MTH-1102-3

STATISTICS AND PROBABILITY

STAT IST ICS AND PROBABILITY

MTH-1102-3

Learning Guide

Common Core Basic Education Curriculum (Secondary Cycle One) Mathematics, Science and Technology Program of study: Mathematics

Secondary I MTH-1101-3 Finance and Arithmetic MTH-1102-3 Statistics and Probability Secondary II MTH-2101-3 Algebraic Modelling MTH 2102-3 Geometric Representations and Transformations

Statistics and Probability This learning guide was produced by the Société de formation à distance des commissions scolaires du Québec (SOFAD). Production Team (French Version) Project Coordinator: Project Coordinator (initial version):

Ronald Côté (SOFAD) Jean-Paul Groleau (SOFAD)

Authors: Content Revisor:

Alain Malouin Gilles Gascon Gilles Gascon

Linguistic Revisors:

Michelle Côté Johanne St-Martin

Graphic Design:

Serge Mercier

Desktop Publishing (initial):

P.P.I. inc.

Cover Page:

Marc Tellier

Production Team (English Version) Project Coordinator:

Jean-Simon Labrecque (SOFAD) Valerie Vucko (i-Edit)

Translator:

Rhonda Sherwood

Content Revisor:

Aalia Persaud

Typesetting:

Lorraine Brown

First Edition:

December 2012

Proof Reading:

Claudia Fulviis

Despite the following statement, SOFAD authorizes all adult education centres that use this learning guide to reproduce the scored activities available at http://cours1.sofad.qc.ca/ressources. © SOFAD All rights for translation and adaptation, in whole or in part, reserved for all countries. Any reproduction by mechanical or electronic means, including microreproduction, is forbidden without the written permission of a duly authorized representative of the Société de formation à distance des commissions scolaires du Québec. This work is financed by the Ministère de l’Éducation, du Loisir et du Sport du Québec. Part of this financing comes from the Canada-Quebec bilateral agreement related to minority language education and second languages instruction. Legal Deposit - 2012 Bibliothèque et Archives nationales du Québec Library and Archives Canada ISBN: 978-2-89493-432-6

December 2012

tdm1 TAB L E OF C ONT E NT S

Table of Contents

This is a preview

of:

n; and - the introductio Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . rn . . .g . . . . . . . . . . . . . . . . . . 7 - the first lea in Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 situation. Organization and Use of the Guide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Additional Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Work Pace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Instructional Support . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Evaluation for Certification Purposes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Essential Knowledge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

Part 1 – Data Collection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Situation 1 – Participating in an Election . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Activity 1.1 – Preparing for an Election . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Activity 1.2 – Preparing for the Election Campaign . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Integration Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Review Activity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17 17 19 27 32 36

Situation 2 – The Sizable Social Issue of Educating the Population . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Activity 2.1 – Conducting a Sample Survey and a Census . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Activity 2.2 – Going Back to School for 16-to 24-Year-Olds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Integration Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Review Activity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39 39 42 51 57 62

Situation 3 – Gathering Data: Why and How? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exploration Activity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Activity 3.1 – Verifying Claims About the Honda Civic’s Popularity in Canada . . . . . . . . . . . . . Activity 3.2 – Verifying Statements in an Article on Eye Colour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Integration Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Review Activity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Instructions for completing Scored Activity 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

65 65 66 68 74 79 81 83

Part 2 – Statistical Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 Situation 4 – Using Tables to Organize Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 Exploration Activity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 Activity 4.1 – Using statistics in Sports . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 Activity 4.2 – Organizing Data to Better Understand It . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 Integration Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 Review Activity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 Situation 5 – Graphically Representing Statistical Distributions . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exploration Activity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Activity 5.1 – Using Circles to Present Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Activity 5.2 – Evolving Over Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Integration Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Review Activity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . © S O FAD

121 121 122 123 131 138 143 5

tdm2 M T H - 11 0 2 -3 – S TAT I S TICS AN D PROBABILITY

TABLE OF CONTENTS

ANSWER KEY

Situation 6 – Assigning Attributes to Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exploration Activity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Activity 6.1 – Using Math in Your Position as Assistant Coach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Activity 6.2 – Numbers or Stars? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Integration Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Review Activity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Instructions for completing Scored Activity 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

147 147 148 149 161 171 174 177

Part 3 – Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 Situation 7 – Rolling, Tossing and Drawing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exploration Activity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Activity 7.1 – Using Dice to Study Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Activity 7.2 – Depending on Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Integration Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Review Activity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

181 181 182 184 205 211 215

Situation 8 – Going from Theory to Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exploration Activity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Activity 8.1 – Differentiating Practice from Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Activity 8.2 – Using Statistics to Calculate Probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Integration Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Review Activity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

219 219 220 222 227 231 233

Situation 9 – Living Your Dreams to the Max! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exploration Activity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Activity 9.1 – Choosing Lotto 6/49 or Lotto Max . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Activity 9.2 – Saying Yes to Extra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Integration Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Review Activity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Instructions for completing Scored Activity 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

237 237 238 239 248 255 259 261

Self-Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 Answer Key . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 – Participating in an Election . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 – The Sizeable Social Issue of Educating the Population . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 – Gathering Data: Why and How? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 – Using Tables to Organize Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 – Graphically Representing Statistical Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 – Assigning Attributes to Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 – Rolling, Tossing and Drawing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 – Going from Theory to Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 – Living Your Dream to the Max! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Self-Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

