Probability and Statistics Workbook

Statistics

Welcome to the Ready, Set, Go!

Probability & Statistics Workbook!

About This Book

This book will help high school math students at all learning levels understand basic geometry. Students will develop the skills, confidence, and knowledge they need to succeed on high school math exams with emphasis on passing high school graduation exams.

More than 20 easy-to-follow lessons break down the material into the basics. In-depth, step-by-step examples and solutions reinforce student learning, while the Math Flash feature provides useful tips and strategies, including advice on common mistakes to avoid.

Students can take drills and quizzes to test themselves on the subject matter, then review any areas in which they need improvement or additional reinforcement. The book concludes with a final exam, designed to comprehensively test what students have learned.

The Ready, Set, Go! Probability & Statistics Workbook will help students master the basics of mathematics—and help them face their next math test—with confidence!

Icons Explained

Icons make navigating through the book easier. The icons, explained below, highlight tips and strategies, where to review a topic, and the drills found at the end of each lesson.

Look for the Math Flash feature for helpful tips and strategies, including advice on how to avoid common mistakes.

When you see the Let’s Review icon, you know just where to look for more help with the topic on which you are currently working.

The Test Yourself! icon, found at the end of every lesson, signals a short drill that reviews the skills you have studied in that lesson.

To the Student

This workbook will help you master the fundamentals of Probability & Statistics. It offers you the support you need to boost your skills and helps you succeed in school and beyond!

It takes the guesswork out of math by explaining what you most need to know in a step-by-step format. When you apply what you learn from this workbook, you can

do better in class;

raise your grades, and

score higher on your high school math exams.

Each compact lesson in this book introduces a math concept and explains the method behind it in plain language. This is followed with lots of examples with fully worked-out solutions that take you through the key points of each problem.

The book gives you two tools to measure what you learn along the way:

Short drills that followeachlesson

Quizzes that test you onmultiplelessons

These tools are designed to comfortably build your test-taking confidence.

Meanwhile, the Math Flash feature throughout the book offers helpful tips and strategies—including advice on how to avoid common mistakes.

When you complete the lessons, take the final exam at the end of the workbook to see how far you’ve come. If you still need to strengthen your grasp on any concept, you can always go back to the related lesson and review at your own pace.

To the Parent

For many students, math can be a challenge—but with the right tools and support, your child can master the basics of algebra. As educational publishers, our goal is to help all students develop the crucial math skills they’ll need in school and beyond.

This Ready, Set, Go! Workbook is intended for students who need to build their basic geometry skills. It was specifically created and designed to assist students who need a boost in understanding and learning the math concepts that are most tested along the path to graduation. Through a series of easy-to-follow lessons, students are introduced to the essential mathematical ideas and methods, and then take short quizzes to test what they are learning.

Each lesson is devoted to a key mathematical building block. The concepts and methods are fully explained, then reinforced with examples and detailed solutions. Your child will be able to test what he or she has learned along the way, and then take a cumulative exam found at the end of the book.

Whether used in school with teachers, for home study, or with a tutor, the Ready, Set, Go! Workbook is a great support tool. It can help improve your child’s math proficiency in a way that’s fun and educational!

To the Teacher

As you know, not all students learn the same, or at the same pace. And most students require additional instruction, guidance, and support in order to do well academically.

Using the Curriculum Focal Points of the National Council of Teachers of Mathematics, this workbook was created to help students increase their math abilities and succeed on high school exams with special emphasis on high school proficiency exams. The book’s easy-to-follow lessons offer a review of the basic material, supported by examples and detailed solutions that illustrate and reinforce what the students have learned.

To accommodate different pacing for students, we provide drills and quizzes throughout the book to enable students to mark their progress. This approach allows for the mastery of smaller chunks of material and provides a greater opportunity to build mathematical competence and confidence.

When we field-tested this series in the classroom, we made every effort to ensure that the book would accommodate the common need to build basic math skills as effectively and flexibly as possible. Therefore, this book can be used in conjunction with lesson plans, stand alone as a single teaching source, or be used in a group-learning environment. The practice quizzes and drills can be given in the classroom as part of the overall curriculum or used for independent study. A cumulative exam at the end of the workbook helps students (and their instructors) gauge their mastery of the subject matter.

We are confident that this workbook will help your students develop the necessary skills and build the confidence they need to succeed on high school math exams.

About Research & Education Association

Founded in 1959, Research & Education Association (REA) is dedicated to publishing the finest and most effective educational materials—including software, study guides, and test preps—for students in elementary school, middle school, high school, college, graduate school, and beyond.

