Understanding Basic Statistics,

Understanding Basic Statistics

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Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

UnderstandingBasic Statistics

Charles Henry BraseRegis University

Corrinne Pellillo BraseArapahoe Community College

F I F T H E D I T I O N

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Understanding Basic Statistics, Fifth EditionCharles Henry Brase and Corrinne Pellillo Brase

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Printed in Canada

1 2 3 4 5 6 7 12 11 10 09 08

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This book is dedicated to the memory of

a great teacher, mathematician, and friend

Burton W. Jones

Professor Emeritus, University of Colorado

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vii

Contents

Preface xix

Table of Prerequisite Material 1

1 Getting Started 2

FOCUS PROBLEM: Where Have All the Fireflies Gone? 31.1 What Is Statistics? 41.2 Random Samples 121.3 Introduction to Experimental Design 20

Summary 28Important Words & Symbols 28Chapter Review Problems 29Data Highlights: Group Projects 31Linking Concepts: Writing Projects 31

USING TECHNOLOGY 32

2 Organizing Data 34

FOCUS PROBLEM: Say It with Pictures 352.1 Frequency Distributions, Histograms, and Related Topics 362.2 Bar Graphs, Circle Graphs, and Time-Series Graphs 482.3 Stem-and-Leaf Displays 56


USING TECHNOLOGY 70

3 Averages and Variation 72

FOCUS PROBLEM: The Educational Advantage 733.1 Measures of Central Tendency: Mode, Median, and Mean 743.2 Measures of Variation 833.3 Percentiles and Box-and-Whisker Plots 98


USING TECHNOLOGY 114

CUMULATIVE REVIEW PROBLEMS: Chapters 1–3 115

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viii Contents

4 Correlation and Regression 118

FOCUS PROBLEM: Changing Populations and Crime Rate 1194.1 Scatter Diagrams and Linear Correlation 1204.2 Linear Regression and the Coefficient of Determination 135



5 Elementary Probability Theory 156

FOCUS PROBLEM: How Often Do Lie Detectors Lie? 1575.1 What Is Probability? 1585.2 Some Probability Rules—Compound Events 1675.3 Tree Diagrams and Counting Techniques 184



6 The Binomial Probability Distribution and Related Topics 202

FOCUS PROBLEM: Personality Preference Types: Introvert or Extrovert? 2036.1 Introduction to Random Variables and Probability Distributions 2046.2 Binomial Probabilities 2146.3 Additional Properties of the Binomial Distribution 226




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Contents ix

7 Normal Curves and Sampling Distributions 242

FOCUS PROBLEM: Impulse Buying 2437.1 Graphs of Normal Probability Distributions 2447.2 Standard Units and Areas Under the Standard Normal Distribution 2517.3 Areas Under Any Normal Curve 2617.4 Sampling Distributions 2747.5 The Central Limit Theorem 2797.6 Normal Approximation to the Binomial Distribution 289



8 Estimation 306

FOCUS PROBLEM: Trick or Treat!!! 3078.1 Estimating mWhen s Is Known 3088.2 Estimating mWhen s Is Unknown 3198.3 Estimating p in the Binomial Distribution 331



9 Hypothesis Testing 350

FOCUS PROBLEM: Benford’s Law: The Importance of Being Number 1 3519.1 Introduction to Statistical Tests 3529.2 Testing the Mean m 3679.3 Testing a Proportion p 383




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x Contents

10 Inferences About Differences 404

FOCUS PROBLEM: The Trouble with Wood Ducks 40510.1 Tests Involving Paired Differences (Dependent Samples) 40610.2 Inferences About the Difference of Two Means �1 � �2 41810.3 Inferences About the Difference of Two Proportions p1 � p2 438



11 Additional Topics Using Inference 456

FOCUS PROBLEM: Archaeology in Bandelier National Monument 457

Part I: Inferences Using the Chi-Square Distribution 458Overview of the Chi-Square Distribution 458

11.1 Chi-Square: Tests of Independence and of Homogeneity 45911.2 Chi-Square: Goodness of Fit 47311.3 Testing a Single Variance or Standard Deviation 482

Part II: Inferences Relating to Linear Regression 49011.4 Inferences for Correlation and Regression 490




Appendix: Tables A1

Table 1: Random Numbers A1Table 2: Binomial Probability Distribution Cn,rp

rqn�r A3Table 3: Areas of a Standard Normal Distribution A8Table 4: Critical Values for Student’s t Distribution A10Table 5: The �2 Distribution A11

Photo Credits A12

Answers and Key Steps to Odd-Numbered Problems A13

Index I1

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xi Chapter 2 Organizing Data

xi

� NEW! Critical ThinkingCritical thinking is an importantskill for students to develop in order to avoid reachingmisleading conclusions. TheCritical Thinking feature providesadditional clarification onspecific concepts as a safeguard against incorrect evaluation of information.

NEW! Interpretation �

Increasingly, calculators andcomputers are used to generatethe numeric results of a statisticalprocess. However, the studentstill needs to correctly interpretthose results in the context of a particular application. TheInterpretation feature callsattention to this important step.

� NEW! Critical Thinking ExercisesIn every section and chapter problem set, CriticalThinking problems provide students with theopportunity to test their understanding of theapplication of statistical methods and theirinterpretation of their results.

(b) Assuming the milk is not contaminated, what is the probability that the aver-age bacteria count for one day is between 2350 and 2650 bacteria permilliliter?

SOLUTION: We convert the interval

to a corresponding interval on the standard z axis.

Therefore,

The probability is 0.9988 that is between 2350 and 2650.

(c) INTERPRETATION At the end of each day, the inspector must decide to acceptor reject the accumulated milk that has been held in cold storage awaitingshipment. Suppose the 42 samples taken by the inspector have a mean bacte-ria count that is not between 2350 and 2650. If you were the inspector,what would be your comment on this situation?