277 279 283 287 290 297 303 310 322 326 335

Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339 Feedback Sheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343 6

© SO FA D

TABLE OF CONTENTS

ANSWER KEY

Introduction

Introduction

S

tatistics is a field of mathematics that involves several aspects. First, there is the observation of facts that have been compiled and classified according to the rules of this field. Then, the graphical representation of these facts allow them to be analyzed and compared with theoretical

models. Finally, this approach leads to interpreting these facts in order to make decisions or predictions. The Statistics and Probability course is divided into three parts. In the first part, “Data Collection,” you will learn how to distinguish between a population and a sample, determine whether it is better to conduct a study, a census or a sample survey, and to indicate whether a chosen sample is representative of the population. You will also learn how to identify possible sources of bias in a statistical study and how to create data collection forms. In the second part, “Statistical Distributions,” you will learn the difference between a quantitative and qualitative statistical variable, whether a quantitative variable is discrete or continuous, and how to construct frequency and relative frequency tables. You will also learn how to calculate the mean using the frequency or relative frequency of a statistical distribution, how to determine the mode of a distribution of data containing qualitative data and how to identify the minimum, maximum and range of a distribution of data. You will also learn how to construct circle graphs and broken-line graphs. In the third part, “Probability,” you will learn how to construct a tree diagram, how to represent the sample space, and how to calculate the probability of a given event using random experiments made up of dependent or independent events. You will also learn how to identify whether a probability is theoretical or experimental, how to calculate theoretical and experimental probabilities, how to construct a probability tree diagram and how to calculate probability for an experiment involving no more than three steps.

© S O FAD

7

M T H - 11 0 2 -3 – S TAT I S TICS AN D PROBABILITY

TABLE OF CONTENTS

ANSWER KEY

Organization and Use of the Guide This learning guide is organized according to the main characteristics of individualized learning and the principles of learning through concrete situations. It is therefore designed to: • • • •

make you as active a participant as possible, make you responsible for your own learning, accommodate your personal work pace, and allow you to make the most of your experience and knowledge.

As you make your way through the course, you will be able to recognize your successes or failures, determine the reasons for them and identify what you can do to continue learning. You can consult your tutor at any time during the course. If you find a particular topic especially difficult, don’t hesitate to ask him or her for help. Your tutor will be glad to provide you with the advice, guidance, constructive criticism and feedback you need.

Learning Situations This guide contains nine learning situations designed to help you learn new concepts and apply them competently. All the learning situations are organized in the same way. Each one begins with an introduction that describes the assignment you will be required to carry out at the end of the learning situation. Each learning situation is divided into learning activities. In each activity, you will be presented with a problem and questions. As the questions deal with new concepts, you may not be able to provide satisfactory answers to all of them; however, you are encouraged to do your best. The correct answers and additional explanations are given after each set of questions. It is important that you try to understand all of the new concepts that are explained to you. Following the explanations is a summary of the new concepts as well as exercises. These exercises will allow you to test your understanding of the newly learned concepts. The answers to these exercises are found at the end of the guide. You can then do the integration exercises dealing with all of the concepts covered in the learning situation. The answers to these supplementary exercises can also be found at the end of the guide. Once you have completed the integration exercises, you can do the review activity at the end of each learning situation. It will require you to put your reasoning and communication skills into practice. Certain learning situations end with a list of new knowledge. You can refer to this list as needed to make sure you have understood all of the concepts.

8

© SO FA D

TABLE OF CONTENTS

ANSWER KEY

int rod uc t i o n

Throughout this guide, different pictograms will guide you in your work.

Remem

The paper clip attached to the corner of a page indicates

ber

items that are important to remember.

The binder clip indicates the last pages of each learning List of New Knowledge

situation. These pages present a summary of the essential knowledge covered in the learning situation.

Tip

A light bulb appears in boxes containing tips to make your work easier.

Reminder

Boxes with a pushpin contain reminders of concepts covered in previous courses.

In the Glossary

Did you know

?

Words and expressions with dotted underlining are defined in the glossary at the end of the guide. The magnifying glass indicates additional information. This information is not, strictly speaking, part of the course material, and none of the questions on the final examination will deal with the information found in these sections.

The last section of the guide summarizes what you have covered in the course. It also contains a selfevaluation activity to help you determine whether you fully understand what you have learned and whether you are ready to sit for the final examination. This section also includes the answer key for the exercises in the guide and the self-evaluation activity as well as the glossary.

© S O FAD

9

TABLE OF CONTENTS

M T H - 11 0 2 -3 – S TAT I S TICS AN D PROBABILITY

ANSWER KEY

Scored Activities The Statistics and Probability guide is accompanied by 3 separate scored activities. You are required to do these activities after learning situations 3, 6 and 9 and send them to your tutor for correction. Evaluation Situations Scored Activity 1

Topics Covered Data Collection (Learning Situations 1, 2 and 3)

Scored Activity 2

Statistical Distributions (Learning Situations 4, 5 and 6)

Scored Activity 3

Probabilities (Learning Situations 7, 8 and 9)

Self-Evaluation Completing the self-evaluation activity is an important step in preparing for the final examination. Be sure to do it without referring to the learning guide or the answer key. Then compare your answers with those in the answer key and review or continue, as necessary. The self-evaluation grid that accompanies this activity will help you identify the concepts you have mastered and those you should review before sitting for the certification exam. Instructions regarding the concepts to be reviewed are also given in the grid.

