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To access your course, test yourself and track your progress simply go to learn.dummies.com and enter your unique access code to start learning now! ��������������������������������������������������� ����������������� ����������������� 1,001 Statistics Practice Problems 1,001 Statistics Practice Problems For Dummies® Published by: John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030-5774, www.wiley.com Copyright © 2014 by John Wiley & Sons, Inc., Hoboken, New Jersey Published simultaneously in Canada No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except as permitted under Sections 107 or 108 of the 1976 United States Copyright Act, without the prior written permission of the Publisher. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, or online at http://www.wiley.com/go/permissions. Trademarks: Wiley, For Dummies, the Dummies Man logo, Dummies.com, Making Everything Easier, and related trade dress are trademarks or registered trademarks of John Wiley & Sons, Inc., and may not be used without written permission. All other trademarks are the property of their respective owners. John Wiley & Sons, Inc., is not associated with any product or vendor mentioned in this book. LIMIT OF LIABILITY/DISCLAIMER OF WARRANTY: WHILE THE PUBLISHER AND AUTHOR HAVE USED THEIR BEST EFFORTS IN PREPARING THIS BOOK, THEY MAKE NO REPRESENTATIONS OR WARRANTIES WITH RESPECT TO THE ACCURACY OR COMPLETENESS OF THE CONTENTS OF THIS BOOK AND SPECIFICALLY DISCLAIM ANY IMPLIED WARRANTIES OF MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE. NO WARRANTY MAY BE CREATED OR EXTENDED BY SALES REPRESENTATIVES OR WRITTEN SALES MATERIALS. THE ADVICE AND STRATEGIES CONTAINED HEREIN MAY NOT BE SUITABLE FOR YOUR SITUATION. YOU SHOULD CONSULT WITH A PROFESSIONAL WHERE APPROPRIATE. NEITHER THE PUBLISHER NOR THE AUTHOR SHALL BE LIABLE FOR DAMAGES ARISING HEREFROM. For general information on our other products and services, please contact our Customer Care Department within the U.S. at 877-762-2974, outside the U.S. at 317-572-3993, or fax 317-572-4002. For technical support, please visit www.wiley.com/techsupport. Wiley publishes in a variety of print and electronic formats and by print-on-demand. Some material included with standard print versions of this book may not be included in e-books or in print-on-demand. If this book refers to media such as a CD or DVD that is not included in the version you purchased, you may download this material at http://booksupport.wiley.com. For more information about Wiley products, visit www.wiley.com. Library of Congress Control Number: 2013954236 ISBN 978-1-118-77604-9 (pbk); ISBN 978-1-118-77605-6 (ebk); ISBN 978-1-118-77616-2 (ebk) Manufactured in the United States of America 10 9 8 7 6 5 4 3 2 1 Contents at a Glance Introduction ............................................................................ 1 Part I: The Questions................................................................ 5 Chapter 1: Basic Vocabulary............................................................................................................... 7 Chapter 2: Descriptive Statistics....................................................................................................... 11 Chapter 3: Graphing ........................................................................................................................... 21 Chapter 4: Random Variables and the Binomial Distribution....................................................... 33 Chapter 5: The Normal Distribution................................................................................................. 43 Chapter 6: The t-Distribution............................................................................................................. 51 Chapter 7: Sampling Distributions and the Central Limit Theorem............................................. 57 Chapter 8: Finding Room for a Margin of Error .............................................................................. 71 Chapter 9: Confidence Intervals: Basics for Single Population Means and Proportions........... 75 Chapter 10: Confidence Intervals for Two Population Means and Proportions ........................ 91 Chapter 11: Claims, Tests, and Conclusions ................................................................................... 97 Chapter 12: Hypothesis Testing Basics for a Single Population Mean: z- and t-Tests.............. 105 Chapter 13: Hypothesis Tests for One Proportion, Two Proportions, or Two Population Means.................................................................................................................. 117 Chapter 14: Surveys.......................................................................................................................... 127 Chapter 15: Correlation.................................................................................................................... 131 Chapter 16: Simple Linear Regression ........................................................................................... 139 Chapter 17: Two-Way Tables and Independence ......................................................................... 153 Part II: The Answers ............................................................ 165 Chapter 18: Answers......................................................................................................................... 167 Appendix: Tables for Reference ............................................. 469 Index .................................................................................. 481 Table of Contents Introduction................................................................. 1 What You’ll Find ..............................................................................................1 How This Workbook Is Organized.................................................................1 Part I: The Questions.............................................................................2 Part II: The Answers ..............................................................................3 Beyond the Book .............................................................................................3 What you’ll find online ..........................................................................3 How to register.......................................................................................3 Where to Go for Additional Help ...................................................................4 Part I: The Questions .................................................... 5 Chapter 1: Basic Vocabulary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7 The Problems You’ll Work On .......................................................................7 What to Watch Out For...................................................................................7 Picking Out the Population, Sample, Parameter, and Statistic..................8 Distinguishing Quantitative and Categorical Variables..............................8 Getting a Handle on Bias, Variables, and the Mean ....................................9 Understanding Different Statistics and Data Analysis Terms....................9 Using Statistical Techniques........................................................................10 Working with the Standard Deviation.........................................................10 Chapter 2: Descriptive Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .11 The Problems You’ll Work On .....................................................................11 What to Watch Out For.................................................................................11 Understanding the Mean and the Median ..................................................12 Surveying Standard Deviation and Variance..............................................13 Employing the Empirical Rule......................................................................15 Measuring Relative Standing with Percentiles...........................................16 Delving into Data Sets and Descriptive Statistics......................................17 Chapter 3: Graphing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .21 The Problems You’ll Work On .....................................................................21 What to Watch Out For.................................................................................21 Interpreting Pie Charts .................................................................................22 Considering Three-Dimensional Pie Charts ...............................................22 Interpreting Bar Charts.................................................................................23 Introducing Other Graphs ............................................................................24 Interpreting Histograms ...............................................................................24 Digging Deeper into Histograms..................................................................25 Comparing