3.7 2.4 1.7 1.3 1.3 1.1 12.9 100.0% © 2019 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Fund 050 – GENERAL FUND 980 – CAPITAL PROJECTS Other (< 1 % each) 701 – METROPOLITAN SEWER DISTRICT OF… 704 – METROPOLITAN SEWER DISTRICT… 101 – WATER WORKS 711 – RISK MANAGEMENT 759 – INCOME TAX – TRANSIT 151 – BOND RETIREMENT – CITY 202 – FLEET SERVICES 898 – WATER WORKS IMPROVEMENT 12 897 – WATER WORKS IMPROVEMENT 11 302 – INCOME TAX – INFRASTRUCTURE 0.00% 5.00% 10.00% 15.00% 20.00% 25.00% 30.00% % of Total Spending Other Points Of 5,427 records of expenditures in the database, 235 (4.3%) are negative. Case Problem 4 Cut-Rate Machining, Inc. A scatter diagram of the results for Hole-Maker in the order the holes were drilled shows that this machine consistently overdrills and is moderately consistent. A scatter diagram of the results for Shafts & Slips in the order the holes were drilled shows that this machine consistently underdrills and is moderately consistent. © 2019 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. A scatter diagram of the results for Judge’s Jigs in the order the holes were drilled shows that on average this machine this machine consistently underdrills and is extremely consistent. A scatter diagram of the results for Drill-for-Bits in the order the holes were drilled shows that an average diameter of approximately 3 centimeters. However, this machine this machine is extremely inconsistent. © 2019 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. If we focus solely on the average performance of a drill, we would purchase Drill-for-Bits as the diameters of holes drilled by this vendor’s drill appear to be centered at approximately 3 centimeters. However, the diameters of the holes drilled by Drill-for-Bits’ machine are extremely inconsistent—several are more than ½ centimeter too wide and several are more than ½ centimeter to narrow. The diameters of holes drilled by the machine provided by Hole-Maker are more consistent than those drilled by the machine provided by Drill-for-Bits, and this machine did not drill a single hole that is too narrow. If holes that are slightly too wide are acceptable, we should consider purchasing our drill from Hole-Maker. The diameters of holes drilled by the machine provided by Shafts & Slips are similar in consistency to the holes by the machine provided by Hole-Maker, and this machine did not drill a single hole that is too wide. If holes that are slightly too small are acceptable, we should consider purchasing our drill from Shafts & Slips. The diameters of holes drilled by the machine provided by Judge’s Jigs are far more consistent than holes by the machine provided any of the other vendors, but these holes are far too narrow. We should determine if this drill can be recalibrated to that, then the mean size of holes drilled is approximately 3 centimeters. If this can be done, we should consider purchasing our drill from Judge’s Jigs and recalibrating the drill; this would give us a machine that consistently drills holes of approximately 3 centimeters. However, we should scrutinize the way these data were collected before we make a decision. We were told that Weideman started all four machines at 8 A.M. and let them warm up for two hours. We also see from the data that the drill provided by Hole-Maker was tested from 10 A.M. to noon, the drill provided by Shafts & Slips, Inc. was tested from noon to 2 P.M., the drill provided by Judge’s Jigs was tested from 2 P.M. to 4 P.M., and the drill provided by Drill-for-Bits was © 2019 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. tested from 4 P.M. to 6 P.M. Were all drills allowed to keep running after the 8 A.M. to 10 A.M. warm-up period? Either way, this could bias the results. We also see from the data that Ms. Ames ran the test drills from 10 A.M. to 4 P.M. when the drills provided by Hole-Maker, Shafts & Slips, and Judge’s Jigs were tested. Mr. Silver ran the test drill from 4 P.M. to 6 P.M. when the drill provided by Drill-for- Bits was tested. If these two employees are not equally competent, then this could bias the results. Furthermore, did Ms. Ames become fatigued as the day progressed? Did she take a break for lunch or take a break at any other time? We also note that we only tested one drill for each vendor. If the drill provided by a vendor is not representative of the drills that vendor produced, then this also could bias the results. The data for this test should have been collected through an experimental study in which the four machine were all warmed up for the same amount of time and then left running as eight holes were drilled by each employee using the drill provided by each vendor in a random order. A design such as this would have eliminated the potential sources of bias we have identified and led to the collection of more reliable data, which would lead to a superior decision. © 2019 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.