SOLUTION: The probability that is between 2350 and 2650 is very high. If theinspector finds that the average bacteria count for the 42 samples is not between2350 and 2650, then it is reasonable to conclude that there is something wrongwith the milk. If is less than 2350, you might suspect someone added chemi-cals to the milk to artificially reduce the bacteria count. If is above 2650, youmight suspect some other kind of biologic contamination.

xx

x

x

x

= 0.9988

= 0.9994 - 0.0006

P(2350 … x … 2650) = P(-3.24 … z … 3.24)

x = 2650 converts to z =2650 - 2500

46.3� 3.24

x = 2350 converts to z =2350 - 2500

46.3� -3.24

z =x - ms�1n

�x - 2500

46.3

2350 … x … 2650

x

Chapter 7 NORMAL CURVES AND SAMPLING DISTRIBUTIONS

Critical Thinking

Students need to develop critical thinking skills in order to understand and evaluate the limitations of statistical methods. Understanding Basic Statistics makes students aware ofmethod appropriateness, assumptions, biases, and justifiable conclusions.

5. Critical Thinking Consider the numbers

2 3 4 5 5

(a) Compute the mode, median, and mean.(b) If the numbers represented codes for the colors of T-shirts ordered from a

catalog, which average(s) would make sense?(c) If the numbers represented one-way mileages for trails to different lakes,

which average(s) would make sense?(d) Suppose the numbers represent survey responses from 1 to 5, with 1 � disagree

strongly, 2 � disagree, 3 � agree, 4 � agree strongly, and 5 � agree verystrongly. Which averages make sense?

6. Critical Thinking Consider a data set of 15 distinct measurements with mean A andmedian B.(a) If the highest number were increased, what would be the effect on the me-

dian and mean? Explain.(b) If the highest number were decreased to a value still larger than B, what

would be the effect on the median and mean?(c) If the highest number were decreased to a value smaller than B, what would

be the effect on the median and mean?

We see from the central limit theorem that the sample mean is an unbiasedestimator of the mean m when The variability of decreases as the sam-ple size increases.

In Section 8.3, we will see that the sample proportion is an unbiased estima-tor of the population proportion of successes p in binomial experiments withsufficiently large numbers of trials n. Again, we will see that the variability of decreases with increasing numbers of trials.

The sample variance is an unbiased estimator for the population variance s2.s2

pN

pN

xn Ú 30.x

CRITICAL THINKING Bias and Variability

Whenever we use a sample statistic as an estimate of a population parameter, weneed to consider both bias and variability of the statistic.

A sample statistic is unbiased if the mean of its sampling distribution equalsthe value of the parameter being estimated.

The spread of the sampling distribution indicates the variability of thestatistic. The spread is affected by the sampling method and the sample size.Statistics from larger random samples have spreads that are smaller.

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xii Chapter 2 Organizing Data

� NEW! StatisticalLiteracy ProblemsIn every section and chapterproblem set, Statistical Literacyproblems test student under-standing of terminology, statisti-cal methods, and theappropriate conditions for useof the different processes.

Statistical Literacy

No language can be spoken without learning the vocabulary, including statistics.Understanding Basic Statistics introduces statistical terms with deliberate care.

Box-and-Whisker PlotsThe quartiles together with the low and high data values give us a very usefulfive-number summary of the data and their spread.Five-number summary

Five-number summary

Lowest value, Q1, median, Q3, highest value

We will use these five numbers to create a graphic sketch of the data called abox-and-whisker plot. Box-and-whisker plots provide another useful techniquefrom exploratory data analysis (EDA) for describing data.

Box-and-whisker plot

Section 5.1Probability of an event A, P(A)

Relative frequency

Law of large numbers

Equally likely outcomes

Statistical experiment

Simple event

Sample space

Complement of event A

Section 5.2Independent events

Dependent events

A B

Conditional probability

Multiplication rules of probability (for

independent and dependent events)

A and BMutually exclusive events

Addition rules (for mutually exclusive and

general events)

A or B

Section 5.3Multiplication rule of counting

Tree diagram

Permutations rule

Combinations rule�

IMPORTANTWORDS &SYMBOLS

Definition Boxes �

Whenever important termsare introduced in text,yellow definition boxesappear within thediscussions. These boxesmake it easy to referenceor review terms as theyare used further.

� Important Words & SymbolsThe Important Words & Symbols within the Chapter Review feature at theend of each chapter summarizes the terms introduced in the DefinitionBoxes for student review at a glance.

SECTION 7.1 PROBLEMS

1. Statistical Literacy Which, if any, of the curves in Figure 7-7 look(s) like a

normal curve? If a curve is not a normal curve, tell why.

2. Statistical Literacy Look at the normal curve in Figure 7-8, and find

and s.

m, m + s,

FIGURE 7-7

16 18 20 22

FIGURE 7-8

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xiiixiii xiii

g p p y g

Discuss each of the following topics in class or review the topics on your own. Then

write a brief but complete essay in which you summarize the main points. Please in-

clude formulas and graphs as appropriate.

1. An average is an attempt to summarize a collection of data into just one number.

Discuss how the mean, median, and mode all represent averages in this context.

Also discuss the differences among these averages. Why is the mean a balance

point? Why is the median a midway point? Why is the mode the most common

data point? List three areas of daily life in which you think one of the mean,

median, or mode would be the best choice to describe an “average.”

2. Why do we need to study the variation of a collection of data? Why isn’t the

average by itself adequate? We have studied three ways to measure variation.