Histograms .................................................................................26 Describing the Center of a Distribution......................................................27 Interpreting Box Plots...................................................................................27 1,001 Statistics Practice Problems For Dummies vi Comparing Two Box Plots............................................................................28 Comparing Three Box Plots .........................................................................29 Interpreting Time Charts..............................................................................30 Getting More Practice with Histograms .....................................................31 Chapter 4: Random Variables and the Binomial Distribution . . . . . . .33 The Problems You’ll Work On .....................................................................33 What to Watch Out For.................................................................................33 Comparing Discrete and Continuous Random Variables.........................34 Understanding the Probability Distribution of a Random Variable........35 Determining the Mean of a Discrete Random Variable.............................35 Digging Deeper into the Mean of a Discrete Random Variable................35 Working with the Variance of a Discrete Random Variable.....................36 Putting Together the Mean, Variance, and Standard Deviation of a Random Variable...............................................................36 Digging Deeper into the Mean, Variance, and Standard Deviation of a Random Variable..............................................37 Introducing Binomial Random Variables....................................................37 Figuring Out the Mean, Variance, and Standard Deviation of a Binomial Random Variable..............................................38 Finding Binomial Probabilities with a Formula..........................................38 Digging Deeper into Binomial Probabilities Using a Formula..................39 Finding Binomial Probabilities with the Binomial Table..........................39 Digging Deeper into Binomial Probabilities Using the Binomial Table.....................................................................................40 Using the Normal Approximation to the Binomial....................................40 Digging Deeper into the Normal Approximation to the Binomial...........................................................................................41 Getting More Practice with Binomial Variables.........................................41 Chapter 5: The Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . .43 The Problems You’ll Work On .....................................................................43 What to Watch Out For.................................................................................43 Defining and Describing the Normal Distribution.....................................44 Working with z-Scores and Values of X ......................................................45 Digging Deeper into z-Scores and Values of X ...........................................45 Writing Probability Notations......................................................................46 Introducing the Z-Table ................................................................................47 Finding Probabilities for a Normal Distribution ........................................47 Digging Deeper into z-Scores and Probabilities ........................................47 Figuring Out Percentiles for a Normal Distribution ..................................48 Digging Deeper into Percentiles for a Normal Distribution .....................49 Getting More Practice with Percentiles......................................................49 Chapter 6: The t-Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .51 The Problems You’ll Work On .....................................................................51 What to Watch Out For.................................................................................51 Understanding the t-Distribution and Comparing It to the Z-Distribution ..................................................................................52 Using the t-Table............................................................................................53 Using the t-Distribution to Calculate Confidence Intervals......................55 Table of Contents vii Chapter 7: Sampling Distributions and the Central Limit Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .57 The Problems You’ll Work On .....................................................................57 What to Watch Out For.................................................................................57 Introducing the Basics of Sampling Distributions.....................................58 Checking Out Random Variables and Sample Means ...............................59 Examining Standard Error............................................................................59 Surveying Notation and Symbols ................................................................60 Understanding What Affects Standard Error.............................................61 Digging Deeper into Standard Error............................................................61 Connecting Sample Means and Sampling Distributions...........................62 Digging Deeper into Sampling Distributions of Sample Means................. 64 Looking at the Central Limit Theorem........................................................65 Getting More Practice with Sample Mean Calculations............................66 Finding Probabilities for Sample Means.....................................................66 Digging Deeper into Probabilities for Sample Means................................67 Adding Proportions to the Mix ....................................................................68 Figuring Out the Standard Error of the Sample Proportion.....................68 Using the Central Limit Theorem for Proportions ....................................68 Matching z-Scores to Sample Proportions.................................................69 Finding Approximate Probabilities .............................................................69 Getting More Practice with Probabilities ...................................................69 Digging Deeper into Approximate Probabilities........................................70 Chapter 8: Finding Room for a Margin of Error . . . . . . . . . . . . . . . . . . . .71 The Problems You’ll Work On .....................................................................71 What to Watch Out For.................................................................................71 Defining and Calculating Margin of Error...................................................72 Using the Formula for Margin of Error When Estimating a Population Mean..................................................................72 Finding Appropriate z*-Values for Given Confidence Levels...................73 Connecting Margin of Error to Sample Size ...............................................73 Getting More Practice with the Formula for Margin of Error ..................73 Linking Margin of Error and Population Proportion.................................74 Chapter 9: Confidence Intervals: Basics for Single Population Means and Proportions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .75 The Problems You’ll Work On .....................................................................75 What to Watch Out For.................................................................................75 Introducing Confidence Intervals................................................................76 Checking Out Components of Confidence Intervals .................................76 Interpreting Confidence Intervals................................................................79 Spotting Misleading Confidence Intervals ......................................................80 Calculating a Confidence Interval for a Population Mean ........................83 Determining the Needed Sample Size .........................................................86 Introducing a Population Proportion..........................................................87 Connecting a Population Proportion to a Survey......................................88 Calculating a Confidence Interval for a Population Proportion ..............88 Digging Deeper into Population Proportions.............................................89 Getting More Practice with Population Proportions ................................90 1,001 Statistics Practice Problems For Dummies viii viii Chapter 10: Confidence Intervals for Two Population Means and Proportions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .91 The Problems You’ll Work On .....................................................................91 What to Watch Out For.................................................................................91 Working with Confidence Intervals and Population Proportions.............................................................................92 Digging Deeper into Confidence Intervals and Population Proportions.............................................................................92 Working with Confidence Intervals and Population Means ...........................93 Making Calculations When Population Standard Deviations Are Known..................................................................................... 