Answer Key The answer key for the exercises in the guide is found after the self-evaluation activity. Refer to it after each set of exercises to make sure you have fully understood all of the concepts, before continuing the activity or going on to the next learning situation. This section also includes the answer key for the self-evaluation activity.

Glossary The glossary found at the end of the guide gives the definitions of the terms with dotted underlining found in the learning situations. These terms are listed in alphabetical order. Don’t hesitate to refer to the glossary to help you better understand the terms you encounter in the guide.

10

© SO FA D

TABLE OF CONTENTS

ANSWER KEY

int rod uc t i o n

Additional Materials Have all the materials you need handy. • Your learning guide and a notebook in which to summarize all of the key concepts found in the list of essential knowledge given in this introduction. • A dictionary, a calculator, a pencil to write your answers and notes in your guide, a ballpoint pen to correct your answers, a highlighter to underline key concepts, an eraser, etc. For certain exercises, you will need a geometry set (a ruler graduated in centimetres, a protractor, a set square and a compass).

Work Pace Here are some tips on how to organize your study time. This course involves approximately 75 hours of work. • Draw up a study schedule, taking into account your availability and needs, as well as your family, work and other obligations. • Try to devote a few hours a week to your studies, preferably setting aside two hours at a time. • Stick to your schedule as much as possible.

Instructional Support Your tutor will help you throughout this course: he or she will be available to answer any questions you may have, and will correct your scored activities. This is the resource person you must call if you need any kind of help. If his or her availability and contact information have not been provided with this guide, you will receive them shortly. Do not hesitate to consult your tutor if you are having any difficulties with the explanations or the exercises, or if you need encouragement to continue with your work. Make a note of any questions in writing and contact your instructor at the appropriate time and, if necessary, write to him or her. Your tutor will guide you throughout the learning process and provide you with advice, constructive criticism and feedback that will help you succeed in your studies.

© S O FAD

11

TABLE OF CONTENTS

M T H - 11 0 2 -3 – S TAT I S TICS AN D PROBABILITY

ANSWER KEY

Evaluation for Certification Purposes In order to earn the 3 credits for this course, you must obtain a mark of at least 60% on the final examination that will be held in an adult education centre. To be able to write this examination, you should have an average of at least 60% on the scored activities accompanying this guide. The final examination for the Statistics and Probability course has two sections. Both sections are included in the same booklet and must be completed during the same exam session. The first section of the examination is intended to evaluate your knowledge of the material. It consists of short-answer questions and questions requiring more elaborate answers. The second section is designed to evaluate competencies. This section consists of problems presented in one or more real-life situations. Authorized materials for both sections of the examination are: • a regular or scientific calculator; • a geometry set; • a graduated ruler; • a checklist inserted in the examination. The examination lasts 2 hours 30 minutes.

Essential Knowledge The knowledge listed below is considered compulsory because you will need it to deal with several situations in the class Predicting Random Events. Learning Situations

Essential Knowledge

1. Participating in an Election

Data Collection

2. The Sizable Social Issue of

• Population

Educating the Population 3. Gathering Data: Why and How?

• Representative sample • Sample survey, census and study • Sources of bias • Sampling methods (random and systematic) • Creating data collection forms • Establishing a representative sample or defining a population • Collecting data

12

© SO FA D

TABLE OF CONTENTS

ANSWER KEY

4. Using Tables to Organize Data 5. Graphically Representing Statistical Distributions 6. Assigning Attributes to Data

int rod uc t i o n

Statistical Distributions • Data (continuous quantitative) • Maximum, minimum, range • Reading statistical representations (relative frequency tables, broken-line graphs and circle graphs) • Constructing relative frequency tables • Graphing statistical distributions (broken-line and circle graphs) • Determining the mode of a statistical distribution involving qualitative data • Calculating the mean using frequencies or relative frequencies from a statistical distribution

7. Rolling, Tossing and Drawing 8. Going from Theory to Practice 9. Living Your Dreams to the Max!

Probability • Chance • Random experiment • Event • Equiprobable and non-equiprobable events • Sample space • Favourable outcomes • Theoretical probability and experimental probability • Determining experimental probability • Calculating theoretical probability • Probable, certain and impossible events • Relationship between two events (dependent, independent, complementary, compatible, incompatible) • Calculating the number of possible outcomes and the number of favourable outcomes • Enumeration for an experiment involving no more than three steps, using representation methods (grids, tables, tree diagrams, networks)

© S O FAD

13

TABLE OF CONTENTS

ANSWER KEY

Part 1

Data Collection Collecting data is the first step in carrying out a statistical study. This step is crucial for everything that follows in the study. In fact, the credibility of the study’s published results rests entirely on the reliability of the data that is collected. When it comes to selecting the respondents (sample), ensuring the clarity of the questions they will be asked or choosing the method to be used for collecting the data, it is important to proceed in a rigorous and orderly manner. This is exactly what you will have to do in this first part. The three learning situations in this first part deal with the basic concepts and specific vocabulary used in the study of statistics.

1

Participating in an Election

2

The Sizable Social Issue of Educating the Population

3

Gathering Data: Why and How?

After completing these three situations, you will be able to perform data collection in preparation for a statistical study.

