# Statistics Success in 20 Minutes a Day

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STATISTICSSUCCESS

in 20 Minutesa Day

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STATISTICSSUCCESSin 20 Minutesa Day

Linda J.Young

N E W Y O R K

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Copyright 2006 LearningExpress, LLC.

All rights reserved under International and Pan-American Copyright Conventions.

Published in the United States by LearningExpress, LLC, New York.

Library of Congress Cataloging-in-Publication Data:

Young, Linda J., 1952-

Statistics success in 20 minutes a day / Linda J.Young.

p. cm.

Includes bibliographical references.

ISBN 1-57685-535-X

1. Mathematical statisticsProblems, exercises, etc. I. Title.

QA276.2.Y68 2005

519.5dc22 2005027521

Printed in the United States of America

9 8 7 6 5 4 3 2 1

ISBN 1-57685-535-X

For information on LearningExpress, other LearningExpress products, or bulk sales, please write to us at:

LearningExpress

55 Broadway

8th Floor

New York, NY 10006

Or visit us at:

www.learnatest.com

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Linda J. Youngis professor of statistics and director of biostatistics at the University of Florida. She has

previously served on the faculties at Oklahoma State University and the University of Nebraska.

About the Author

v

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Introduction How to Use This Book ix

Pretest 1Lesson 1 Populations, Samples, and Variables 13

Lesson 2 Studies 17

Lesson 3 Describing and Displaying Categorical Data 25

Lesson 4 Dotplots and Stem-and-Leaf Plots 33

Lesson 5 Measures of Central Tendency for Numerical Data 41

Lesson 6 Measures of Dispersion for Numerical Data 47

Lesson 7 Histograms and Boxplots 55

Lesson 8 Describing and Displaying Bivariate Data 65

Lesson 9 Basic Ideas in Probability 73

Lesson 10 Discrete Probability Distributions 85Lesson 11 Continuous Probability Distributions 91

Lesson 12 Sampling Distributions and the t-Distribution 101

Lesson 13 The Law of Large Numbers and the Central Limit Theorem 111

Lesson 14 Sample Surveys 117

Lesson 15 Confidence Intervals for Proportions 123

Lesson 16 Hypothesis Testing for Proportions 129

Lesson 17 Confidence Intervals and Tests of Hypotheses for Means 137

Lesson 18 The Matched-Pairs Design and Comparing Two Treatment Means 145

Lesson 19 Confidence Intervals for Comparing Two Treatment or Population Means 157

Lesson 20 Analyzing Categorical Data 169Posttest 181

Answer Key 193

Appendix How to Prepare for a Test 207

Contents

vi i

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Introduction

ix

If you have never taken a statistics course, and now find that you need to know the basics of statistics

this is the book for you. If you have already taken a statistics course, but felt like you never understood

what the teacher was trying to tell youthis book can teach you what you need to know. If it has been

a while since you have taken a statistics course, and you need to refresh your skillsthis book will review the

basics and reteach you the skills you may have forgotten. Whatever your reason for needing to know statis-

tics, Statistics Success in 20 Minutes a Daywill teach you what you need to know.It gives you the statistics basics

in clear and straightforward lessons that you can do at your own pace.

How to Use This Book

Statistics Success teaches basic concepts in 20 self-paced lessons. The book includes a pretest, a posttest, and

tips on how to prepare for a standardized test. Before you begin Lesson 1, take the pretest to assess your cur-

rent statistics abilities. The answer key follows the pretest. This will be helpful in determining your strengths

and weaknesses. After taking the pretest, move on to Lesson 1.Each lesson offers detailed explanations of a new concept. There are numerous examples with step-by-

step solutions.As you proceed through a lesson, you will find tips and shortcuts that will help you learn a con-

cept. Each new concept is followed by a set of practice problems. The answers to the practice problems are

in the answer key located at the end of the book.

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When you have completed all 20 lessons, take the posttest at the end of the book. The posttest has the

same format as the pretest, but the questions are different. Compare the results of the posttest with the results

of the pretest. What are your strengths? Do you have weak areas? Do you need to spend more time on some

concepts, or are you ready to go to the next level?