93 Digging Deeper into Calculations When Population Standard Deviations Are Known..............................................................94 Working with Unknown Population Standard Deviations and Small Sample Sizes..........................................................95 Digging Deeper into Unknown Population Standard Deviations and Small Sample Sizes ............................................................... 95 Chapter 11: Claims, Tests, and Conclusions . . . . . . . . . . . . . . . . . . . . . .97 The Problems You’ll Work On .....................................................................97 What to Watch Out For.................................................................................97 Knowing When to Use a Hypothesis Test...................................................98 Setting Up Null and Alternative Hypotheses..............................................98 Finding the Test Statistic and the p-Value................................................100 Making Decisions Based on Alpha Levels and Test Statistics ...............101 Making Conclusions ....................................................................................101 Understanding Type I and Type II Errors.................................................103 Chapter 12: Hypothesis Testing Basics for a Single Population Mean: z- and t-Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .105 The Problems You’ll Work On ...................................................................105 What to Watch Out For...............................................................................105 Knowing What You Need to Run a z-Test.................................................106 Determining Null and Alternative Hypotheses ........................................106 Introducing p-Values ...................................................................................107 Calculating the z-Test Statistic ..................................................................107 Finding p-Values by Doing a Test of One Population Mean ...................108 Drawing Conclusions about Hypotheses..................................................108 Digging Deeper into p-Values.....................................................................109 Digging Deeper into Conclusions about Hypotheses..............................109 Digging Deeper into Null and Alternative Hypotheses ...........................110 Knowing When to Use a t-Test ...................................................................111 Connecting Hypotheses to t-Tests.............................................................111 Calculating Test Statistics ..........................................................................112 Working with Critical Values of t ...............................................................112 Linking p-Values and t-Tests.......................................................................113 Drawing Conclusions from t-Tests.............................................................114 Performing a t-Test for a Single Population Mean...................................115 Drawing More Conclusions from t-Tests ..................................................115 Table of Contents ixix Chapter 13: Hypothesis Tests for One Proportion, Two Proportions, or Two Population Means . . . . . . . . . . . . . . . . . . . .117 The Problems You’ll Work On ...................................................................117 What to Watch Out For...............................................................................117 Testing One Population Proportion..........................................................118 Comparing Two Independent Population Means....................................120 Digging Deeper into Two Independent Population Means.....................121 Getting More Practice with Two Independent Population Means ........121 Using the Paired t-Test................................................................................122 Digging Deeper into the Paired t-Test .......................................................123 Comparing Two Population Proportions .................................................124 Digging Deeper into Two Population Proportions ..................................124 Chapter 14: Surveys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .127 The Problems You’ll Work On ...................................................................127 What to Watch Out For...............................................................................127 Planning and Designing Surveys................................................................128 Selecting Samples and Conducting Surveys.............................................128 Chapter 15: Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .131 The Problems You’ll Work On ...................................................................131 What to Watch Out For...............................................................................131 Interpreting Scatter Plots ...........................................................................132 Creating Scatter Plots .................................................................................132 Understanding What Correlations Indicate .............................................133 Digging Deeper into Scatter Plots..............................................................133 Digging Deeper into What Correlations Indicate.....................................134 Calculating Correlations.............................................................................134 Noting How Correlations Can Change ......................................................135 Looking at the Properties of Correlations................................................135 Digging Deeper into How Correlations Can Change................................136 Making Conclusions about Correlations ..................................................136 Getting More Practice with Scatter Plots and Correlation Changes.....136 Making More Conclusions about Correlations ........................................137 Chapter 16: Simple Linear Regression . . . . . . . . . . . . . . . . . . . . . . . . . .139 The Problems You’ll Work On ...................................................................139 What to Watch Out For...............................................................................139 Introducing the Regression Line...............................................................140 Knowing the Conditions for Regression...................................................140 Examining the Equation for Calculating the Least-Squares Regression Line ...............................................................140 Finding the Slope and y-Intercept of a Regression Line..........................141 Seeing How Variables Can Change in a Regression Line........................141 Finding a Regression Line...........................................................................141 Digging Deeper into Finding a Regression Line .......................................142 Connecting to Correlation and Linear Relationships..............................144 Determining Whether Variables Are Candidates for a Linear Regression Analysis............................................................144 1,001 Statistics Practice Problems For Dummies x Digging Deeper into Correlations and Linear Relationships..................145 Describing Linear Relationships................................................................145 Getting More Practice with Finding a Regression Line...........................146 Making Predictions......................................................................................147 Figuring Out Expected Values and Differences........................................148 Digging Deeper into Expected Values and Differences...........................149 Digging Deeper into Predictions................................................................150 Getting More Practice with Expected Values and Differences ..............150 Chapter 17: Two-Way Tables and Independence . . . . . . . . . . . . . . . .153 The Problems You’ll Work On ...................................................................153 What to Watch Out For...............................................................................153 Introducing Variables and Two-Way Tables............................................154 Reading a Two-Way Table ..........................................................................154 Interpreting a Two-Way Table by Using Percentages.............................156 Interpreting a Two-Way Table by Using Counts......................................156 Connecting Conditional Probabilities to Two-Way Tables ........................... 