© S O FAD

15

TABLE OF CONTENTS

1

ANSWER KEY

Participating in an Election

Introduction

H

ave you ever voted? Did you know that the Municipal Elections Act stipulates that general elections shall be held in all municipalities of Québec on the first Sunday in November every four years? If you have never voted, you’ll soon have

the chance. As a citizen aged 18 years or older, you will be asked to exercise this right to choose the mayor of your city. You will also be asked to vote for the parties that will form the provincial and federal governments. But how do you go about making an informed choice? Television news programs and newspapers regularly report on public voting intentions and trends. But are the statistical tools they use to communicate this information reliable? Is the data collected being properly interpreted? In this first learning situation, you will learn how to distinguish between the concepts of sample and population. You will have to

describe

the

specific

characteristics

of various types of statistical studies: census, sample survey and study. You will also have to determine the necessary criteria for obtaining a representative sample of a given population in order to conduct a reliable sample survey. You will be asked to use the sampling method best suited to a given situation. Lastly, you will learn to identify the sources that can bias a sample survey by briefly describing the elements that can introduce false data.

© S O FAD

17

M T H - 11 0 2 -3 – S TAT I S TICS AN D PROBABILITY

Your task

TABLE OF CONTENTS

ANSWER KEY

Your brother-in-law, a well-known businessman who is liked by everyone in your town, tells you that he has decided to run in the upcoming election—not for the position of counsellor, but for that of mayor. Since you have been his faithful business partner for several years now, he asks you to help him in this venture as a researcher, analyst and advisor. All on a voluntary basis, of course! Your first task, therefore, is to distinguish not only between a sample survey, a census and a study, but also between a population and a sample. Moreover, you have to be able to eliminate any sources of bias by using the correct sampling method.

Exploration Activity We often think that mathematicians, particularly statisticians, use specific and complicated terminology; and yet, they use the same ordinary words you often use. The difference mainly lies in the precision mathematicians assign to the definition of words to avoid any possible confusion. • You read in your favourite newspaper that Québec’s population has surpassed 8 000 000 people. • A household products representative gave you a sample of an extraordinary soap. Are the words population and sample familiar? Are you able to explain their meaning? Of course you are! Statistics also uses these two words, but gives them a much more specific definition. Statistics will teach you that a population is a group of people or entities with one or more characteristics in common. A sample (such as the soap the representative gave you) should represent a population as accurately as possible. • Researchers conduct a study to determine the effects of a new drug on patients. • A caller asks you if you would be kind enough to answer a few questions for a sample survey. • The government has changed the census rules. Census, sample survey and study are words you already know that are also used in statistics. Statistics simply describes their meaning and the specific rules for their use, more precisely. If you looked up the definitions of these five words in a dictionary you would see that their general meaning is the same, but that mathematics, and statistics in particular, uses them in a much more restricted and precise way. The following activities will help you become familiar with this terminology and, more importantly, to use it properly.

18

© SO FA D

TABLE OF CONTENTS

ANSWER KEY

Sit uation 1 – PARTIC IPATING IN AN E L E C TION

Activity 1.1 – Preparing for an Election Goal

• To choose an appropriate sampling method • To master the relevant terminology

For starters, your brother-in-law gives you the voter list for your municipality from the last census conducted in the country. This allows you to obtain the names of all individuals who will be 18 years of age or older on election day. The first mandate he gives you is to conduct a sample survey using this list to quickly find out the voting intentions of the population. Since he wants the results to be fairly accurate, he asks that you avoid any sources of bias to ensure your sample is representative of the population. To do this, you will have to choose an appro-

For Mayor

priate sampling method. But you can’t decide between the random sampling method and the systematic sampling method. Moreover, your results must be accurate and obtained quickly; your brother-in-law has not asked you to do a study. You’re starting to wish you had never offered to help your brother-in-law! You are not really clear on the distinction between the existing types of statistical studies. What is a representative sample? What are these sources of bias your brother-in-law mentioned? What is the difference between the random sampling method and the systematic sampling method? Let’s start by looking at the difference between a sample and a population.

• A population is the set of all individuals or elements included in a study. • A sample is a small group of individuals or elements selected to represent the target

mber

Reme

population as faithfully as possible.

© S O FAD

19

M T H - 11 0 2 -3 – S TAT I S TICS AN D PROBABILITY

TABLE OF CONTENTS

ANSWER KEY

Since there are 16 000 people who are 18 years of age or older in the town where you live, it’s clear that to satisfy your brother-in-law’s request, you will have to make do with a sample to do your survey. When we question the entire population, our statistical study is a census. Since we do not need to be experts to conduct the statistical study, we will be conducting a sample survey rather than a study.

• A census is a statistical study that includes all the individuals or elements of a population.

mber

Reme

• A sample survey is a statistical study that includes all the individuals or elements of a sample, from which we draw conclusions about the population. • A study is an in-depth statistical study involving experts to obtain specific information through various data collection techniques.

A research team wants to know the effect of a new drug on diabetics in Québec. Will it commission a sample survey or a study ? A study should be commissioned. Even though both statistical studies include all the individuals or elements of the exact same sample, the study mentioned above is conducted by experts; this is what makes it different from a sample survey. Test your new knowledge with the following exercises.

Exercises for Activity 1.1 1.1

From among the examples of groups of individuals or elements described below, determine which ones form a population and which ones form a sample. Check (√) the box that corresponds to your choice.