The range, the standard deviation, and, to a large extent, a box-and-whisker plot

all indicate the variation within a data collection. Discuss similarities and differ-

ences among these ways of measuring data variation. Why would it seem reason-

able to pair the median with a box-and-whisker plot and to pair the mean with

the standard deviation? What are the advantages and disadvantages of each

method of describing data spread? Comment on statements such as the follow-

ing: (a) The range is easy to compute, but it doesn’t give much information;

(b) although the standard deviation is more complicated to compute, it has some

significant applications; (c) the box-and-whisker plot is fairly easy to construct,

and it gives a lot of information at a glance.

LINKING CONCEPTS:WRITING PROJECTS

Statistical Literacy

period be (to the nearest month)?

31. Expand Your Knowledge: Estimating the Standard Deviation ConsumerReports gave information about the ages at which various household products

are replaced. For example, color TVs are replaced at an average age of

years after purchase, and the (95% of data) range was from 5 to 11 years. Thus,

the range was years. Let x be the age (in years) at which a color TV

is replaced. Assume that x has a distribution that is approximately normal.

(a) The empirical rule (Section 7.1) indicates that for a symmetrical and bell-

shaped distribution, approximately 95% of the data lies within two standard

deviations of the mean. Therefore, a 95% range of data values extending

from to is often used for “commonly occurring” data val-

ues. Note that the interval from to is in length. This leads

to a “rule of thumb” for estimating the standard deviation from a 95% range

of data values.

4sm + 2sm - 2s

m + 2sm - 2s

11 - 5 = 6

m = 8

pp ymal distributions, we can get a much more precise result, which is given by theempirical rule.

Empirical rule

For a distribution that is symmetrical and bell-shaped (in particular, for anormal distribution):

Approximately 68% of the data values will lie within 1 standarddeviation on each side of the mean.

Approximately 95% of the data values will lie within 2 standarddeviations on each side of the mean.

Approximately 99.7% (or almost all) of the data values will lie within 3 standard deviations on each side of the mean.

Empirical rule

Problems 31 through 35 of Section 7.3show how to use the empirical rule toestimate the standard deviation whenwe know the low and high values ofsample data drawn from a distributionthat is approximately normal.

� Expand YourKnowledge ProblemsExpand Your Knowledgeproblems present optionalenrichment topics that gobeyond the material introducedin a section. Vocabulary andconcepts needed to solve theproblems are included at point-of-use, expanding students’statistical literacy.

Linking Concepts:Writing Projects �

Much of statistical literacyis the ability to communi-cate concepts effectively.The Linking Concepts:Writing Projects feature at the end of each chaptertests both statistical literacyand critical thinking by asking the student toexpress their understand-ing in words.

NEW! ExpandYour Knowledge:Foreshadowing �

The Foreshadowing featurealerts students to additionalmaterial presented in ExpandYour Knowledge problems.Students can skip ahead tolearn more, or investigate later.

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Chapter Preview �

QuestionsPreview Questions at thebeginning of each chaptergive the student a taste ofwhat types of questions can be answered with anunderstanding of the knowledge to come.

F O C U S P R O B L E M

Estimation

P R E V I E W Q U EST I O N S

How do you estimate the expected value of a random variable? Whatassumptions are needed? How much confidence should beplaced in such estimates? (SECTION 8.1)

At the beginning design stage of a statistical project, how large asample size should you plan to get? (SECTION 8.1)

What famous statistician worked for Guinness brewing company inIreland? What has this to do with constructing estimates fromsample data? (SECTION 8.2)

How do you estimate the proportion p of successes in a binomialexperiment? How does the normal approximation fit into thisprocess? (SECTION 8.3)

Direction and Purpose

Real knowledge is delivered through direction, not just facts. UnderstandingBasic Statistics ensures the student knows what is being covered and why atevery step along the way to statistical literacy.

F O C U S P R O B L E M

Trick or Treat!!!

About 28% of U.S. households turn out the lights and pretend not to be athome on Halloween (Source: Are You Normal About Money? by BerniceKanner, Bloomberg Press).

Alice is a sociology student who is studying theaffluent Cherry Creek neighborhood in Denver. As partof a larger survey, Alice interviewed a random sample of35 households. One of the questions she asked waswhether the resident turned out the lights and pretendednot to be at home on Halloween. It was found that11 of the 35 residents actually did this practice.

(a) Compute a 90% confidence interval for p, theproportion of all households in Cherry Creek thatpretend not to be at home on Halloween.

(b) What assumptions are necessary to calculate theconfidence interval of part (a)? Do you think theseassumptions are met in this case? Explain.

(c) The national proportion is about 0.28. Is 0.28 in theconfidence interval you computed? Based on youranswer, does it seem that the Cherry Creek neigh-borhood is much different (either higher or lowerproportion) from the population of all U.S. house-holds? Explain.

(See Problem 12 of Section 8.3.)

307

� Chapter Focus ProblemsThe Preview Questions in each chapter are followed by FocusProblems, which serve as more specific examples of whatquestions the student will soon be able to answer. The FocusProblems are set within appropriate applications and are incor-porated into the end-of-section exercises, giving students theopportunity to test their understanding.

12. Focus Problem: Trick or Treat In a survey of a random sample of 35 households

in the Cherry Creek neighborhood of Denver, it was found that 11 households

turned out the lights and pretended not to be home on Halloween.

(a) Compute a 90% confidence interval for p, the proportion of all households

in Cherry Creek that pretend not to be home on Halloween.

(b) What assumptions are necessary to calculate the confidence interval of part (a)?

(c) Interpretation: The national proportion of all households in the United States

that turn out the lights and pretend not to be home on Halloween is 0.28. Is

0.28 in the confidence interval you computed? Based on your answer, does it

seem that the Cherry Creek neighborhood is much different (either higher or

lower proportion) from the population of all U.S. households? Explain.