157 Investigating Independent Variables ........................................................158 Calculating Marginal Probability and More .............................................159 Adding Joint Probability to the Mix ..........................................................160 Digging Deeper into Conditional and Marginal Probabilities ................161 Figuring Out the Number of Cells in a Two-Way Table...........................162 Including Conditional Probability .............................................................162 Digging into Research Designs...................................................................163 Digging Deeper into Two-Way Tables.......................................................163 Part II: The Answers ................................................. 165 Chapter 18: Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .167 Appendix: Tables for Reference.................................. 469 Index....................................................................... 481 Introduction O ne thousand and one practice problems for statistics! That’s probably more than a professor would assign you in one semester (we hope!). And it’s more than you’d ever want to tackle in one sitting (and we don’t recommend you try). So why so many practice problems, and why this book? Many textbooks are pretty thin on exercises, and even those that do contain a fair number of problems can’t focus on all aspects of each topic. With so many problems available in this book, you get to choose how many problems you want to work on. And the way these problems are organized helps you quickly find and dig into problems on particular topics you need to study at the time. Whether you’re into the normal distribution, hypothesis tests, the slope of a regression line, or histograms, it’s all here and easy to find. Then there’s the entertainment factor. What better way to draw a crowd than to invite people over for a statistics practice problems marathon! What You’ll Find This book contains 1,001 statistics problems divided into 17 chapters, organized by the major statistical topics in a first-semester introductory course. The problems basically take on three levels: ✓ Statistical literacy: Understanding the basic concepts of the topic, including terms and notation ✓ Reasoning: Applying the ideas within a context ✓ Thinking: Putting ideas and concepts together to solve more difficult problems In addition to providing plenty of problems to work on in each chapter, this book also pro- vides worked-out solutions with detailed explanations, so you aren’t left high and dry if you get a wrong answer. So you can rest assured that when you work for 30 minutes on a prob- lem, get an answer of 1.25, and go to the back of the book to see that the correct answer is actually 1,218.31, you’ll find a detailed explanation to help you figure out what went wrong with your calculations. How This Workbook Is Organized This book is divided into two main parts: the questions and the answers. 2 1,001 Statistics Practice Problems For Dummies Part 1: The Questions The questions in this book center on the following areas: ✓ Descriptive statistics and graphs: After you collect and review data, your first job is to make sense of it. The way to do that is two-fold: (1) Organize the data in a visual manner so you can see it, and (2) crank out some numbers that describe it in a basic way. ✓ Random variables: A random variable is a characteristic of interest that varies in a random fashion. Each type of random variable has its own pattern in which the data falls (or is expected to fall), along with its own mean and standard deviation for the data. The pattern of a random variable is called its distribution. The random variables in this book include the binomial, the normal (or Z), and the t. For each random variable, you practice identifying its characteristics, seeing what its pattern (distribution) looks like, determining its mean and standard deviation, and, most commonly, finding probabilities and percentiles for it. ✓ Inference: This term can seem complex (and word on the street says it is), but infer- ence basically just means taking the information from your data (your sample) and using it to draw conclusions about the group you’re interested in (your population). The two basic types of statistical inferences are confidence intervals and hypothesis testing: • You use confidence intervals when you want to make an estimate regarding the population — for example, “What percentage of all kindergarteners in the United States are obese?” • You use a hypothesis test when someone has a supposed value regarding the pop- ulation, and you’re putting it to the test. For example, a researcher claims that 14 percent of today’s kindergarteners are obese, but you question whether it’s really that high. The underpinnings needed for both types of inference are margin of error, standard error, sampling distributions, and the central limit theorem. All of them play a major role in statistics and can be somewhat complex, so make sure you spend time on these elements as a backdrop for confidence intervals and hypothesis tests. ✓ Relationships: One of the most important and common uses of statistics is to look for relationships between two random variables. If variables are categorical (such as gender), you explore relationships by using two-way tables containing rows and col- umns, and you examine relationships by looking at and comparing percentages among and within groups. If both variables are numerical, you explore relationships graphi- cally by using scatter plots, quantify them by using correlation, and use them to make predictions (one variable predicting the other) by using regression. Studying relation- ships helps you get at the essence of how statistics is applied in the real world. ✓ Surveys: Before you analyze data in all the ways mentioned in this list, you have to col- lect the data. Surveys are one of the most common means of data collection; the main ideas of surveys to address with practice are planning a survey, selecting a representa- tive sample of individuals to survey, and carrying out the survey properly. The main goal in all of these areas is to avoid bias (systematic favoritism). Many types of bias exist, and in this book, you practice identifying and seeing ways to minimize them. 3 Introduction Part II: The Answers This part provides detailed answers to every question in this book. You see how to set up and work through each problem and how to interpret the answer. Beyond the Book This book of 1,001 practice problems will keep you busy with pencil and paper for a while, but like the infomercials say, “Wait! There’s more!” Your book purchase also comes with a free, one-year subscription to all 1,001 practice problems online. Track your progress and view personalized reports that show where you need to study the most and what you’re pretty comfortable with. What you’ll find online The online practice that comes free with this book offers you the same 1,001 questions and answers that are available here, presented in a multiple-choice format. Multiple-choice ques- tions force you to zoom in on the details that can make or break your correct solution to the problem. Sometimes one of the possible wrong answers will catch you in the act of making a certain error. But that’s great because after you identify a particular error (often a common error that many others make as well), you’ll know not to fall into that trap again. The beauty of the online problems is that you can customize your online practice — that is, you can select the types of problems and the number of problems you want to work on. The online program keeps track of your performance so you can focus on the areas where you need the biggest boost. You can access this online tool by using a PIN code, as described in the next section. Keep in mind that you can create only one login with your PIN. Once the PIN is used, it’s no longer valid and is nontransferable. So you can’t share your PIN with others after you’ve estab- lished your login credentials. In other words, the problems are yours and only yours! How to register Purchasing this book entitles you to one year of free access to the online, multiple-choice version of all 1,001 problems in this book. All you have to do to access them is register by following these simple steps: 1. Find your PIN code. • Print-book users: If you purchased a hard copy of this book, turn to the front of this book to find your PIN. • E-book users: If you purchased this book as an e-book, you can get your PIN by registering your e-book at www.dummies.com/go/getaccess. Simply select your book from the drop-down menu, fill in your personal information, and then answer the security question to verify your purchase. You’ll then receive an e-mail with your PIN.