20

Population Sample

a) The hockey players in the National Hockey League



b) The letters of the alphabet



c) The students enrolled in an adult education centre



d) People randomly selected to answer a questionnaire



e) Customers who came on the opening day of a hair salon



f) People having to vote in a referendum



g) The passengers on a plane



h) The winners of a lottery



© SO FA D

TABLE OF CONTENTS

1.2

ANSWER KEY

Sit uation 1 – PARTIC IPATING IN AN E L E C TION

Indicate whether a study, a census or a sample survey should be conducted in each of the following situations. a) A car dealership wants to know what percentage of its customers from the last year are satisfied with its customer service.

b) We want to know what percentage of the 2500 employees working at a large company are single.

c) We want to get people’s opinion before making a decision on the construction of a sports centre in a small village.

d) The manager of a sports store wants to know how many ladies’ ski boots, in the colour white, are currently available.

e) We want to know how many students enrolled in a vocational centre are receiving employment insurance benefits.

f) A sociologist wants to know what percentage of Québec women are employed full-time.

g) A team of medical specialists would like to know the effects of an experimental drug in people suffering from acid reflux.

vvv

Did you know

?

The word “statistic” comes from the Latin word status (statement, state or situation). When used in the singular, the word “statistic” refers to an observation or piece of data from a study of a large quantity of numerical data. When used in the plural, the word “statistics” refers to the branch of mathematics that deals with the collection and analysis of data.

© S O FAD

21

M T H - 11 0 2 -3 – S TAT I S TICS AN D PROBABILITY

TABLE OF CONTENTS

ANSWER KEY

You have the voter list of the 16 000 people in your town. Your brother-in-law wants you to produce a representative sample of this population.

• A sample is said to be representative if it is a faithful portrait of the population; that is, when the sample has all the same characteristics as the population.

mber

Reme

• A sample that is not representative of a population is said to be biased.

Statisticians have developed several methods for ensuring a sample is representative of the population. The most commonly-used ones are systematic sampling and random sampling.

• The systematic sampling method requires having a list of all the elements of the population. This method consists in randomly choosing a starting point and then

mber

Reme

using the same procedure to select the other individuals in the sample. • The random sampling method consists in choosing the elements of a sample randomly, without any guidelines.

Using the list of 16 000 names your brother-in-law gave you, you decide to select every 25th person, starting with the first name on the list. Is this method random or systematic? Since you established a specific rule (every 25th name), we can say that the sampling method is systematic. Moreover, since you will be selecting 640 people ( 16 000 ÷ 25 = 640 ), we can say that the sample will be large enough to be representative.

Did you know

?

The size of a sample does not depend on the size of the population, but rather on the desired degree of precision. Obviously, a larger sample will lead to a greater degree of precision in the sample survey results. When only a few characteristics are being studied, a sample of 500 to 1000 individuals is sufficient for obtaining results with a reliability of 95%.

22

© SO FA D

TABLE OF CONTENTS

ANSWER KEY

Sit uation 1 – PARTIC IPATING IN AN E L E C TION

When you ask the town for an official voter list, the clerk tells you there is a list that specifies voters’ name and gender by borough. Table 1.1 summarizes this information. Table 1.1 – Voter List by Borough Borough

Men

Women

De Sève

1500

500

Verschelden

1000

500

Morris

2000

1000

Chapleau

500

500

Lonergan

1000

1000

Ducharme

1500

1000

Blanchard

1000

1000

Marie-Thérèse

1500

500

Total

10 000

6000

You take advantage of this list to improve the representativeness of your sample. You notice that the men:women ratio is not the same for all boroughs.

Reminder A ratio is the quotient (result of a division) of two quantities being compared. It can be expressed in the form of a fraction. The ratio of 1500 to 500, which is written as 1500:500, gives

1500 3 = 500 1

oral t c e El List

© S O FAD

23

TABLE OF CONTENTS

M T H - 11 0 2 -3 – S TAT I S TICS AN D PROBABILITY

ANSWER KEY

In the De Sève borough, the ratio is 3 to 1. This means you should choose 3 times more men than women in this borough to ensure your sample is representative. To respect the rule you established regarding the size of the sample (choosing every 25th name), you should therefore choose 60 men and 20 women in the De Sève borough. In the Verschelden borough, you will need 40 men (1000 25) and 20 women (500 25). For the borough of Morris, you’ll need 80 men and 20 women. The following table presents the number of people you will need for your new sample by borough and by gender. Table 1.2 – Voter List by Borough Borough

Men

Women

De Sève

60

20

Verschelden

40

20

Morris

80

40

Chapleau

20

20

Lonergan

40

40

Ducharme

60

40

Blanchard

20

40

Marie-Thérèse

60

20

Total

400

240

Now that you know the number of people you will need from each borough for your sample survey, you can: • Apply the rule of selecting every 25th name in each category; your sampling method will therefore be systematic. or • Randomly select 25 names in each category, without any specific rule; your sampling method will therefore be random. Test your new knowledge with the following exercises.