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Focus Points �

Each section opens withbulleted Focus Pointsdescribing the primarylearning objectives ofthe section.

S EC T I O N 3 . 1 Measures of Central Tendency: Mode, Median, and Mean

FOCUS POINTS

• Compute mean, median, and mode from raw data.• Interpret what mean, median, and mode tell you.• Explain how mean, median, and mode can be affected by extreme data values.• What is a trimmed mean? How do you compute it?• Compute a weighted average.

The average price of an ounce of gold is $920. The Zippy car averages 39 miles pergallon on the highway. A survey showed the average shoe size for women is size 8.

In each of the preceding statements, one number is used to describe the entiresample or population. Such a number is called an average. There are many waysto compute averages, but we will study only three of the major ones.

The easiest average to compute is the mode.

The mode of a data set is the value that occurs most frequently.

EXAMPLE 1 ModeCount the letters in each word of this sentence and give the mode. The numbersof letters in the words of the sentence are

5 3 7 2 4 4 2 4 8 3 4 3 4

Scanning the data, we see that 4 is the mode because more words have 4 lettersthan any other number. For larger data sets, it is useful to order—or sort—thedata before scanning them for the mode.

Direction and Purpose

Organizing and presenting data are the main

purposes of the branch of statistics called de-

scriptive statistics. Graphs provide an important

way to show how the data are distributed.

• Frequency tables show how the data are dis-

tributed within set classes. The classes are

chosen so that they cover all data values and

so that each data value falls within only one

class. The number of classes and the class

width determine the class limits and class

boundaries. The number of data values

falling within a class is the class frequency.

• A histogram is a graphical display of the in-

formation in a frequency table. Classes are

shown on the horizontal axis, with corre-

sponding frequencies on the vertical axis.

Relative-frequency histograms show relative

frequencies on the vertical axis. Dotplots

are like histograms except that the classes

are individual data values.

• Bar graphs, Pareto charts, and pie charts

are useful for showing how quantitative or

qualitative data are distributed over chosen

categories.

• Time-series graphs show how data change

over set intervals of time.

• Stem-and-leaf displays are an effective means

of ordering data and showing important

features of the distribution.

Graphs aren’t just pretty pictures. They help

reveal important properties of the data distribu-

tion, including the shape and whether or not

there are any outliers.

Chapter Review

SUMMARY

� REVISED! Chapter Summaries The Summary within each Chapter Review feature now alsoappears in bulleted form, so students can see what they needto know at a glance.

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Real-World Skills

Statistics is not done in a vacuum. Understanding Basic Statistics gives studentsvaluable skills for the real world with technology instruction, genuine applications,actual data, and group projects.

Using Technology

Binomial DistributionsAlthough tables of binomial probabilities can be found in

most libraries, such tables are often inadequate. Either the

value of p (the probability of success on a trial) you are

looking for is not in the table, or the value of n (the num-

ber of trials) you are looking for is too large for the table.

In Chapter 7, we will study the normal approximation to

the binomial. This approximation is a great help in many

practical applications. Even so, we sometimes use the for-

mula for the binomial probability distribution on a com-

puter or graphing calculator to compute the probability

we want.

ApplicationsThe following percentages were obtained over many years

of observation by the U.S. Weather Bureau. All data listed

are for the month of December.

Long-Term Mean % Location of Clear Days in Dec.

Juneau, Alaska 18%

Seattle, Washington 24%

Hilo, Hawaii 36%

Honolulu, Hawaii 60%

Las Vegas, Nevada 75%

Phoenix, Arizona 77%

Adapted from Local Climatological Data, U.S. Weather Bureau publication,“Normals, Means, and Extremes” Table.

In the locations listed, the month of December is a rel-

atively stable month with respect to weather. Since

weather patterns from one day to the next are more or

less the same, it is reasonable to use a binomial probabil-

ity model.

1. Let r be the number of clear days in December. Since

December has 31 days, Using appropri-

ate computer software or calculators available to you,

find the probability P(r) for each of the listed locations

when 1, 2, . . . , 31.r = 0,

0 … r … 31.

2. For each location, what is the expected value of the

probability distribution? What is the standard

deviation?

You may find that the use of cumulative probabilities and

appropriate subtraction of probabilities, rather than

adding probabilities, will make finding the solutions to

Applications 3 to 7 easier.

3. Estimate the probability that Juneau will have at most

7 clear days in December.

4. Estimate the probability that Seattle will have from 5

to 10 (including 5 and 10) clear days in December.

5. Estimate the probability that Hilo will have at least 12

clear days in December.

6. Estimate the probability that Phoenix will have 20 or

more clear days in December.

7. Estimate the probability that Las Vegas will have

from 20 to 25 (including 20 and 25) clear days in

December.

Technology Hints

TI-84Plus/TI-83Plus, Excel, MinitabThe Tech Notes in Section 6.2 give specific instructions

for binomial distribution functions on the TI-84Plus and

TI-83Plus calculators, Excel, and Minitab.

SPSSIn SPSS, the function PDF.BINOM(q,n,p) gives the proba-

bility of q successes out of n trials, where p is the proba-

bility of success on a single trial. In the data editor, name

a variable r and enter values 0 through n. Name another

variable Prob_r. Then use the menu choices Transform �Compute. In the dialogue box, use Prob_r for the target

variable. In the function box, select PDF.BINOM(q,n,p).Use the variable r for q and appropriate values for n and

p. Note that the function CDF.BINOM(q,n,p) gives the

cumulative probability of 0 through q successes.

� REVISED! Using TechnologyFurther technology instruction is available atthe end of each chapter in the Using Technologysection. Problems arepresented with real-worlddata from a variety ofdisciplines that can be solved by using TI-84 Plus and TI-83 Pluscalculators, Microsoft Excel,Minitab, and SPSS.