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To access your course, test yourself and track your progress simply go to learn.dummies.com and enter your unique access code to start learning now! ��������������������������������������������������� ����������������� ����������������� 1,001 Statistics Practice Problems 1,001 Statistics Practice Problems For Dummies® Published by: John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030-5774, www.wiley.com Copyright © 2014 by John Wiley & Sons, Inc., Hoboken, New Jersey Published simultaneously in Canada No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except as permitted under Sections 107 or 108 of the 1976 United States Copyright Act, without the prior written permission of the Publisher. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, or online at http://www.wiley.com/go/permissions. Trademarks: Wiley, For Dummies, the Dummies Man logo, Dummies.com, Making Everything Easier, and related trade dress are trademarks or registered trademarks of John Wiley & Sons, Inc., and may not be used without written permission. All other trademarks are the property of their respective owners. John Wiley & Sons, Inc., is not associated with any product or vendor mentioned in this book. LIMIT OF LIABILITY/DISCLAIMER OF WARRANTY: WHILE THE PUBLISHER AND AUTHOR HAVE USED THEIR BEST EFFORTS IN PREPARING THIS BOOK, THEY MAKE NO REPRESENTATIONS OR WARRANTIES WITH RESPECT TO THE ACCURACY OR COMPLETENESS OF THE CONTENTS OF THIS BOOK AND SPECIFICALLY DISCLAIM ANY IMPLIED WARRANTIES OF MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE. NO WARRANTY MAY BE CREATED OR EXTENDED BY SALES REPRESENTATIVES OR WRITTEN SALES MATERIALS. THE ADVICE AND STRATEGIES CONTAINED HEREIN MAY NOT BE SUITABLE FOR YOUR SITUATION. YOU SHOULD CONSULT WITH A PROFESSIONAL WHERE APPROPRIATE. NEITHER THE PUBLISHER NOR THE AUTHOR SHALL BE LIABLE FOR DAMAGES ARISING HEREFROM. For general information on our other products and services, please contact our Customer Care Department within the U.S. at 877-762-2974, outside the U.S. at 317-572-3993, or fax 317-572-4002. For technical support, please visit www.wiley.com/techsupport. Wiley publishes in a variety of print and electronic formats and by print-on-demand. Some material included with standard print versions of this book may not be included in e-books or in print-on-demand. If this book refers to media such as a CD or DVD that is not included in the version you purchased, you may download this material at http://booksupport.wiley.com. For more information about Wiley products, visit www.wiley.com. Library of Congress Control Number: 2013954236 ISBN 978-1-118-77604-9 (pbk); ISBN 978-1-118-77605-6 (ebk); ISBN 978-1-118-77616-2 (ebk) Manufactured in the United States of America 10 9 8 7 6 5 4 3 2 1 Contents at a Glance Introduction ............................................................................ 1 Part I: The Questions................................................................ 5 Chapter 1: Basic Vocabulary............................................................................................................... 7 Chapter 2: Descriptive Statistics....................................................................................................... 11 Chapter 3: Graphing ........................................................................................................................... 21 Chapter 4: Random Variables and the Binomial Distribution....................................................... 33 Chapter 5: The Normal Distribution................................................................................................. 43 Chapter 6: The t-Distribution............................................................................................................. 51 Chapter 7: Sampling Distributions and the Central Limit Theorem............................................. 57 Chapter 8: Finding Room for a Margin of Error .............................................................................. 71 Chapter 9: Confidence Intervals: Basics for Single Population Means and Proportions........... 75 Chapter 10: Confidence Intervals for Two Population Means and Proportions ........................ 91 Chapter 11: Claims, Tests, and Conclusions ................................................................................... 97 Chapter 12: Hypothesis Testing Basics for a Single Population Mean: z- and t-Tests.............. 105 Chapter 13: Hypothesis Tests for One Proportion, Two Proportions, or Two Population Means.................................................................................................................. 117 Chapter 14: Surveys.......................................................................................................................... 127 Chapter 15: Correlation.................................................................................................................... 131 Chapter 16: Simple Linear Regression ........................................................................................... 139 Chapter 17: Two-Way Tables and Independence ......................................................................... 153 Part II: The Answers ............................................................ 165 Chapter 18: Answers......................................................................................................................... 167 Appendix: Tables for Reference ............................................. 469 Index .................................................................................. 481 Table of Contents Introduction................................................................. 1 What You’ll Find ..............................................................................................1 How This Workbook Is Organized.................................................................1 Part I: The Questions.............................................................................2 Part II: The Answers ..............................................................................3 Beyond the Book .............................................................................................3 What you’ll find online ..........................................................................3 How to register.......................................................................................3 Where to Go for Additional Help ...................................................................4 Part I: The Questions .................................................... 5 Chapter 1: Basic Vocabulary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7 The Problems You’ll Work On .......................................................................7 What to Watch Out For...................................................................................7 Picking Out the Population, Sample, Parameter, and Statistic..................8 Distinguishing Quantitative and Categorical Variables..............................8 Getting a Handle on Bias, Variables, and the Mean ....................................9 Understanding Different Statistics and Data Analysis Terms....................9 Using Statistical Techniques........................................................................10 Working with the Standard Deviation.........................................................10 Chapter 2: Descriptive Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .11 The Problems You’ll Work On .....................................................................11 What to Watch Out For.................................................................................11 Understanding the Mean and the Median ..................................................12 Surveying Standard Deviation and Variance..............................................13 Employing the Empirical Rule......................................................................15 Measuring Relative Standing with Percentiles...........................................16 Delving into Data Sets and Descriptive Statistics......................................17 Chapter 3: Graphing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .21 The Problems You’ll Work On .....................................................................21 What to Watch Out For.................................................................................21 Interpreting Pie Charts .................................................................................22 Considering Three-Dimensional Pie Charts ...............................................22 Interpreting Bar Charts.................................................................................23 Introducing Other Graphs ............................................................................24 Interpreting Histograms ...............................................................................24 Digging Deeper into Histograms..................................................................25 Comparing Histograms .................................................................................26 Describing the Center of a Distribution......................................................27 Interpreting Box Plots...................................................................................27 1,001 Statistics Practice Problems For Dummies vi Comparing Two Box Plots............................................................................28 Comparing Three Box Plots .........................................................................29 Interpreting Time Charts..............................................................................30 Getting More Practice with Histograms .....................................................31 Chapter 4: Random Variables and the Binomial Distribution . . . . . . .33 The Problems You’ll Work On .....................................................................33 What to Watch Out For.................................................................................33 Comparing Discrete and Continuous Random Variables.........................34 Understanding the Probability Distribution of a Random Variable........35 Determining the Mean of a Discrete Random Variable.............................35 Digging Deeper into the Mean of a Discrete Random Variable................35 Working with the Variance of a Discrete Random Variable.....................36 Putting Together the Mean, Variance, and Standard Deviation of a Random Variable...............................................................36 Digging Deeper into the Mean, Variance, and Standard Deviation of a Random Variable..............................................37 Introducing Binomial Random Variables....................................................37 Figuring Out the Mean, Variance, and Standard Deviation of a Binomial Random Variable..............................................38 Finding Binomial Probabilities with a Formula..........................................38 Digging Deeper into Binomial Probabilities Using a Formula..................39 Finding Binomial Probabilities with the Binomial Table..........................39 Digging Deeper into Binomial Probabilities Using the Binomial Table.....................................................................................40 Using the Normal Approximation to the Binomial....................................40 Digging Deeper into the Normal Approximation to the Binomial...........................................................................................41 Getting More Practice with Binomial Variables.........................................41 Chapter 5: The Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . .43 The Problems You’ll Work On .....................................................................43 What to Watch Out For.................................................................................43 Defining and Describing the Normal Distribution.....................................44 Working with z-Scores and Values of X ......................................................45 Digging Deeper into z-Scores and Values of X ...........................................45 Writing Probability Notations......................................................................46 Introducing the Z-Table ................................................................................47 Finding Probabilities for a Normal Distribution ........................................47 Digging Deeper into z-Scores and Probabilities ........................................47 Figuring Out Percentiles for a Normal Distribution ..................................48 Digging Deeper into Percentiles for a Normal Distribution .....................49 Getting More Practice with Percentiles......................................................49 Chapter 6: The t-Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .51 The Problems You’ll Work On .....................................................................51 What to Watch Out For.................................................................................51 Understanding the t-Distribution and Comparing It to the Z-Distribution ..................................................................................52 Using the t-Table............................................................................................53 Using the t-Distribution to Calculate Confidence Intervals......................55 Table of Contents vii Chapter 7: Sampling Distributions and the Central Limit Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .57 The Problems You’ll Work On .....................................................................57 What to Watch Out For.................................................................................57 Introducing the Basics of Sampling Distributions.....................................58 Checking Out Random Variables and Sample Means ...............................59 Examining Standard Error............................................................................59 Surveying Notation and Symbols ................................................................60 Understanding What Affects Standard Error.............................................61 Digging Deeper into Standard Error............................................................61 Connecting Sample Means and Sampling Distributions...........................62 Digging Deeper into Sampling Distributions of Sample Means................. 64 Looking at the Central Limit Theorem........................................................65 Getting More Practice with Sample Mean Calculations............................66 Finding Probabilities for Sample Means.....................................................66 Digging Deeper into Probabilities for Sample Means................................67 Adding Proportions to the Mix ....................................................................68 Figuring Out the Standard Error of the Sample Proportion.....................68 Using the Central Limit Theorem for Proportions ....................................68 Matching z-Scores to Sample Proportions.................................................69 Finding Approximate Probabilities .............................................................69 Getting More Practice with Probabilities ...................................................69 Digging Deeper into Approximate Probabilities........................................70 Chapter 8: Finding Room for a Margin of Error . . . . . . . . . . . . . . . . . . . .71 The Problems You’ll Work On .....................................................................71 What to Watch Out For.................................................................................71 Defining and Calculating Margin of Error...................................................72 Using the Formula for Margin of Error When Estimating a Population Mean..................................................................72 Finding Appropriate z*-Values for Given Confidence Levels...................73 Connecting Margin of Error to Sample Size ...............................................73 Getting More Practice with the Formula for Margin of Error ..................73 Linking Margin of Error and Population Proportion.................................74 Chapter 9: Confidence Intervals: Basics for Single Population Means and Proportions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .75 The Problems You’ll Work On .....................................................................75 What to Watch Out For.................................................................................75 Introducing Confidence Intervals................................................................76 Checking Out Components of Confidence Intervals .................................76 Interpreting Confidence Intervals................................................................79 Spotting Misleading Confidence Intervals ......................................................80 Calculating a Confidence Interval for a Population Mean ........................83 Determining the Needed Sample Size .........................................................86 Introducing a Population Proportion..........................................................87 Connecting a Population Proportion to a Survey......................................88 Calculating a Confidence Interval for a Population Proportion ..............88 Digging Deeper into Population Proportions.............................................89 Getting More Practice with Population Proportions ................................90 1,001 Statistics Practice Problems For Dummies viii viii Chapter 10: Confidence Intervals for Two Population Means and Proportions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .91 The Problems You’ll Work On .....................................................................91 What to Watch Out For.................................................................................91 Working with Confidence Intervals and Population Proportions.............................................................................92 Digging Deeper into Confidence Intervals and Population Proportions.............................................................................92 Working with Confidence Intervals and Population Means ...........................93 Making Calculations When Population Standard Deviations Are Known..................................................................................... 