Today, REA’s wide-ranging catalog is a leading resource for teachers, students, and professionals.

We invite you to visit us at www.rea.com to find out how REA is making the world smarter.

About the Author

Author Mel Friedman is a former classroom teacher and test-item writer for Educational Testing Service and ACT, Inc.

Acknowledgments

We would like to thank Larry Kling, Vice President, Editorial, for his editorial direction; Pam Weston, Vice President, Publishing, for setting the quality standards for production integrity and managing the publication to completion; Alice Leonard, Senior Editor, for project management and preflight editorial review; Diane Goldschmidt, Senior Editor, for post-production quality assurance; Rachel DiMatteo, Graphic Artist, for page design; Christine Saul, Senior Graphic Artist, for cover design; and Jeff LoBalbo, Senior Graphic Artist, for post-production file mapping

We also gratefully acknowledge Heather Brashear for copyediting, and Kathy Caratozzolo of Caragraphics for typesetting.

A special thank you to Robin Levine-Wissing for her technical review of the math content.

Lesson One

Organizing Data—Part I

In this lesson, we will explore the major categories of data, using statistics. Statistics is a science that deals with analyzing data. We will also explore a way in which data can be organized into picture form. Different types of data are present in almost every part of our lives. As examples, you can find data in (a) baseball batting averages, (b) the numbers of people insured by various insurance companies, (c) the most popular brand of designer jeans, and (d) the highest salaries for school principals.

Your Goal: When you have completed this lesson, you should be able to classify data into different categories and be able to present data in a popular pictorial form.

There are two major classifications of data: qualitative and quantitative. Qualitative data is data that can be identified as nonnumerical. Examples would be (a) gender, (b) subjects taught in high school, (c) leading causes of cancer, and (d) nationally recognized holidays.

Quantitative data is data that can be identified as numerical, which implies that it can be ranked in some way. Examples would be (a) grade point averages, (b) bowling scores, (c) rankings of major cities in crime prevention, and (d) prices of gasoline at different service stations.

Also, quantitative data can be split into two subcategories, namely, discrete and continuous. Discrete data can be assigned a specific value and can be counted. Examples would be (a) the number of children in a classroom; (b) the annual salary of a person, in whole dollars; (c) the number of months in a year; and (d) attendance at a concert.

Continuous data is data that is not discrete. This type of data includes (a) the temperature at different times of the day, (b) the weight of each player on a football team, (c) the number of gallons of water in different swimming pools, and (d) the heights of adults at a party. Continuous data is obtained by measuring, not by counting. For this reason, continuous data really is an estimation. Thus, if we claim that a person weighs 160 pounds, we recognize that our accuracy is only as good as the scale that is being used. That is, the 160-pound person might really weigh 159.85 pounds or possibly 160.003 pounds. We realize that the scale is only accurate to the nearest pound.

As another example, let’s suppose that you ran a 100-yard dash. Suppose that your time was clocked at 11.2 seconds. It is possible that your time was closer to 11.22 seconds or even 11.19 seconds. However, the watch that was used to record your time was only accurate to the nearest tenth of a second.

When continuous data is used, there are presumed boundaries associated with a given value. In the above examples, we saw that a weight of 160 pounds would have boundaries of 159.5 pounds and 160.5 pounds. We would include the lower boundary of 159.5 but not include the upper boundary of 160.5. The reason is that to the nearest pound, 159.5 rounds off to 160, but 160.5 would actually round off to 161. For the number 11.2, the lower boundary would be 11.15, and the upper boundary would be 11.25. As you can see, the number 11.15 would be included as a possible value, but 11.25 would not be included.

It is important to remember that boundaries are only used for continuous data.

Boundaries for continuous data are commonly written as intervals. An interval will indicate all allowable values. Instead of writing the lower boundary as 159.5 and the upper boundary as 160.5, the boundaries may be written as 159.5–160.5. This means that values such as 159.7 and 160.233 would be included.

Are you curious about the technique that is used to establish boundaries for continuous data?

If the given data x is an integer, the lower boundary is x–0.5, and the upper boundary is x + 0.5.

If the given data x is written in tenths, the lower boundary is x–0.05, and the upper boundary is x + 0.05.

If the given data x is written in hundredths, the lower boundary is x–0.005, and the upper boundary is x + 0.005.

This process can be extended to any decimal number. Basically, we are subtracting and adding one-half unit to arrive at the boundaries.