Tech Notes �

Tech Notes appearing throughout thetext give students helpful hints on usingTI-84 Plus and TI-83 Plus calculators,Microsoft Excel, and Minitab to solve aproblem. They include display screens tohelp students visualize and better under-stand the solution.

Minitab Release 14 Stem-and-Leaf Display (for Data inGuided Exercise 4)

The values shown in the left column represent depth. Numbers above the value in

parentheses show the cumulative number of values from the top to the stem of the

middle value. Numbers below the value in parentheses show the cumulative number of

values from the bottom to the stem of the middle value. The number in parentheses

shows how many values are on the same line as the middle value.

TECH NOTES Stem-and-leaf display

TI-84Plus/TI-83Plus Does not support stem-and-leaf displays. You can sort the data by

using keys Stat � Edit � 2:SortA.

Excel Does not support stem-and-leaf displays. You can sort the data by using the

menu choices Data � Sort.

Minitab Use the menu selections Graph � Stem-and-Leaf and fill in the dialogue box.

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g y g g g g

are 2 for new contacts, 3 for successful contacts, 3 for total contacts, 5 for dollar

value of sales, and 3 for reports. What would the overall rating be for a sales rep-

resentative with ratings of 5 for new contacts, 8 for successful contacts, 7 for

total contacts, 9 for dollar volume of sales, and 7 for reports?

Old Faithful Geyser, YellowstoneNational Park

DATA H IGHLIGHTS:GROUP PROJECTS

Break into small groups and discuss the following topics. Organize a brief outline in

which you summarize the main points of your group discussion.

1. The Story of Old Faithful is a short book written by George Marler and pub-

lished by the Yellowstone Association. Chapter 7 of this interesting book talks

about the effect of the 1959 earthquake on eruption intervals for Old Faithful

Geyser. Dr. John Rinehart (a senior research scientist with the National Oceanic

and Atmospheric Administration) has done extensive studies of the eruption in-

tervals before and after the 1959 earthquake. Examine Figure 3-11. Notice the

general shape. Is the graph more or less symmetrical? Does it have a single mode

frequency? The mean interval between eruptions has remained steady at about

65 minutes for the past 100 years. Therefore, the 1959 earthquake did not signif-

icantly change the mean, but it did change the distribution of eruption intervals.

Examine Figure 3-12. Would you say there are really two frequency modes, one

shorter and the other longer? Explain. The overall mean is about the same for

both graphs, but one graph has a much larger standard deviation (for eruption

intervals) than the other. Do no calculations, just look at both graphs, and then

explain which graph has the smaller and which has the larger standard devia-

tion. Which distribution will have the larger coefficient of variation? In everyday

terms, what would this mean if you were actually at Yellowstone waiting to see

the next eruption of Old Faithful? Explain your answer.

Typical Behavior of Old Faithful Geyser Before 1959 Quake

FIGURE 3-11

Typical Behavior of Old Faithful Geyser After 1959 Quake

FIGURE 3-12

Real-World Skills

� UPDATED! ApplicationsReal-world applications are used from the beginning to introduce eachstatistical process. Rather than justcrunching numbers, students come toappreciate the value of statisticsthrough relevant examples.

Most exercises in each section �

are applications problems.

� Data Highlights:Group ProjectsUsing Group Projects,students gain experienceworking with others bydiscussing a topic,analyzing data, andcollaborating to formulatetheir response to thequestions posed in theexercise.

(a) estimate a range of years centered about the mean in which about 68% of

the data (tree-ring dates) will be found.

(b) estimate a range of years centered about the mean in which about 95% of

the data (tree-ring dates) will be found.

(c) estimate a range of years centered about the mean in which almost all the

data (tree-ring dates) will be found.

10. Vending Machine: Soft Drinks A vending machine automatically pours soft

drinks into cups. The amount of soft drink dispensed into a cup is normally dis-

tributed with a mean of 7.6 ounces and standard deviation of 0.4 ounce. Exam-

ine Figure 7-3 and answer the following questions.

(a) Estimate the probability that the machine will overflow an 8-ounce cup.

(b) Estimate the probability that the machine will not overflow an 8-ounce cup.

(c) The machine has just been loaded with 850 cups. How many of these do you

expect will overflow when served?

11. Pain Management: Laser Therapy “Effect of Helium-Neon Laser Auriculotherapy

on Experimental Pain Threshold” is the title of an article in the journal PhysicalTherapy (Vol. 70, No. 1, pp. 24–30). In this article, laser therapy was discussed as

a useful alternative to drugs in pain management of chronically ill patients. To

EXAMPLE 11 Central limit theoremA certain strain of bacteria occurs in all raw milk. Let x be the bacteria count per milliliter of milk. The health department has found that if the milk is notcontaminated, then x has a distribution that is more or less mound-shaped andsymmetrical. The mean of the x distribution is and the standard devi-ation is In a large commercial dairy, the health inspector takes 42 random samples of the milk produced each day. At the end of the day, thebacteria count in each of the 42 samples is averaged to obtain the sample meanbacteria count

(a) Assuming the milk is not contaminated, what is the distribution of

SOLUTION: The sample size is Since this value exceeds 30, the centrallimit theorem applies, and we know that will be approximately normal withmean and standard deviation

sx = s�1n = 300�142 � 46.3

mx = m = 2500

xn = 42.

x?

x.

s = 300.m = 2500,

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xviii

PROCEDURE HOW TO COMPUTE QUARTILES

1. Order the data from smallest to largest.

2. Find the median. This is the second quartile.

3. The first quartile Q1 is then the median of the lower half of the data;that is, it is the median of the data falling below the Q2 position (andnot including Q2).