93 Digging Deeper into Calculations When Population Standard Deviations Are Known..............................................................94 Working with Unknown Population Standard Deviations and Small Sample Sizes..........................................................95 Digging Deeper into Unknown Population Standard Deviations and Small Sample Sizes ............................................................... 95 Chapter 11: Claims, Tests, and Conclusions . . . . . . . . . . . . . . . . . . . . . .97 The Problems You’ll Work On .....................................................................97 What to Watch Out For.................................................................................97 Knowing When to Use a Hypothesis Test...................................................98 Setting Up Null and Alternative Hypotheses..............................................98 Finding the Test Statistic and the p-Value................................................100 Making Decisions Based on Alpha Levels and Test Statistics ...............101 Making Conclusions ....................................................................................101 Understanding Type I and Type II Errors.................................................103 Chapter 12: Hypothesis Testing Basics for a Single Population Mean: z- and t-Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .105 The Problems You’ll Work On ...................................................................105 What to Watch Out For...............................................................................105 Knowing What You Need to Run a z-Test.................................................106 Determining Null and Alternative Hypotheses ........................................106 Introducing p-Values ...................................................................................107 Calculating the z-Test Statistic ..................................................................107 Finding p-Values by Doing a Test of One Population Mean ...................108 Drawing Conclusions about Hypotheses..................................................108 Digging Deeper into p-Values.....................................................................109 Digging Deeper into Conclusions about Hypotheses..............................109 Digging Deeper into Null and Alternative Hypotheses ...........................110 Knowing When to Use a t-Test ...................................................................111 Connecting Hypotheses to t-Tests.............................................................111 Calculating Test Statistics ..........................................................................112 Working with Critical Values of t ...............................................................112 Linking p-Values and t-Tests.......................................................................113 Drawing Conclusions from t-Tests.............................................................114 Performing a t-Test for a Single Population Mean...................................115 Drawing More Conclusions from t-Tests ..................................................115 Table of Contents ixix Chapter 13: Hypothesis Tests for One Proportion, Two Proportions, or Two Population Means . . . . . . . . . . . . . . . . . . . .117 The Problems You’ll Work On ...................................................................117 What to Watch Out For...............................................................................117 Testing One Population Proportion..........................................................118 Comparing Two Independent Population Means....................................120 Digging Deeper into Two Independent Population Means.....................121 Getting More Practice with Two Independent Population Means ........121 Using the Paired t-Test................................................................................122 Digging Deeper into the Paired t-Test .......................................................123 Comparing Two Population Proportions .................................................124 Digging Deeper into Two Population Proportions ..................................124 Chapter 14: Surveys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .127 The Problems You’ll Work On ...................................................................127 What to Watch Out For...............................................................................127 Planning and Designing Surveys................................................................128 Selecting Samples and Conducting Surveys.............................................128 Chapter 15: Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .131 The Problems You’ll Work On ...................................................................131 What to Watch Out For...............................................................................131 Interpreting Scatter Plots ...........................................................................132 Creating Scatter Plots .................................................................................132 Understanding What Correlations Indicate .............................................133 Digging Deeper into Scatter Plots..............................................................133 Digging Deeper into What Correlations Indicate.....................................134 Calculating Correlations.............................................................................134 Noting How Correlations Can Change ......................................................135 Looking at the Properties of Correlations................................................135 Digging Deeper into How Correlations Can Change................................136 Making Conclusions about Correlations ..................................................136 Getting More Practice with Scatter Plots and Correlation Changes.....136 Making More Conclusions about Correlations ........................................137 Chapter 16: Simple Linear Regression . . . . . . . . . . . . . . . . . . . . . . . . . .139 The Problems You’ll Work On ...................................................................139 What to Watch Out For...............................................................................139 Introducing the Regression Line...............................................................140 Knowing the Conditions for Regression...................................................140 Examining the Equation for Calculating the Least-Squares Regression Line ...............................................................140 Finding the Slope and y-Intercept of a Regression Line..........................141 Seeing How Variables Can Change in a Regression Line........................141 Finding a Regression Line...........................................................................141 Digging Deeper into Finding a Regression Line .......................................142 Connecting to Correlation and Linear Relationships..............................144 Determining Whether Variables Are Candidates for a Linear Regression Analysis............................................................144 1,001 Statistics Practice Problems For Dummies x Digging Deeper into Correlations and Linear Relationships..................145 Describing Linear Relationships................................................................145 Getting More Practice with Finding a Regression Line...........................146 Making Predictions......................................................................................147 Figuring Out Expected Values and Differences........................................148 Digging Deeper into Expected Values and Differences...........................149 Digging Deeper into Predictions................................................................150 Getting More Practice with Expected Values and Differences ..............150 Chapter 17: Two-Way Tables and Independence . . . . . . . . . . . . . . . .153 The Problems You’ll Work On ...................................................................153 What to Watch Out For...............................................................................153 Introducing Variables and Two-Way Tables............................................154 Reading a Two-Way Table ..........................................................................154 Interpreting a Two-Way Table by Using Percentages.............................156 Interpreting a Two-Way Table by Using Counts......................................156 Connecting Conditional Probabilities to Two-Way Tables ........................... 