Make a Commitment

Success does not come without effort. If you truly want to be successful, make a commitment to spend the

time you need to improve your statistics skills. When you achieve statistics success, you have laid the foun-

dation for future challenges and opportunities.

So sharpen your pencil and get ready to begin the pretest!

INTRODUCTION

x

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Before you begin Lesson 1, you may want to get an idea of what you know and what you need to learn.

The pretest will answer some of these questions for you. The pretest consists of 50 multiple-choice

questions covering the topics in this book. Although 50 questions cant cover every concept, skill, or

shortcut taught in this book, your performance on the pretest will give you a good indication of your strengths

and weaknesses. Keep in mind, the pretest does not test all the skills taught in this statistics book.

If you score high on the pretest, you have a good foundation and should be able to work your way

through the book quickly. If you score low on the pretest, dont despair. This book will take you through the

statistics concepts step by step. If you get a low score, you may need to take more than 20 minutes a day to

work through a lesson. However, this is a self-paced program, so you can spend as much time on a lesson as

you need. You decide when you fully comprehend the lesson and are ready to go on to the next one.

Take as much time as you need to do the pretest.You will find that the level of difficulty increases as you

work your way through the pretest.

Pretest

1

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AN SW ER SH EE T

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Pretest

1. The time it takes an employee to drive to work isthe variable of interest. What type of variable is

being observed?

a. categorical variable

b. continuous variable

c. discrete variable

d. explanatory variable

2. A study was conducted to compare two different

approaches to preparing for an exam. Twenty

high school students taking chemistry volun-teered to participate. Ten were randomly

assigned to use the first approach; the other ten

used the second approach. Each ones perform-

ance on the next chemistry exam was recorded.

What type of study is this?

a. experiment with a broad scope of inference

b. experiment with a narrow scope of inference

c. sample survey

d. observational study

3. Random digit dialing was used to select house-

holds in a particular state. An adult in each

household contacted was asked whether the

household had adequate health insurance. A

critic of the poll said that the results were biased

because households without telephones were not

included in the survey. As a consequence, the

estimated percentage of households that had

adequate health insurance was biased upward.

What type of bias was the critic concerned

about?

a. measurement bias

b. nonresponse bias

c. response bias

d. selection bias

4. A study was conducted to determine whether a

newly developed rose smelled better than the

rose of the standard variety. Twenty students

were randomly selected from a large high schoolto participate in a smell study. Each selected

student smelled both roses in a random order

and selected the one that smelled best. What is

the population of interest and what are the

response and explanatory variables?

a. The population is all students at the large

high school; the response variable is the rose

choice; and the explanatory variable is the

type of rose.

b. The population is all students at the largehigh school; the response variable is the type

of rose; and the explanatory variable is the

rose choice.

c. The population is all roses of these two types;

the response variable is the rose choice; and

the explanatory variable is the type of rose.

d. The population is all roses of these two types;

the response variable is the type of rose; and

the explanatory variable is the rose choice.

For problems 5 and 6, consider the following 12 data

points: 10, 12, 10, 18, 16, 15, 9, 14, 11, 13, 12, and 16.

5. What is the median of these data?

a. 9

b. 12

c. 12.5

d. 13

6. What is the interquartile range of these data?

a. 4

b. 5

c. 6

d. 9

P R E T E S T

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7. What does the length of the box in a boxplot

represent?

a. the range

b.the interquartile range

c. the median

d. the mean

8. How does one standardize a random variable?

a. Add the mean.

b. Subtract the mean.

c. Add the mean and divide by the standard

deviation.

d. Subtract the mean and divide by the standard

deviation.

For problems 9 and 10, consider this information: On

any given day, the probability it will rain is 0.32; the

probability the wind will blow is 0.2; and the proba-

bility that it will rain and the wind will blow is 0.1.

9. For a randomly selected day, what is the proba-

bility that it will rain or the wind will blow?

a. 0.42

b. 0.52c. 0.58

d. 0.62

10. For a randomly selected day, what is the proba-

bility that it will NOT rain and the wind will

NOT blow?

a. 0.38

b. 0.48

c. 0.58

d. 0.90

Use the following information for problems 11, 12,

and 13. The students in a small high school were sur-

veyed. Each student was asked whether he or she used

a safety belt whenever driving. This information andthe gender of the student was recorded as follows:

11. What is the probability that a randomly selected

student is a male who does not use his seat belt?

a.

b.

c.

d.