4. The third quartile Q3 is the median of the upper half of the data;that is, it is the median of the data falling above the Q2 position(and not including Q2).

In short, all we do to find the quartiles is find three medians.The median, or second quartile, is a popular measure of the center utilizing

relative position. A useful measure of data spread utilizing relative position is theinterquartile range (IQR). It is simply the difference between the third and firstquartiles.

Interquartile range = Q3 - Q1

The interquartile range tells us the spread of the middle half of the data. Now let’slook at an example to see how to compute all of these quantities.

Interquartile range

G U I D E D E X E R C I S E 10 Probability regarding x

In mountain country, major highways sometimes use tunnels instead of long, winding roads

over high passes. However, too many vehicles in a tunnel at the same time can cause a

hazardous situation. Traffic engineers are studying a long tunnel in Colorado. If x represents

the time for a vehicle to go through the tunnel, it is known that the x distribution has mean

minutes and standard deviation minutes under ordinary traffic conditions.

From a histogram of x values, it was found that the x distribution is mound-shaped with some

symmetry about the mean.

Engineers have calculated that, on average, vehicles should spend from 11 to 13 minutes in the tun-

nel. If the time is less than 11 minutes, traffic is moving too fast for safe travel in the tunnel. If the

time is more than 13 minutes, there is a problem of bad air quality (too much carbon monoxide

and other pollutants).

Under ordinary conditions, there are about 50 vehicles in the tunnel at one time. What is the prob-

ability that the mean time for 50 vehicles in the tunnel will be from 11 to 13 minutes?

We will answer this question in steps.

s = 3.8m = 12.1

(a) Let represent the sample mean based on

samples of size 50. Describe the distribution.

(b) Find

(c) Interpret your answer to part (b).

From the central limit theorem, we expect the

distribution to be approximately normal with mean

and standard deviation

We convert the interval

to a standard z interval and use the standard normal

probability table to find our answer. Since

converts to

and converts to

Therefore,

It seems that about 93% of the time there should be

no safety hazard for average traffic flow.

= 0.9318

= 0.9525 - 0.0207

P(11 6 x 6 13) = P(-2.04 6 z 6 1.67)

z �13 - 12.1

0.54= 1.67x = 13

z �11 - 12.1

0.54= -2.04x = 11

z =x - ms�1n

�x - 12.1

0.54

11 6 x 6 13

s

1n=

3.8

150� 0.54sx =mx = m = 12.1

x

P(11 6 x 6 13).

xx

Making the Jump

Get to the “Aha!” moment faster. Understanding Basic Statistics provides thepush students need to get there through guidance and example.

� Procedures Procedure display boxessummarize simple step-by-step strategies for carryingout statistical proceduresand methods as they areintroduced. Students canrefer back to these boxesas they practice using theprocedures.

Guided Exercises �

Students gain experiencewith new procedures andmethods through GuidedExercises. Beside eachproblem in a GuidedExercise, a completelyworked-out solutionappears for immediatereinforcement.

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Preface

Welcome to the exciting world of statistics! We have written this text to makestatistics accessible to everyone, including those with a limited mathematics

background. Statistics affects all aspects of our lives. Whether we are testing newmedical devices or determining what will entertain us, applications of statisticsare so numerous that, in a sense, we are limited only by our own imagination indiscovering new uses for statistics.

Overview

The fifth edition of Understanding Basic Statistics continues to emphasize conceptsof statistics. Statistical methods are carefully presented with a focus on understandingboth the suitability of the method and the meaning of the result. Statistical methodsand measurements are developed in the context of applications.

We have retained and expanded features that made the first four editions ofthe text very readable. Definition boxes highlight important terms. Proceduredisplays summarize steps for analyzing data. Examples, exercises, and problemstouch on applications appropriate to a broad range of interests.

New with the fifth edition is the Online Study Center, encompassing all inter-active online products and services with this text. Online homework powered byWeb-Assign® is now available through Cengage Learning’s course managementsystem. Also available in the Online Study Center are over 100 data sets (inMicrosoft Excel, Minitab, SPSS, and TI-84Plus/TI-83Plus ASCII file formats),lecture aids, a glossary, statistical tables, intructional video (also available onDVDs), an Online Multimedia eBook, and interactive tutorials.

Understanding Basic Statistics, Fifth Edition, is carefully designed to presentcore topics of an introductory course in statistics. Our other text, UnderstandableStatistics, Ninth Edition, is a more comprehensive text with additional topics.Both books feature the same effective pedagogy and instructional design.

Major Changes in the

Fifth Edition

With each new edition, the authors reevaluate the scope, appropriateness, andeffectiveness of the text’s presentation and reflect on extensive user feedback.Revisions have been made throughout the text to clarify explanations of impor-tant concepts and to update problems.

Critical Thinking and Statistical LiteracyCritical thinking is essential in understanding and evaluating information. Thereare more than a few situations in statistics in which the lack of critical thinkingcan lead to conclusions that are misleading or incorrect. Throughout the text,critical thinking is emphasized and highlighted. In each section and chapter prob-lem set students are asked to apply their critical thinking abilities.

Statistical literacy is fundamental for applying and interpreting statisticalresults. Students need to know correct statistical terminology. The knowledge ofcorrect terminology helps students focus on correct analysis and processes. Eachsection and chapter problem set has questions designed to reinforce statisticalliteracy.

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More Emphasis on InterpretationCalculators and computers are very good at providing the numerical results of sta-tistical processes. It is up to the user of statistics to interpret the results in the con-text of an application. Were the correct processes used to analyze the data? Whatdo the results mean? Students are asked these questions throughout the text.

New ContentIn Chapter 1 there is more emphasis on experimental design.