24

© SO FA D

TABLE OF CONTENTS

ANSWER KEY

Sit uation 1 – PARTIC IPATING IN AN E L E C TION

Exercises for Activity 1.1 1.3

In each of the following situations, indicate whether the selected sample from which we want to obtain specific information is representative of the population. Explain why. a) The coach of an NHL hockey team wants to increase the number of practices. To find out what the players think of this idea, he asks 5 players on the team.



b) The principal of a secondary school with 1800 students, of whom 500 are girls, wants to change the rules regarding the dress code for boys. To do this, she asks 300 students, 150 boys and 150 girls.

c) The director of human resources at a large company wants to know what percentage of the company’s 3000 employees are smokers so that a shelter can be built outside the factory. He asks 500 randomly selected employees if they are smokers.

d) A car manufacturer wants to know the satisfaction level of Québec men and women who have purchased one of its vehicles. It commissions a sample survey of 3000 people living in Québec.

e) The Formula 1 Grand Prix du Canada organizing committee wants to know the level of satisfaction of the 300 000 people who are attending the three-day event, regarding the number and location of restaurants on site. A questionnaire is handed out to every fifth customer who comes to the ticket booth.

© S O FAD

25

TABLE OF CONTENTS

M T H - 11 0 2 -3 – S TAT I S TICS AN D PROBABILITY

1.4

ANSWER KEY

Identify the sampling method used in each of the following cases. a) Flight attendants on a flight from Montréal to Paris want to know if the passengers on the Boeing 747 enjoyed the supper that was served. They decide to ask everyone sitting in an aisle seat.

b) The mayor of a small town in the Laurentians wants to know citizens’ level of satisfaction regarding his accomplishments during his last term. He randomly chooses 300 people from the voter list to answer a questionnaire.

c) A large sporting goods chain wants to know if the customers who bought a bike are satisfied with their purchase. From a list of these customers, it picks every 10th name.

d) The Société des alcools du Québec wants to improve its website to make searching for products easier. It takes the list of people who subscribe to its promotional activities and contacts one out of every 10 subscribers.

e) A cable company has just added three new channels for subscribers. After a trial period of three months, it wants to know how many clients would want to keep them if they had to pay a small fee. The company then randomly selects 500 of its subscribers to answer a short questionnaire.

vvv Secondary V students with a science profile at a private school are divided into eight classes of 30 students. The guidance counselor wants to know how many of these students intend to pursue studies in engineering. To form the sample, she randomly chooses two students from each of these eight classes; 16 students in total are questioned. Do you think the sample obtained is representative of the population? The random sampling method is correct, but because the sample contains only 16 students, we can say that it is biased and not representative of the population. In the next activity, you will see that there are many sources that can skew the results of a sample survey.

26

© SO FA D

TABLE OF CONTENTS

ANSWER KEY

Sit uation 1 – PARTIC IPATING IN AN E L E C TION

Activity 1.2 – Preparing for the Election Campaign Goal

To prepare for the collection of data.

It’s all good and fine to want to enter politics and to be popular and well-liked by your community, but without the right ideas to attract voters, the candidate of the opposing party could win with a catchy slogan and enticing ideas. To avoid this trap, your brother-in-law tells you that you now need to perform data collection and that you will first need to create a data collection form which should be exempt of all sources of bias. Since your brother-in-law has not given you any specific instructions, several questions come to mind. • Will you collect data by telephone interview, in-person interview, mailed questionnaire or through electronic means? • Can the attitude of the person conducting the sample survey bias the results? • Will the rejection of too many questionnaires make the sample non-representative? • Are there other sources of bias that can influence the results of a sample survey? The following key concepts will help clarify matters.

• The sources of bias that can skew the results of a sample survey when creating a data collection form are:

mber

Reme

• a) poor wording of the goal of the sample survey; • b) poor wording of questions. • The sources of bias when performing data collection are: • a) a sample that is not representative of the population; • b) a poor attitude in the person conducting the sample survey; • c) misinterpretation of results; • d) misrepresentation of results.

© S O FAD

27

TABLE OF CONTENTS

M T H - 11 0 2 -3 – S TAT I S TICS AN D PROBABILITY

ANSWER KEY

Since the goal of our sample survey is to elicit interest in voters, we have to ask questions that will tell us what issues citizens would like the future mayor to discuss, while avoiding any biased questions. For example, the question: “Would you like sports and recreation services to be improved?” would be better worded as: “Which municipal service should be improved?” while offering a choice of answers similar to the following: • Arts and culture

• Public safety

• Municipal library

• Technical services

• Communication

• Sports and recreation

• General administration

• Purification station

• Finances

• Public works

• Laws and regulations

• Urban planning

• Human resources It’s also very important to word questions properly to avoid sources of bias. Questions should: • 1st use vocabulary that respondents can easily understand; • 2nd be neutral; that is, they should not suggest an answer; • 3rd focus on only one aspect; • 4th provide a choice of all possible answers while ensuring the respondent can give only one answer; • 5th be varied (multiple choice, essay, closed; that is, can only be answered by yes or no).

