Solutions CD-ROM Roland W. Meisel, B.Sc., B.Ed., M.Sc. Port Colborne, Ontario Toronto Montréal Boston Burr Ridge, IL Dubuque, IA Madison, WI New York San Francisco St. Louis Bangkok Bogotá Caracas Kuala Lumpur Lisbon London Madrid Mexico City Milan New Delhi Santiago Seoul Singapore Sydney Taipei McGraw-Hill Ryerson Limited McGraw-Hill Ryerson Mathematics of Data Management Solutions CD-ROM Copyright © 2003, McGraw-Hill Ryerson Limited, a Subsidiary of The McGraw-Hill Companies. All rights reserved. McGraw-Hill Ryerson Limited shall not be held responsible for content if any revisions, aditions or deletions are made to any material provided in editable format on the enclosed CD-ROM. ISBN 0-07-560813-8 http://www.mcgrawhill.ca 1 2 3 4 5 6 7 8 9 0 MRS 0 9 8 7 6 5 4 3 Care has been taken to trace ownership of copyright material contained in this text. The publishers will gladly take any information that will enable them to rectify any reference or credit in subsequent printings. National Library of Canada Cataloguing In Publication Meisel, Roland W McGraw-Hill Ryerson mathematics of data management Solutions CD-ROM [electronic resource] / Roland W. Meisel. For use in grade 12. ISBN 0-07-560813-8 1. Probabilities—Problems, exercises, etc. 2. Mathematical statistics—Problems, exercises, etc. 3. Permutations—Problems, exercises, etc. 4. Combinations—Problems, exercises, etc. I.Title. II. Title: Mathematics of data management. QA273.M344 2002 Suppl. 3 519 C2002-903178-8 PUBLISHER: Diane Wyman DEVELOPMENTAL EDITOR: Jean Ford SENIOR SUPERVISING EDITOR: Carol Altilia COPY EDITOR: Rosina Daillie EDITORIAL ASSISTANT: Erin Parton PRODUCTION COORDINATOR: Paula Brown ELECTRONIC PAGE MAKE-UP: Roland W. Meisel COVER DESIGN: Dianna Little COVER IMAGE: John Warden/Getty Images/Stone. Chart: From "The Demographic Population Viability of Algonquin Wolves," by John Vucetich and Paul Paquet, prepared for the Algonquin Wolf Advisory Committee. SOFTWARE DEVELOPER: Media Replication Services CONTENTS Using McGraw-Hill Ryerson Mathematics of Data Management, Solutions v CHAPTER 1 Tools for Data Management 1 CHAPTER 2 Statistics of One Variable 38 CHAPTER 3 Statistics of Two Variables 56 CHAPTER 4 Permutations and Organized Counting 77 CHAPTER 5 Combinations and the Binomial Theorem 95 CHAPTER 6 Introduction to Probability 114 CHAPTER 7 Probability Distributions 139 CHAPTER 8 The Normal Distribution 163 Course Review 193 Using McGraw-Hill Ryerson Mathematics of Data Management, Solutions McGraw-Hill Ryerson Mathematics of Data Management, Solutions provides complete model solutions to all the even-numbered questions for: - each Review of Prerequisite Skills - each numbered section, except for Achievement Check questions - each Technology Extension - each Review of Key Concepts - each Chapter Test - each Cumulative Review - the Course Review Teachers will find the McGraw-Hill Ryerson Mathematics of Data Management, Solutions helpful in planning students’ assignments. Depending on their level of ability, the time available, and local curriculum constraints, students will probably only be able to work on a selection of the questions presented in the McGraw-Hill Ryerson Mathematics of Data Management Student text. A review of the solutions provides a valuable tool in deciding which problems might be better choices in any particular situation. The solutions will also be helpful in determining which questions might be suitable for extra practice of a particular skill. In mathematics, for all but the most routine of practice questions, multiple solutions exist. The methods used in McGraw-Hill Ryerson Mathematics of Data Management, Solutions are generally modelled after the examples presented in the student text. Although only one solution is presented per question, teachers and students are encouraged to develop as many different solutions as possible. Discussion and comparison of different methods can be highly rewarding. This leads to a deeper understanding of the concepts involved in solving the problem and to a greater appreciation of the many connections among topics. Occasionally different approaches are used. This is done deliberately to enrich and extend the reader’s insight or to emphasize a particular concept. In such cases, the foundations of the approach are supplied. There are many complex numerical expressions that are evaluated in a single step. The solutions are developed with the understanding that the reader has access to a scientific calculator, and that one has been used to achieve the result. Despite access to calculators, some problems offer irresistible challenges to develop solutions in a manner that avoids the need for the calculator. Such challenges should be encouraged. There are a number of situations, particularly in the solutions to the Practise questions, where the reader may sense a repetition in the style of presentation. These solutions were developed with an understanding that a solution may, from time to time, be viewed in isolation and as such might require the full treatment. The entire body of McGraw-Hill Ryerson Mathematics of Data Management, Solutions was created on a home computer in Microsoft® Word 2000. Graphics for the solutions were created with the help of spreadsheets (Microsoft® Excel 2000 and Corel® Quattro® Pro 8), Fathom™ 1.1, and TI-83 Plus graphing calculator screens captured to the computer. Some of the traditional elements of the accompanying graphic support are missing in favour of the rapid capabilities provided by the electronic tools. Since many