157 Investigating Independent Variables ........................................................158 Calculating Marginal Probability and More .............................................159 Adding Joint Probability to the Mix ..........................................................160 Digging Deeper into Conditional and Marginal Probabilities ................161 Figuring Out the Number of Cells in a Two-Way Table...........................162 Including Conditional Probability .............................................................162 Digging into Research Designs...................................................................163 Digging Deeper into Two-Way Tables.......................................................163 Part II: The Answers ................................................. 165 Chapter 18: Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .167 Appendix: Tables for Reference.................................. 469 Index....................................................................... 481 Introduction O ne thousand and one practice problems for statistics! That’s probably more than a professor would assign you in one semester (we hope!). And it’s more than you’d ever want to tackle in one sitting (and we don’t recommend you try). So why so many practice problems, and why this book? Many textbooks are pretty thin on exercises, and even those that do contain a fair number of problems can’t focus on all aspects of each topic. With so many problems available in this book, you get to choose how many problems you want to work on. And the way these problems are organized helps you quickly find and dig into problems on particular topics you need to study at the time. Whether you’re into the normal distribution, hypothesis tests, the slope of a regression line, or histograms, it’s all here and easy to find. Then there’s the entertainment factor. What better way to draw a crowd than to invite people over for a statistics practice problems marathon! What You’ll Find This book contains 1,001 statistics problems divided into 17 chapters, organized by the major statistical topics in a first-semester introductory course. The problems basically take on three levels: ✓ Statistical literacy: Understanding the basic concepts of the topic, including terms and notation ✓ Reasoning: Applying the ideas within a context ✓ Thinking: Putting ideas and concepts together to solve more difficult problems In addition to providing plenty of problems to work on in each chapter, this book also pro- vides worked-out solutions with detailed explanations, so you aren’t left high and dry if you get a wrong answer. So you can rest assured that when you work for 30 minutes on a prob- lem, get an answer of 1.25, and go to the back of the book to see that the correct answer is actually 1,218.31, you’ll find a detailed explanation to help you figure out what went wrong with your calculations. How This Workbook Is Organized This book is divided into two main parts: the questions and the answers. 2 1,001 Statistics Practice Problems For Dummies Part 1: The Questions The questions in this book center on the following areas: ✓ Descriptive statistics and graphs: After you collect and review data, your first job is to make sense of it. The way to do that is two-fold: (1) Organize the data in a visual manner so you can see it, and (2) crank out some numbers that describe it in a basic way. ✓ Random variables: A random variable is a characteristic of interest that varies in a random fashion. Each type of random variable has its own pattern in which the data falls (or is expected to fall), along with its own mean and standard deviation for the data. The pattern of a random variable is called its distribution. The random variables in this book include the binomial, the normal (or Z), and the t. For each random variable, you practice identifying its characteristics, seeing what its pattern (distribution) looks like, determining its mean and standard deviation, and, most commonly, finding probabilities and percentiles for it. ✓ Inference: This term can seem complex (and word on the street says it is), but infer- ence basically just means taking the information from your data (your sample) and using it to draw conclusions about the group you’re interested in (your population). The two basic types of statistical inferences are confidence intervals and hypothesis testing: • You use confidence intervals when you want to make an estimate regarding the population — for example, “What percentage of all kindergarteners in the United States are obese?” • You use a hypothesis test when someone has a supposed value regarding the pop- ulation, and you’re putting it to the test. For example, a researcher claims that 14 percent of today’s kindergarteners are obese, but you question whether it’s really that high. The underpinnings needed for both types of inference are margin of error, standard error, sampling distributions, and the central limit theorem. All of them play a major role in statistics and can be somewhat complex, so make sure you spend time on these elements as a backdrop for confidence intervals and hypothesis tests. ✓ Relationships: One of the most important and common uses of statistics is to look for relationships between two random variables. If variables are categorical (such as gender), you explore relationships by using two-way tables containing rows and col- umns, and you examine relationships by looking at and comparing percentages among and within groups. If both variables are numerical, you explore relationships graphi- cally by using scatter plots, quantify them by using correlation, and use them to make predictions (one variable predicting the other) by using regression. Studying relation- ships helps you get at the essence of how statistics is applied in the real world. ✓ Surveys: Before you analyze data in all the ways mentioned in this list, you have to col- lect the data. Surveys are one of the most common means of data collection; the main ideas of surveys to address with practice are planning a survey, selecting a representa- tive sample of individuals to survey, and carrying out the survey properly. The main goal in all of these areas is to avoid bias (systematic favoritism). Many types of bias exist, and in this book, you practice identifying and seeing ways to minimize them. 3 Introduction Part II: The Answers This part provides detailed answers to every question in this book. You see how to set up and work through each problem and how to interpret the answer. Beyond the Book This book of 1,001 practice problems will keep you busy with pencil and paper for a while, but like the infomercials say, “Wait! There’s more!” Your book purchase also comes with a free, one-year subscription to all 1,001 practice problems online. Track your progress and view personalized reports that show where you need to study the most and what you’re pretty comfortable with. What you’ll find online The online practice that comes free with this book offers you the same 1,001 questions and answers that are available here, presented in a multiple-choice format. Multiple-choice ques- tions force you to zoom in on the details that can make or break your correct solution to the problem. Sometimes one of the possible wrong answers will catch you in the act of making a certain error. But that’s great because after you identify a particular error (often a common error that many others make as well), you’ll know not to fall into that trap again. The beauty of the online problems is that you can customize your online practice — that is, you can select the types of problems and the number of problems you want to work on. The online program keeps track of your performance so you can focus on the areas where you need the biggest boost. You can access this online tool by using a PIN code, as described in the next section. Keep in mind that you can create only one login with your PIN. Once the PIN is used, it’s no longer valid and is nontransferable. So you can’t share your PIN with others after you’ve estab- lished your login credentials. In other words, the problems are yours and only yours! How to register Purchasing this book entitles you to one year of free access to the online, multiple-choice version of all 1,001 problems in this book. All you have to do to access them is register by following these simple steps: 1. Find your PIN code. • Print-book users: If you purchased a hard copy of this book, turn to the front of this book to find your PIN. • E-book users: If you purchased this book as an e-book, you can get your PIN by registering your e-book at www.dummies.com/go/getaccess. Simply select your book from the drop-down menu, fill in your personal information, and then answer the security question to verify your purchase. You’ll then receive an e-mail with your PIN.

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