If data collection is done through interviews, the attitude of the person conducting the sample survey should not influence the respondent; nor should this person suggest answers, or show approval or disapproval of the topic being addressed by the sample survey. When interpreting the data, you should be careful when the time comes to eliminate certain results. If too many people were unable to make a clear choice, and if eliminating these results makes our sample too small, it will no longer be representative of the population. For example, we ask voters whom they intend to vote for as mayor in the next election and get the following results:

28

© SO FA D

TABLE OF CONTENTS

ANSWER KEY

Sit uation 1 – PARTIC IPATING IN AN E L E C TION

Table 1.3 – Voting Intentions for Mayor Candidate

Number of Votes

Result (%)

Marcel Vallée

280

38%

Guy Lamontagne

295

40%

Do not know

165

22%

If we decide to reject the 165 undecided votes, we may reduce the size of our sample by too much and it will no longer be representative of the population. In addition, since these undecided voters represent 22% of all votes, we cannot draw a conclusion from this sample survey because the gap between the two candidates is too small (2%) and the undecided voters (22%) could change the outcome. Lastly, a misrepresentation of the results obtained during data collection can also be a source of bias. Imagine that in the previous example you decide to present the results in the following manner: Table 1.4 – Voting Intentions for Mayor Undecided Voters Not Taken Into Account Candidate

Number of Votes

Result (%)

Marcel Vallée

280

49%

Guy Lamontagne

295

51%

By omitting the undecided voters and by calculating the results without this number, we get a different view of the situation. Before going any further, complete the following exercises to see if you have clearly understood the different sources of bias.

Did you know

?

Two Frenchmen, Blaise Pascal (1623–1662) and Pierre de Fermat (1601–1665), were the first to become interested in probabilities and went on to create probability theory. Today, this theory is the basis of all statistical studies and is used by all statisticians.

© S O FAD

29

M T H - 11 0 2 -3 – S TAT I S TICS AN D PROBABILITY

TABLE OF CONTENTS

ANSWER KEY

Exercises for Activity 1.2 In each of the following statistical studies, determine the possible sources of bias. 1.5

A plant that manufacturers coffee makers removes the last 10 coffee makers made each day from its assembly line for quality control.

1.6

A mayor wants to impose a special tax on property owners in one of the city’s districts for sidewalk repairs. To find out whether citizens support the project, he holds a referendum of the city’s entire population.

1.7

A company that makes high-end chips conducts a sample survey of 800 people to find out what their favourite chips are. Surprised to find that another brand is almost as popular, it decides to weight the result; that is, by multiplying the votes favouring the company conducting the sample survey by 2.

1.8

A pharmaceutical company is conducting a study of 300 people suffering from migraines and invites them to test a new drug. It plans to present the results at an upcoming conference. Even though only half of the participants were able to take the medication, it decides to disclose the results of the study as planned.

1.9

A restaurant in the Îles-de-la-Madeleine that specializes in lobster commissions a sample survey of 500 tourists to find out whether they prefer steamed or broiled lobster. The survey firm receives the results at its Montréal office where it converts the responses into numerical codes as required by its computer program, but the codes are accidentally inverted.

30

© SO FA D

TABLE OF CONTENTS

1.10

ANSWER KEY

Sit uation 1 – PARTIC IPATING IN AN E L E C TION

A travel agency in the Montréal area specializing in bus trips to New York has been having a hard time filling its buses for this destination. To encourage Montrealers to travel to New York, it conducts a sample survey of 500 people randomly selected on the streets of Montréal, asking them the following question: “Among large American cities, is New York your favourite destination?”

vvv

Did you know

?

The sampling methods we have just studied belong to the same family called probability sampling method; that is, individuals are randomly selected. There is a second family of non-probability sampling methods in which individuals are not randomly chosen. Among the non-probability sampling methods, the main ones are: • quota sampling; • accidental sampling; • voluntary sampling. However, these methods are often challenged by experts because neither the reliability of the sampling nor the possible sources of bias can be determined.

To speed up the nomination of the student who will serve as a representative on the governing board of an adult education centre, the director gathers all the students together in the centre’s cafeteria and decides to choose the candidate based on the applause following each candidate’s speech. In your opinion, is this approach correct?

If you claimed that the data collection method is biased, you have clearly understood the concept of source bias. In fact, if one of the candidates had asked his supporters to applaud very loudly in an attempt to mislead the audience, he would surely have been chosen as the most popular candidate. However, if voting had been done by conventional written ballot, the result might have been different. The following integration exercises will allow you to put all the knowledge you have acquired in this learning situation into practice.

© S O FAD

31

M T H - 11 0 2 -3 – S TAT I S TICS AN D PROBABILITY

TABLE OF CONTENTS

ANSWER KEY

Integration Exercises 1.11

Complete the following sentences. a) The set of all individuals or elements included in a study is called

.

b) Conducting a statistical study of all the individuals or elements of a population, amounts to conducting

.

c) Consulting experts during a statistical study to obtain specific information is called conducting

.

d) When we choose a small group of individuals or elements to represent the target population as faithfully as possible, our statistical study is on

.

e) Selecting a representative sample of a population to find out people’s opinion or to gather information on certain characteristics, is called conducting 1.12

.

In each of the following situations, indicate whether it is better to conduct a study, a sample survey or a census. Give the reasons for your choice of statistical study. a) A very large soft drink company has recently changed its cola recipe and wants to know if people prefer the new recipe or the old one.

b) A team of radiologists want to know the side effects of X-rays on people with asthma.

c) Employees do a monthly inventory of goods at a computer store.

32

© SO FA D

TABLE OF CONTENTS

ANSWER KEY

Sit uation 1 – PARTIC IPATING IN AN E L E C TION

d) A non-profit organization wants to know if it should sell cookies or chocolate bars for its fundraising campaign.

e) We want to know the favourite subject of Secondary V students in Québec.

f) The ministère des Transports would like to know what percentage of drivers in Québec wear contact lenses for driving.

g) A vacuum cleaner salesman would like to know the percentage of homes in Québec with a central vacuum system.

h) During a car show, an antique car club would like to know what model is the public’s favourite.

i) A pharmaceutical company wants to test a new drug in people with high cholesterol.

j) A radio station would like to know which of its shows has the highest listenership.