1 Example: What are the boundaries for 52.81?

Solution:

A unit for this number is 0.01, so we need to subtract and add one-half of 0.01, which is 0.005.

Thus, the boundaries are 52.805–52.815.

Remember that 52.805 is included in this interval, but 52.815 is not included.

2 Example: What are the boundaries for 33.0?

Solution:

Be careful here! Even though 33.0 has the same numerical value as 33, the given measurement is shown to be accurate to the nearest tenth. We must subtract and add 0.05 to 33.0.

The answer is 32.95–33.05.

3 Example:

The length of a stapler is measured to be 4.85 inches. How many of the following numbers would be included in the interval containing the associated boundaries?

4.848 inches, 4.855 inches, 4.8449 inches, 4.9 inches, 4.8452 inches

Solution:

The correct boundaries are given by subtracting and adding 0.005, so that the interval is 4.845–4.855. Thus, two of these five numbers, namely, 4.848 and 4.8452, are included in this interval. Be sure you understand that the upper boundary of 4.855 is not included.

4 Example:

Which one(s) of the following are representative of qualitative data?

time needed to complete a project, number of houses in a city block, depth of an ocean, blood type

Solution:

Only blood type qualifies as qualitative data. Each of the other three represents a numerical value.

Quantities such as Social Security numbers, house addresses, and company badge numbers are also considered qualitative data. Even though they contain numbers, there is no ranking system involved. For example, we cannot say that one Social Security number has a greater value than another Social Security number.

One popular way to organize data is to represent it in a pie graph form. (Some books use the term pie chart or circle graph.) It is best to use this type of graph when we are looking at the percent contribution of the component parts of a particular category.

5 Example:

In the Growing Strong Hospital, a survey was taken of the marital status of each employee. The results showed that 20% are married with no children, 25% are married with at least one child, 15% are single, 30% are divorced, and the remaining 10% are widowed. Create an appropriate pie graph.

Solution:

Recall from your knowledge of geometry that there are 360° in a circle. In order to create the correct portions to represent each of these categories, we start in the center of the circle and construct an angle proportional to the percent assigned to each category. Here are the computations:

Married with No Children, (0.20)(360°) = 72°

Married with at Least One Child, (0.25)(360°) = 90°

Single, (0.15)(360°) = 54°

Divorced, (0.30)(360°) = 108°

Widowed, (0.10)(360°) = 36°

The pie graph should appear as follows:

GROWING STRONG HOSPITAL EMPLOYEES

Notice that the percents add up to 100%. This is a necessity for all pie graphs.

6 Example:

The mayor of the town of Peopleville has mailed a survey to each adult resident. The resident was asked to rate the quality of the mayor’s ability to govern the town. The five ratings given were (a) excellent, (b) very good, (c) average, (d) below average, and (e) poor. The results showed that

of the population ranked the mayor as excellent

ranked him as very good

ranked him as average

ranked him as below average

the remainder ranked him as poor

Create an appropriate pie graph.

Solution:

In order to determine the fraction of the population that ranked the mayor as poor, we calculate

Now, we must convert each fraction to the correct central angle of a circle. Again, remember that 360° represents a full circle.

The central angles corresponding to these five categories are as follows:

Excellent,

Very Good,

Average,

Below Average,

Poor,

The pie graph should appear as follows:

RANKING OF THE MAYOR OF PEOPLEVILLE

Notice that these fractions must add up to 1, which is equivalent to 100%. Sometimes you will be asked to compute an actual number that is associated with a specific sector of the pie graph. (From geometry, you know that a sector is a portion of any circle that is bounded by two radii and an included arc.)

7 Example:

Return to Example 5. Suppose there are a total of 1500 employees in the Growing Strong Hospital. How many of them are either single or divorced?

Solution:

The corresponding percent of single or divorced employees is 15% + 30% = 45%. Then (1500)(0.45) = 675 employees.

8 Example:

Return to Example 6. Assuming that each adult completed the survey, if 1416 adults had ranked the mayor as excellent, how many adults live in Peopleville?

Solution:

Let x represent the number of adults in Peopleville.

Which one of the following represents qualitative data?

Number of golf balls in a golf bag

Days of the week

Amount of salt, in grams, in a slice of cake

Number of motorists in Pennsylvania

Which one of the following represents discrete data?

Street addresses

Weights of cats in an animal shelter

Number of quarts of milk in a large bottle

Number of dollar bills in a bank teller’s drawer

If a horse runs a race in 123.8 seconds, what is the lower boundary for this number when it is written in interval form?

123.75 seconds

123.85 seconds