12. What is the probability that a randomly selected

student is a female given that the person is a seat

belt user?

a.

b.

c.

d.151205

82

205

82

151

82

105

54

205

31205

23

105

31

100

P R E T E S T

3

USE OF SAFETY BELTS

USE SAFETY BELT?

GENDER YES NO TOTALS

Female 82 23 105

Male 69 31 100

Totals 151 54 205

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13. Is the use of a safety belt independent of gender?

a. no, because the probability that a randomly

chosen student is a female does not equal the

probability of female given safety belt useb. no, because the number of females who use a

safety belt is not equal to the number of

males who use a seat belt

c. yes, because the sample was randomly

selected

d. yes, because both genders do use seat belts

more than they do notuse seat belts

For problems 14 and 15, consider that 1% of a popu-

lation has a particular disease. A new test for identify-ing the disease has been developed. If the person has

the disease, the test is positive 94% of the time. If the

person does not have the disease,the test is positive 2%

of the time.

14. What is the probability that a randomly selected

person from this population tests positive?

a. 0.0292

b. 0.0094

c. 0.096d. 0.96

15. A person is randomly selected from this popula-

tion and tested. She tests positive. Which of the

following best represents the probability that she

has the disease?

a. 0.0094

b. 0.32

c. 0.34

d. 0.94

Use the following information for problems 16, 17,

and 18. On any given day, the probability that Megan

will be late for work is 0.2. Whether or not she is late

to work is independent from day to day.

16. Megan was late to work today.What is the

probability that she will NOT be late to work

tomorrow?

a.0.16

b. 0.2

c. 0.6

d. 0.8

17. Which of the following is closest to the probabil-

ity that Megan will be late to work at least one of

the five days next week?

a. 0.00032

b. 0.33

c. 0.41d. 0.67

18. What is the probability that Megan will be on

time exactly three days and then be late on the

fourth one?

a. 0.0064

b. 0.1024

c. 0.16

d. 0.512

19. A train is scheduled to leave the station at 3 P.M.

However, it is equally likely to actually leave the

station any time from 2:55 to 3:15 P.M. What is

the probability it will depart the station early?

a. 0.25

b. 0.33

c. 0.67

d. 0.75

20. Let zbe a standard normal random variable.

Find the probability that a randomly selected

value ofzis between2.1 and 0.4.

a. 0.1079

b. 0.3446

c. 0.5475

d. 0.6554

P R E T E S T

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21. Let zbe a standard normal random variable.

Find z* such that the probability that a randomly

selected value ofzis greater than z* is 0.2.

a. 0.84

b. 0.4207

c. 0.5793

d. 0.84

22. LetXbe a normal random variable with mean

20 and standard deviation 5.What is the proba-

bility that a randomly selected value ofXis

between 15 and 25?

a. 0.32

b. 0.68c. 0.95

d. 0.997

23. A random sample of size 25 is selected from a

population that is normally distributed with a

mean of 15 and a standard deviation of 4. What

is the sampling distribution of the sample mean?

a. normal with a mean of 0 and a standard

deviation of 1

b. normal with a mean of 15 and a standarddeviation of 0.16

c. normal with a mean of 15 and a standard

of 0.8

d. normal with a mean of 15 and a standard

deviation of 4

24. Find t* such that the probability that a randomly

selected observation from a t-distribution with

16 degrees of freedom is less than t* is 0.1.

a. 1.746

b. 1.337

c. 1.337

d. 1.746

25. A researcher decides to study the bite strength of

alligators. She believes that if she takes a large

enough random sample, she will be able to say

that the average of the bite strengths she recordswill be close to the mean bite strength of all the

alligators in the population she is studying. Is

she correct?

a. No. One can never be sure that the sample

mean is close to the population mean.

b. Yes. By the Central Limit Theorem, the sam-

ple mean will be equal to the population

mean ifn 30.

c. Yes. By the Central Limit Theorem, the sam-

ple mean will be...