© S O FAD

33

M T H - 11 0 2 -3 – S TAT I S TICS AN D PROBABILITY

1.13

TABLE OF CONTENTS

ANSWER KEY

Complete the following sentences. a) The results obtained during a sample survey will be a faithful portrait of the population if we have used a

sample.

b) Conducting a statistical study on all the individuals or elements of a population, amounts to doing

.

c) A sample that is not representative of a population is said to be

.

d) When we choose a small group of individuals in order to represent the target population as faithfully as possible, the statistical study is conducted on

.

e) The set of all individuals included in a statistical study is called

.

f) Selecting a representative sample of a population to find out its opinion, is called a

.

g) A statistical study that collects data from different sources and often relies on experts is called 1.14

.

In each of the following situations, indicate whether the selected sample from which we want to obtain specific information is representative of the population. a) An ice cream business has just created a new flavour for the summer season. To find out whether people will like it, the company has reps give away samples in grocery stores to 2000 people and ask their opinion.

b) A radio station decides to only play music from the years 1965 to 1985. To find out whether its listeners agree with this decision, it asks 200 people randomly selected from the telephone book. c) The ministère du Tourisme wants to know what percentage of Quebecers will not be travelling outside the province for their summer holidays. It commissions a sample survey in which 2000 people from the Montréal region are asked to answer a questionnaire.

34

© SO FA D

TABLE OF CONTENTS

ANSWER KEY

Sit uation 1 – PARTIC IPATING IN AN E L E C TION

d) The principal of a college that has 4000 students is planning to redesign the parking lot. To find out the number of users, he asks 350 randomly selected students.

e) To ensure the quality of their products, a company that manufactures bicycle tires has a controller remove one out of every 30 tires from its production line.

1.15

In each of the following statistical studies, determine the possible sources of bias. a) A cigarette company wants to show that the general population does not consider smoking to be dangerous. It conducts a sample survey of 1000 randomly selected smokers.

b) The director of a private high school is planning to increase tuition for the upcoming year, but to do so he must first obtain the consent of more than 50% of parents who send their children to the school. He asks all parents the following question: “In order to provide a much better quality of education for your child, would you consider a very small tuition increase of about $250 to be acceptable?”

c) A dealer specializing in the sale of central air conditioners uses the services of a subcontractor to install the units it sells. To show that the subcontractor takes too long to install its units, the company mails a short questionnaire to 250 randomly selected clients asking their opinion. When it comes time to compile the results, it decides to use only the questionnaires of clients who had their unit installed during the months of June, July, August and September.

© S O FAD

35

TABLE OF CONTENTS

M T H - 11 0 2 -3 – S TATIS TI CS AND PROBABILIT Y

ANSWER KEY

Review Activity In a small village in the Laurentians, 15 000 people are registered on the voter list. The village is known as being a peaceful place and many retirees live there. Recently there have been acts of vandalism in the area and many members of the community are worried. The mayor has asked the police to deal with the situation. At the same time, she has also hired a specialized firm to conduct a sample survey on possible steps to take, not only to deal with this problem that is troubling her fellow citizens, but to also satisfy her voters. 1.16

The police have interviewed several witnesses and citizens who seemed to have pertinent information. What do we call this kind of survey the police conducted? Why?

1.17

The mayor has provided the specialized firm with the voter list as well as the breakdown of individuals in each of the village’s four districts. Breakdown of Voters by District

36

District

Men

Women

A

1200

1300

B

3000

2500

C

1100

1400

D

2000

2500

Total

7300

7700

© SO FA D

TABLE OF CONTENTS

ANSWER KEY

Sit uation 1 – PARTIC IPATING IN AN E L E C TION

The firm creates a sample survey on ways to combat the problem and will interview 500 people. a) What is the population of this village?

b) What sampling method should the firm use?

c) Why do you think the mayor decided to conduct a sample survey and not a census?

d) How should the firm build its sample to ensure it is truly representative?

e) Name three sources of bias that could skew the results.

© S O FAD

37

M T H - 11 0 2 -3

– S TATIS TICS AN D PROBABILITY

TABLE OF CONTENTS

ANSWER KEY

List of New Knowledge • A population is the set of all individuals or elements included in a study. • A sample is a small group of individuals or elements selected to represent the target population as faithfully as possible. • A census is a statistical study that includes all the individuals or elements of a population. • A sample survey is a statistical study that includes all the individuals or elements of a sample, from which we draw conclusions about the population. • A study is an in-depth statistical study involving experts to obtain specific information through various data collection techniques. • The sources of bias that can skew the results of a sample survey when creating a data collection form are: • a) poor wording of the goal of the sample survey; • b) poor wording of questions. • The sources of bias when performing data collection are: • a) a sample that is not representative of the population; • b) a poor attitude in the person conducting the sample survey; • c) misinterpretation of results; • d) misrepresentation of results. • A sample is said to be representative if it is a faithful portrait of the population; that is, if the sample has all the same characteristics as the population. • A sample that is not representative of a population is said to be biased. • The random sampling method consists in choosing the elements of a sample randomly, without any guidelines. • The systematic sampling method requires having a list of all the elements of the population. It consists in randomly choosing a starting point and then using the same procedure to select the other individuals in the sample.

38

© SO FA D