1. Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. 2 - 1 Describing Data requency Distributions f Graphic Presentations Copyright © 2003 by The McGraw-Hill Companies, Inc. All rights reserved.

2. Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. 2 - 2 Organize raw data into frequency distribution Produce a histogram, a frequency polygon, and a cumulative frequency polygon from quantitative data Develop and interpret a stem-and-leaf display When you have completed this chapter, you will be able to:

3. Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. 2 - 3 Present qualitative data using such graphical techniques such as a clustered bar chart, a stacked bar chart, and a pie chart Detect graphic deceptions and use a graph to present data with clarity, precision, and efficiency

4. Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. 2 - 4

5. Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. 2 - 5 A Frequency Distribution is a grouping of data into non-overlapping classes (mutually exclusive)… showing the number of observations in each category or class. The range of categories includes all values in the data set (collectively exhaustive classes).

6. Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. 2 - 6 Class Midpoint or Class Mark: A point that divides a class into two equal parts, i.e. the average of the upper and lower class limits. 12.5 Class frequency: The number of observations in each class. Class interval: The class interval is obtained by subtracting the lower limit of a class from the lower limit of the next class, e.g. 5 17.5 22.5 27.5 32.5

7. Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. 2 - 7 Dr. Tillman is Dean of the School of Business. He wishes to prepare a report showing the number of hours per week students spend studying. He selects a random sample of 30 students and determines the number of hours each student studied last week. 15.0, 23.7, 19.7, 15.4, 18.3, 23.0, 14.2, 20.8, 13.5, 20.7, 17.4, 18.6, 12.9, 20.3, 13.7, 21.4, 18.3, 29.8, 17.1, 18.9, 10.3, 26.1, 15.7, 14.0, 17.8, 33.8, 23.2, 12.9, 27.1, 16.6. Organize the data into a frequency distribution.

8. Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. 2 - 8 Decide how many classes you wish to use. Frequency Distributions by hand Determine the class width. There are five steps that can be used to Construct a Frequency Distribution: Set up the individual class limits. Tally the items into the classes. Count the number of items in each class.

9. Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. 2 - 9 Decide how many classes you wish to use Use the 2 to the K rule. Choose k so that 2 raised to the power of k is greater than the number of data points (n) or 30. Rule of Thumb: For most data sets, you would want between 3 and 12 classes! Rule of Thumb: For most data sets, you would want between 3 and 12 classes! 2 k = 30 students 2 5 = 32, so use k = about 5 classes In this case…

10. Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. 2 - 10 Determine the class width Generally, the class width should be the same size for all classes. Class width >= Max - Min K (33.8 – 10.3)/ 5 = 4.7 Therefore, use class size of 5 hours 15.0, 23.7, 19.7, 15.4, 18.3, 23.0, 14.2, 20.8, 13.5, 20.7, 17.4, 18.6, 12.9, 20.3, 13.7, 21.4, 18.3, 29.8, 17.1, 18.9, 10.3, 26.1, 15.7, 14.0, 17.8, 33.8, 23.2, 12.9, 27.1, 16.6. 10.3, 33.8, MaxMin K=5

11. Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. 2 - 11 Minimum Value is 10.3, therefore, classes should start at 10 hours 10.0 – 14.9 15.0 – 19.9 20.0 – 24.9 25.0 – 29.9 30.0 – 34.9 10.0 – 14.9 15.0 – 19.9 20.0 – 24.9 25.0 – 29.9 30.0 – 34.9 Lower class limits will be: 10, 15, 20, etc. Classes or 10.0 to under 15 15.0 to under 20 20.0 to under 25 25.0 to under 30 30.0 to under 35 10.0 to under 15 15.0 to under 20 20.0 to under 25 25.0 to under 30 30.0 to under 35 Classes Set up the individual class limits 15.0, 23.7, 19.7, 15.4, 18.3, 23.0, 14.2, 20.8, 13.5, 20.7, 17.4, 18.6, 12.9, 20.3, 13.7, 21.4, 18.3, 29.8, 17.1, 18.9, 10.3, 26.1, 15.7, 14.0, 17.8, 33.8, 23.2, 12.9, 27.1, 16.6. 10.3, 33.8, Class Width 5 hours

12. Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. 2 - 12 15.0, 23.7, 19.7, 15.4, 18.3, 23.0, 14.2, 20.8, 13.5, 20.7, 17.4, 18.6, 12.9, 20.3, 13.7, 21.4, 18.3, 29.8, 17.1, 18.9, 10.3, 26.1, 15.7, 14.0, 17.8, 33.8, 23.2, 12.9, 27.1, 16.6. Tally the items into the classes 10.0 to under 15 15.0 to under 20 20.0 to under 25 25.0 to under 30 30.0 to under 35 10.0 to under 15 15.0 to under 20 20.0 to under 25 25.0 to under 30 30.0 to under 35 Classes Tally …and so on with the remaining hours 10.3, 13.5 14.2 13.7 14.0 12.9 Find

13. Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. 2 - 13 Count the number of items in each class 10.0 to under 15 15.0 to under 20 20.0 to under 25 25.0 to under 30 30.0 to under 35 10.0 to under 15 15.0 to under 20 20.0 to under 25 25.0 to under 30 30.0 to under 35 Hours Studying x Frequency f 7 12 7 3 1 30

14. Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. 2 - 14 Using different limits 7.5 to under 12.5 12.5 to under 17.5 17.5 to under 22.5 22.5 to under 27.5 27.5 to under 32.5 32.5 to under 37.5 7.5 to under 12.5 12.5 to under 17.5 17.5 to under 22.5 22.5 to under 27.5 27.5 to under 32.5 32.5 to under 37.5 Hours Studying x Frequency f 1 12 10 1 1 30 5 …will give you a different distribution, e.g.

15. Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. 2 - 15 Construct a Frequency Distribution Using Excel

16. Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. 2 - 16 Click on MegaStat See Click on Frequency Distributions See… Using Click on Quantitative

17. Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. 2 - 17 Using See INPUT NEEDS $A:$A 5 10 See…

18. Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. 2 - 18 See Using

19. Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. 2 - 19 Relative Frequency Distribution

20. Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. 2 - 20 R elative Frequency Distribution …shows the percent of observations in each class! Hours Studying x f 7 12 7 3 1 Relative f 30 10.0 to under 15 15.0 to under 20 20.0 to under 25 25.0 to under 30 30.0 to under 35 10.0 to under 15 15.0 to under 20 20.0 to under 25 25.0 to under 30 30.0 to under 35 Total 7/30 = 0.2333 12/30 = 0.40 7/30 = 0.2333 3/30 = 0.10 1/30 = 0.0333 30/30 =1 30/30 =1

21. Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. 2 - 21 Using different limits 7.5 to under 12.5 12.5 to under 17.5 17.5 to under 22.5 22.5 to under 27.5 27.5 to under 32.5 32.5 to under 37.5 7.5 to under 12.5 12.5 to under 17.5 17.5 to under 22.5 22.5 to under 27.5 27.5 to under 32.5 32.5 to under 37.5 Hours Studying x f Relative f 30 Total 1/30 = 0.0333 12/30 = 0.40 10/30 = 0.3333 1/30 = 0.0333 30/30 =1 1 12 10 1 1 5 5/30 = 0.1666

22. Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. 2 - 22 Stem-and-leaf Displays Each numerical value is divided into two parts: 1. the leading digits become the stem and 2. the trailing digits become the leaf. …an advantage of the stem-and-leaf display over a frequency distribution is that we retain the value of each observation! A statistical technique for displaying a set of data.

23. Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. 2 - 23 A student achieved the following scores on the twelve accounting quizzes this semester: 86, 79, 92, 84, 69, 88, 91, 83, 96, 78, 82, 85. Construct a stem-and-leaf chart to illustrate the results. Stem-and-leaf Displays

24. Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. 2 - 24 Stem-and-leaf Displays First, find the lowest score 86, 79, 92, 84, 69, 88, 91, 83, 96, 78, 82, 85. Now list the next scores with the highest leading digits. You should now have the following STEMS: 69, 78, 82, 91 6789

25. Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. 2 - 25 Stem 69 78 82 91 7 8 9 6 Split Leaf 6 9 7 8 8 2 9 1 Now, list the remaining ‘leaf’ scores! 9 3 4 5 8 26 6 All 12 Scores Stem-and-leaf Displays 86, 79, 92, 84, 69, 88, 91, 83, 96, 78, 82, 85.

26. Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. 2 - 26 The grades on a statistics exam for a sample of 40 students are as follows: Stem Leaf 3 6 8 4 1 2 7 8 5 0 1 2 5 5 8 9 6 0 1 1 1 2 5 7 8 8 8 9 7 0 0 2 5 6 6 7 8 4 6 8 8 9 9 0 2 4 6 Stem Leaf 3 6 8 4 1 2 7 8 5 0 1 2 5 5 8 9 6 0 1 1 1 2 5 7 8 8 8 9 7 0 0 2 5 6 6 7 8 4 6 8 8 9 9 0 2 4 6 How many students earned an A on this test? How many students earned an A on this test? 5 5 What is the most common letter grade earned? F F A+ = 90%-100% A = 80%-89% B+ = 75%-79% B = 70%-74% C+ = 65%-69% C = 60%-64% D = 55%-59% F = 0%-54% A+ = 90%-100% A = 80%-89% B+ = 75%-79% B = 70%-74% C+ = 65%-69% C = 60%-64% D = 55%-59% F = 0%-54% Alpha-Numeric Grading Alpha-Numeric Grading

27. Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. 2 - 27 Graphic Presentation of a Frequency Distribution

28. Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. 2 - 28 Graphic Presentation of a Frequency Distribution The three commonly used graphic forms are: Histograms Frequency Polygons or Line Charts Cumulative Frequency Distributions The three commonly used graphic forms are: Histograms Frequency Polygons or Line Charts Cumulative Frequency Distributions

29. Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. 2 - 29 The class frequencies are represented by the heights of the bars and the bars are drawn adjacent to each other. A Histogram is a graph in which the classes are marked on the horizontal axis and the class frequencies on the vertical axis Frequency Class Graphic Presentation of a Frequency Distribution

30. Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. 2 - 30 10.0 to under 15 15.0 to under 20 20.0 to under 25 25.0 to under 30 30.0 to under 35 10.0 to under 15 15.0 to under 20 20.0 to under 25 25.0 to under 30 30.0 to under 35 Hours Studying x f 7 12 7 3 1 Graphic Presentation of a Frequency Distribution 0 10 15 20 25 30 35 Hours spent studying 14 12 10 8 6 4 2 Frequency Histogram

31. Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. 2 - 31 A frequency polygon consists of line segments connecting the points formed by the class midpoint and the class frequency. 0 2 4 6 8 10 12 14 7.512.517.522.527.5 0 5 10 15 20 25 30 35 10 1520253035 A cumulative frequency distribution is used to determine how many or what proportion of the data values are below or above a certain value. Graphic Presentation of a Frequency Distribution

32. Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. 2 - 32 Making a Histogram in Excel

33. Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. 2 - 33 Using Click on DATA ANALYSIS See Click on HISTOGRAM

34. Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. 2 - 34 The upper limits of the classes you have determined Using Complete INPUTTING of DATA must now be entered from Column B (Excel calls these “bins”)

35. Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. 2 - 35 To remove the Legend on the right side… Right mouse click and Click on Clear Using To remove the spaces between the bars … Right mouse click on one of the bars and Click on Format Data Series

36. Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. 2 - 36 Using Now, Click on the Options tab; To reduce/remove the spaces between the bars Adjust the Gap width down to 0 and Click on OK.

37. Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. 2 - 37 Using Edit the size of the histogram, titles, etc as appropriate. Note that the upper limit values are included in each class – this explains the difference between this Excel Frequency Distribution and the one we did by hand.

38. Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. 2 - 38 0 10 15 20 25 30 35 Hours spent studying 14 12 10 8 6 4 2 Frequency Frequency Polygon or Line Chart for Hours Spent Studying 0 10 15 20 25 30 35 Hours spent studying 14 12 10 8 6 4 2 Frequency 10.0 to under 15 15.0 to under 20 20.0 to under 25 25.0 to under 30 30.0 to under 35 10.0 to under 15 15.0 to under 20 20.0 to under 25 25.0 to under 30 30.0 to under 35 Hours Studying x f 7 12 7 3 1 Notice that the class midpoints (the plotted points) aren’t as “user friendly” in this distribution choice.

39. Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. 2 - 39 10.0 to under 15 15.0 to under 20 20.0 to under 25 25.0 to under 30 30.0 to under 35 10.0 to under 15 15.0 to under 20 20.0 to under 25 25.0 to under 30 30.0 to under 35 Hours Studying x f 7 12 7 3 1 Cumulative Frequency Distribution For Hours Studying under 15 under 20 under 25 under 30 under 35 under 15 under 20 under 25 under 30 under 35 Hours Studying Cumulative f 19 26 29 30 7 Graph…..

40. Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. 2 - 40 Cumulative Frequency Distribution For Hours Studying 0 10 15 20 25 30 35 Hours spent studying 35 30 25 20 15 10 5 Frequency under 15 under 20 under 25 under 30 under 35 under 15 under 20 under 25 under 30 under 35 19 26 29 30 7 Hours Studying Cumulative f Cumulative f Notice that the limits are the plotted points.

41. Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. 2 - 41 Pie Bar Line Pie Bar Line … used primarily for Qualitative Data

42. Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. 2 - 42 …is useful for displaying a Relative Frequency Distribution Pie A circle is divided proportionally to the relative frequency and portions of the circle are allocated for the different groups.

43. Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. 2 - 43 Pie 200 runners were asked to indicate their favourite type of running shoe. Type Nike 92 Adidas49 Reebok37 Asics13 Other 9 # of runners selecting: Draw a pie chart based on this information.

44. Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. 2 - 44 Nike 92 Adidas 49 Reebok 37 Asics 13 Other 9 Type # # 200 % 46.0 24.5 18.5 6.5 4.5 100 Adidas 24.5% Nike 46.0% Reebok 18.5% Asics 6.5% Other 4.5% Relative Frequency Distribution for the running shoes Pie

45. Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. 2 - 45 Nike 92 Adidas 49 Reebok 37 Asics 13 Other 9 Type # # 200 % 46.0 24.5 18.5 6.5 4.5 100 Using Excel, follow the steps in the Chart Wizard to construct a Pie Chart! Pie

46. Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. 2 - 46 Bar …can be used to depict any of the levels of measurement (nominal, ordinal, interval, or ratio). (also known as a ‘column chart’) Examples of… 3-D

47. Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. 2 - 47 Bar Use bar charts also when the order in which qualitative data are presented is meaningful.

48. Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. 2 - 48 How could we chart this data? Bar

49. Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. 2 - 49 Bar Using Excel we can produce this… Other formats…

50. Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. 2 - 50 Employment Rate Canadian City Victoria 57.7 Halifax 60.5 Montreal 60.4 Sherbrooke 59.2 Quebec 59.7 Toronto 65.1 Hamilton 63.2 Kitchener 66.0 London 63.3 Thunder Bay61.0 Regina 67.4 Saskatoon 63.7 Edmonton 67.1 Vancouver 61.4 Winnipeg 66.7 Bar Halifax Montreal Sherbrooke Quebec Toronto Hamilton Kitchener London Thunder Bay Regina Saskatoon Winnipeg Edmonton Vancouver Victoria % employment 52 54 56 58 60 62 64 66 68 70 Employment Rate in Canadian Cities

51. Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. 2 - 51 Employment Rate Canadian City Victoria 57.7 Halifax 60.5 Montreal 60.4 Sherbrooke 59.2 Quebec 59.7 Toronto 65.1 Hamilton 63.2 Kitchener 66.0 London 63.3 Thunder Bay61.0 Regina 67.4 Saskatoon 63.7 Edmonton 67.1 Vancouver 61.4 Winnipeg 66.7 Bar Employment Rate in Canadian Cities % employment 52 54 56 58 60 62 64 66 68 70 Halifax Montreal Sherbrooke Quebec Toronto Hamilton Kitchener London Thunder Bay Regina Saskatoon Winnipeg Edmonton Vancouver Victoria - by Province

52. Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. 2 - 52 Bar Did any of the previous Bar Charts adequately display all the information that was provided? The following has been modified from that data found by Statistics Canada. Does it do an effective job of displaying the StatCan data?

53. Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. 2 - 53 Real estate and rental and leasing Professional, scientific and technical services Management of companies and enterprises Educational services (private sector) Health care and social assistance (private sector) Administration and support, waste management and remediation services Arts, entertainment and recreation Accommodation and food services All private sector Information and cultural industries Finance and insurance Manufacturing Wholesale trade Retail trade 0 20 40 60 80 100 % of enterprises Clustered Bar Comparison of Internet Use in 2000 and 2001 % of enterprises that use the Internet 2000 % of enterprises that use the Internet 2001 % of enterprises with a Web site 2000 % of enterprises with a Web site 2001 Data Source: Statistics Canada

54. Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. 2 - 54 Full-Time University Faculty By Gender, Canada and Jurisdictions, 1987-88 and 1997-98 Stacked Bar Canadian Full Time University Faculty 0 20 40 60 80 100 120 1987-881997-98 % of Total % males % females Data Source: Statistics Canada Total 34,651 33,925 12,829 13,910 12,650 12,095 9,172 7,817 Full ProfessorAssociate Professor 1987-881997-981987-881997-981987-88 1997-98 Other 1987-881997-98 % Male % Female 17 83 25 75 7 93 13 87 17 83 28 72 32 68 44 56

55. Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. 2 - 55 Make sure that your charts are not overly cluttered

56. Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. 2 - 56 There are four typical shape characteristics Shapes of Histograms M oda l Clas s

57. Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. 2 - 57 …a balanced effect! Both ‘balanced’ or ‘have symmetry’

58. Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. 2 - 58 … occurs when the observations are graphed as being skewed or tilted more to one side of the centre of the observations than the other. The skewness, if on the right side is said to be ‘positive’. The skewness, if on the left side is said to be ‘negative’.

59. Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. 2 - 59 Clas s M oda l A modal class is the one with the largest number of observations This is a uniModal Histogram biModal

60. Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. 2 - 60 Clas s M oda l biModal This is a biModal Histogram

61. Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. 2 - 61 Population distributions are often bell shaped. Drawing a histogram helps verify the shape of the population in question.

62. Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. 2 - 62 Line Line charts are particularly useful when the trend over time is to be emphasized Examples … 3-D In combination

63. Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. 2 - 63 Time Plot Line OSAJJMAMFJDNOSAJJMAMFJDNOSAJJMAMFJ 8.5 7.5 6.5 5.5 Month Monthly Steel Production Millions of Tons 200020012002 M o n t h l y S t e e l P r o d u c t i o n

64. Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. 2 - 64 Employment Rate in Canadian Cities 52 54 56 58 60 62 64 66 68 70 % employment Halifax Montreal Sherbrooke Quebec Toronto Hamilton Kitchener London Thunder Bay Regina Saskatoon Winnipeg Edmonton Vancouver Victoria Line Preparing a Line Chart for this type of data is not overly useful!

65. Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. 2 - 65 Employment Rate in Canadian Cities 52 54 56 58 60 62 64 66 68 70 % employment Halifax Montreal Sherbrooke Quebec Toronto Hamilton Kitchener London Thunder Bay Regina Saskatoon Winnipeg Edmonton Vancouver Victoria Line Is this combination any better for displaying the data?

66. Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. 2 - 66 f requency Polygon and Ogive f requency Polygon 50403020100 0.3 0.2 0.1 0.0 Relative Frequency Sales Ogive 50403020100 1.0 0.5 0.0 Cumulative Relative Frequency Sales Line

67. Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. 2 - 67 Test your learning … www.mcgrawhill.ca/college/lind Click on… Online Learning Centre for quizzes extra content data sets searchable glossary access to Statistics Canada’s E-Stat data …and much more!

68. Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. 2 - 68 This completes Chapter 2