students will be working with such tools in their future careers, discussion of the features and interpretation of these various graphs and tables are encouraged, and will provide a very worthwhile learning experience. CHAPTER 1 Tools for Data Management Review of Prerequisite Skills Review of Prerequisite Skills Page 4 Question 2 a) f(2) = 3(2)2 − 5(2) + 2 b) g(2) = 2(2) − 1 c) f(g(−1)) = f(2(−1) − 1) = 12 − 10 + 2 = 4 − 1 = f(−2 − 1) = 4 = 3 = f(−3) = 3(−3)2 −5(−3) + 2 = 27 + 15 + 2 = 44 d) f(g(1)) = f(2(1) − 1) e) f(f(2)) = f(4) f) g(f(2)) = g(4) = f(1) = 3(4)2 − 5(4) + 2 = 2(4) − 1 = 3(1)2 − 5(1) + 2 = 48 − 20 + 2 = 8 − 1 = 3 − 5 + 2 = 30 = 7 = 0 Review of Prerequisite Skills Page 4 Question 4 The angle required for Type O is calculated by finding 46% of 360° or 166°. Similarly the angle for Type A is 155°, the angle for Type B is 29°, and the angle for Type AB is 11°. The circle graph at the right was generated using a spreadsheet. See Appendix B of the student text for instructions on using spreadsheets. Review of Prerequisite Skills Page 4 Question 6 The double-line graph below was generated using a spreadsheet. See Appendix B of the student text for instructions on using spreadsheets. You can also generate the graph using pencil and paper. Chapter 1●MHR 1 Review of Prerequisite Skills Page 5 Question 8 For similarity, all corresponding pairs of sides of two triangles must be in the same ratio. DF 4 DE 7 The ratio = = 2. The ratio = . Therefore, ∆DFE is not similar to ∆CAB. CA 2 CB 4 GH 6 3 GJ 12 The ratio = = . The ratio = . Therefore, ∆DFE is not similar to ∆GHJ. DF 4 2 DE 7 GH 6 HJ 9 GJ 12 The ratio = = 3. The ratio = = 3. The ratio = = 3. Therefore, ∆GHJ is similar CA 2 AB 3 CB 4 to ∆CAB. Review of Prerequisite Skills Page 5 Question 10 a) The area of square ABCD at the right is 1 unit × 1 unit = 1 unit2. The area of square EFGH at the right is 2 units × 2 units = 4 units2. Therefore, the ratio of the areas is 4:1. b) This ratio can be confirmed by counting the number of grid units in each square, as shown in the diagram at the right. c) The diagram at the right was constructed using The Geometer's Sketchpad®. Review of Prerequisite Skills Page 5 Question 12 To change a fraction into a decimal, divide the numerator by the denominator. To change a percent to a decimal, divide by 100. A shortcut to dividing by 100 is to move the decimal point two places to the left. When the result is rounded, use the approximately equal to sign≈. 5 23 2 a) = 0.25 b) = 0.46 c) ≈ 0.67 20 50 3 138 6 d) = 11.5 e) ≈ 0.857 f) 73% = 0.73 12 7 2 MHR●Chapter 1 CHAPTER 1 Tools for Data Management Section 1.1 The Iterative Process Practise Section 1.1 Page 10 Question 2 The gains and losses of carbon dioxide in the atmosphere are shown using arrows in the following diagram. For example, the eruption of a volcano at the right will result in a gain of carbon dioxide in the atmosphere. On the other hand, photosynthesis in green plants will remove carbon dioxide from the atmosphere, as shown in the centre of the diagram. Section 1.1 Page 11 Question 4 Answers will vary. Section 1.1 Page 11 Question 6 a) Begin with the first birthday. Compare it to the second birthday. If the second is higher than the first, interchange the two. Compare the first to the third. If the third is higher than the first, interchange the two. Continue until the first has been compared to all the birthdays in the list. Next, start with the second birthday compared to the third. If the third is higher than the second, interchange the two. Continue to the bottom of the list. Continue selecting each item and comparing it with the rest of the list until the list has been sorted from highest to lowest. Chapter 1●MHR 3 b) Compare the first birthday to the second. If the second is higher, interchange the two. Compare the second birthday to the third. If the third is higher, interchange the two. Continue down the list to the bottom. Then, start the process again from the top. When you can go through the list with no interchanges, you are finished. This method of sorting is called a "bubble sort." Section 1.1 Page 11 Question 8 Pencil-and-Paper Solution: Sketch each side of your triangle using a length that can be divided by three multiple times, such as 8.1 cm or 24.3 cm. Use this convenient measurement to trisect each side of the triangle, as shown in the diagram at the right. Continue to construct the required second iteration equilateral triangle, using a compass. Continue this process for the third iteration. If you can, perform a fourth iteration. Then, erase the internal lines to obtain the final fractal, as shown in the series of diagrams below. a) The original triangle has three sides. After the second iteration, this has increased to 12, a factor of 4. After the third iteration, it has increased to 48, another factor of 4. b) After the fourth iteration, you may expect another factor of 4, which gives you 192 sides. c) The pattern appears to work, as shown in the table at the right. Iteration Number of Sides Therefore, after the nth iteration, the number of sides is given by the expression 3×4n−1. 1 3 2 3×4=12 3 3×4×4=48 4 3×4×4×4=192 Section 1.1 Page 12 Question 10 a) i) Begin with a vertical line segment of length one unit, as shown in the diagram at the right. Branch off the top at an angle of 90° and construct lengths of one-half unit. Repeat the process from the end of each new branch, making each succeeding set of branches one half the length of the previous set, until the desired fractal tree is drawn. 4 MHR●Chapter 1