Tests of homogeneity are discussed with chi-square tests of independence inSection 11.1

Other ChangesIn general, the material on descriptive statistics has been streamlined, so that aprofessor can move more quickly to topics of inferential statistics.

Chapter 2, Organizing Data, has been rearranged so that the section onfrequency distributions and histograms is the first section. The second sectiondiscusses other types of graphs.

In Chapter 3, the discussion of grouped data has been incorporated in ExpandYour Knowledge problems.

In Chapter 8, Estimation, discussion of sample size for a specified error ofestimate is now incorporated into the sections that introduce confidence intervalsfor the mean and for a proportion.

Continuing Content

Introduction of Hypothesis Testing Using P-ValuesIn keeping with the use of computer technology and standard practice inresearch, hypothesis testing is introduced using P-values. The critical regionmethod is still supported, but not given primary emphasis.

Use of Student’s t Distribution in Confidence Intervalsand Testing of MeansIf the normal distribution is used in confidence intervals and testing of means,then the population standard deviation must be known. If the population stan-dard deviation is not known, then under conditions described in the text, theStudent’s t distribution is used. This is the most commonly used procedure in sta-tistical research. It is also used in statistical software packages such as MicrosoftExcel, Minitab, SPSS, and TI-84Plus/TI-83Plus calculators.

Confidence Intervals and Hypothesis Testsof Difference of MeansIf the normal distribution is used, then both population standard deviations mustbe known. When this is not the case, the Student’s t distribution incorporates anapproximation for t, with a commonly used conservative choice for the degreesof freedom. Satterthwaite’s approximation for the degrees of freedom as used incomputer software is also discussed. The pooled standard deviation is presentedfor appropriate applications (s1 � s2).

xx Preface

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Features in the Fifth Edition

Chapter and Section Lead-ins• Preview Questions at the beginning of each chapter are keyed to the sections.• Focus Problems at the beginning of each chapter demonstrate types of ques-

tions students can answer once they master the concepts and skills presentedin the chapter.

• Focus Points at the beginning of each section describe the primary learningobjectives of the section.

Carefully Developed Pedagogy• Examples show students how to select and use appropriate procedures.• Guided Exercises within the sections give students an opportunity to work

with a new concept. Completely worked-out solutions appear beside eachexercise to give immediate reinforcement.

• Definition boxes highlight important definitions throughout the text.• Procedure displays summarize appropriate conditions and key strategies for

carrying out statistical procedures and methods.• Labels for each example or guided exercise highlight the technique, concept,

or process illustrated by the example or guided exercise. In addition, labels forsection and chapter problems describe the field of application and show thewide variety of subjects in which statistics is used.

• Section and chapter problems require the student to use all the new conceptsmastered in the section or chapter. Problem sets include a variety of real-world applications with data or settings from identifiable sources. Key stepsand solutions to odd-numbered problems appear at the end of the book.

• NEW! Statistical Literacy problems ask students to focus on correct terminol-ogy and processes of appropriate statistical methods. Such problems occur inevery section and chapter problem set.

• NEW! Critical Thinking problems ask students to analyze and comment onvarious issues that arise in the application of statistical methods and in theinterpretation of results. These problems occur in every section and chapterproblem set.

• Expand Your Knowledge problems present enrichment topics such as dotplots; grouped data; estimation of standard deviation from a range of datavalues; residual plots; relationship between confidence intervals and two-tailed hypothesis tests; and more.

• NEW! Expand Your Knowledge Foreshadowing alerts students to optionalmaterial presented in the exercise sets.

• Cumulative review problem sets occur after every third chapter and includekey topics from previous chapters. Answers to all cumulative review problemsare given at the end of the book.

• Data Highlights and Linking Concepts provide group projects and writingprojects.

• Viewpoints are brief essays presenting diverse situations in which statisticsis used.

Technology within the Text• Tech Notes within sections provide brief point-of-use instructions for the

TI-84Plus and TI-83Plus calculators, Microsoft Excel, and Minitab.• Using Technology sections have been revised to show the use of SPSS as well

as the TI-84Plus and TI-83Plus calculators, Microsoft Excel, and Minitab.

Preface xxi

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Alternate Routes Through the Text

Understanding Basic Statistics, Fifth Edition, is designed to be flexible. It offersthe professor a choice of teaching possibilities. In most one-semester courses, it isnot practical to cover all the material in depth. However, depending on theemphasis of the course, the professor may choose to cover various topics. Forhelp in topic selection, refer to the Table of Prerequisite Material on page 1.

• Linear regression. Chapter 4, Correlation and Regression, may be delayeduntil after Chapter 9. The descriptive topics of linear regression may then be followed immediately by the inferential topics of linear regression presentedin Chapter 11.

• Probability. For courses requiring minimal probability, Section 5.1 (What IsProbability?) and the first part of Section 5.2 (Some Probability Rules—Compound Events) will be sufficient.

Acknowledgments

It is our pleasure to acknowledge the prepublication reviewers of this text. All oftheir insights and comments have been very valuable to us. Reviewers of this textinclude:

Reza Abbasian, Texas Lutheran UniversityPaul Ache, Kutztown UniversityKathleen Almy, Rock Valley CollegePolly Amstutz, University of Nebraska at KearneyDelores Anderson, Truett-McConnell CollegeRobert J. Astalos, Feather River CollegeLynda L. Ballou, Kansas State UniversityMary Benson, Pensacola Junior CollegeLarry Bernett, Benedictine UniversityKiran Bhutani, The Catholic University of AmericaKristy E. Bland, Valdosta State UniversityJohn Bray, Broward Community CollegeBill Burgin, Gaston CollegeFerry Butar, Sam Houston State UniversityToni Carroll, Siena Heights UniversityCoskun Cetin, Sacramento State UniversityPinyuen Chen, Syracuse UniversityDavid Cochener, Austin Peay UniversityJennifer M. Dollar, Grand Rapids Community CollegeLarry E. Dunham, Wor-Wic Community CollegeAndrew Ellett, Indiana UniversityMary Fine, Moberly Area Community CollegeRene Garcia, Miami-Dade Community CollegeLarry Green, Lake Tahoe Community CollegeMary Hartz, Mohawk Valley Community CollegeJane Keller, Metropolitan Community CollegeRaja Khoury, Collin County Community CollegeDiane Koenig, Rock Valley CollegeCharles G. Laws, Cleveland State Community CollegeMarc Loizeaux, University of Tennessee at ChattanoogaMichael R. Lloyd, Henderson State UniversityBeth Long, Pellissippi State Technical and Community CollegeLewis Lum, University of Portland

xxii Preface

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Darcy P. Mays, Virginia Commonwealth UniversityCharles C. Okeke, College of Southern Nevada, Las VegasBernard Omolo, University of South Carolina UpstatePeg Pankowski, Community College of Allegheny CountyMazbahur Rahman, Minnesota State University, MankatoAzar Raiszadeh, Chattanooga State Technical Community CollegeMichael L. Russo, Suffolk County Community CollegeJanel Schultz, Saint Mary’s University of MinnesotaSankara Sethuraman, Augusta State UniversityWinson Taam, Oakland UniversityJennifer L. Taggart, Rockford CollegeJanis Todd, Campbell UniversityWilliam Truman, University of North Carolina at PembrokeKaren Watson, Fort Valley State UniversityBill White, University of South Carolina UpstateJim Wienckowski, State University of New York at BuffaloStephen M. Wilkerson, Susquehanna UniversityHongkai Zhang, East Central UniversityShunpu Zhang, University of Alaska, FairbanksCathy Zuccoteveloff, Trinity College

We would especially like to thank George Pasles for his careful accuracyreview of this text. We are especially appreciative of the excellent work bythe editorial and production professionals at Cengage Learning. In particular,we thank Molly Taylor, Andrew Lipsett, Katherine Greig, Erin Timm, RachelD’Angelo Wimberly, Joanna Carter-O’Connell, and Carl Chudyk. Without theircreative insight and attention to detail, a project of this quality and magnitudewould not be possible. Finally, we acknowledge the cooperation of Minitab, Inc.,SPSS, Texas Instruments, and Microsoft Excel.

Charles Henry Brase

Corrinne Pellillo Brase

Preface xxiii

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Additional Resources—Get More from YourTextbook!

Instructor Resources

Instructor’s Annotated Edition (IAE) Answers to all exercises, teachingcomments, and pedagogical suggestions appear in the margin, or at theend of the text in the case of large graphs.

Instructor’s Resource Guide with Complete Solutions Contains completesolutions to all exercises, sample tests for each chapter, Teaching Hints,and Transparency Masters for the tables and frequently used formulas inthe text.

Diploma Testing Provides instructors with a wide array of new algo-rithmic exercises along with improved functionality and ease of use.Instructors can create, author/edit algorithmic questions, customize, anddeliver multiple types of tests.

Student Resources

Student Solutions Manual Provides solutions to the odd-numbered sec-tion and chapter exercises and to all the Cumulative Review exercises inthe student textbook.

Technology Guides Separate Guides exist with information and examplesfor each of four technology tools. Guides are available for the TI-84Plusand TI-83Plus graphing calculators, Minitab software (version 15)Microsoft Excel (2003/2007), and SPSS software (version 16).

Instructional DVDs Hosted by Dana Mosely, these text-specificDVDs cover all sections of the text and provide explanations of key

concepts, examples, exercises, and applications in a lecture-based format.DVDs are close-captioned for the hearing-impaired.

xxv

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MINITAB (Release 15) and SPSS (Release 16) CD-ROMs These statisti-cal software packages manipulate and interpret data to produce textual,graphical, and tabular results. MINITAB and/or SPSS may be packagedwith the textbook. Student versions are available.

The Online Study Center encompasses the interactive online products andservices integrated with Cengage Learning textbook programs. The OnlineStudy Center is available via Cengage Learning’s online course managementsystem. The Online Study Center now includes homework powered byWebAssign®; a new Multimedia eBook, videos, tutorials, and SMAR-THINKING®.

• NEW! Online Multimedia eBook Integrates numerous assets such asvideo explanations and tutorials to expand upon and reinforce con-cepts as they appear in the text.

• SMARTHINKING® Live, Online Tutoring Provides an easy-to-useand effective online, text-specific tutoring service. A dynamicWhiteboard and a Graphing Calculator function enable students ande-structors to collaborate easily.

• The Online Teaching Center Students can continue their learning witha new Multimedia eBook, ACE practice tests, glossary flash cards,online data sets, statistical tables and formulae, and more.

• The Online Teaching Center Instructors can download transparencies,chapter tests, instructor’s solutions, course sequences, a printed testbank, lecture aids (PowerPoint®), and digital art and figures.

Online Course Management Content for Blackboard®, WebCT®, andeCollege® Deliver program- or text-specific Cengage Learning contentonline using your institution’s local course management system. CengageLearning offers homework, tutorials, videos, and other resources format-ted for Blackboard, WebCT, eCollege, and other course management sys-tems. Add to an existing online course or create a new one by selectingfrom a wide range of powerful learning and instructional materials.

For more information, visit www.cengage.com/statistics/Brase/UBS5e orcontact your local Cengage Learning sales representative.

xxvi Additional Resources—Get More from Your Textbook

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Understanding Basic Statistics

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Chapter Prerequisite Sections

1 Getting Started None

2 Organizing Data 1.1, 1.2

3 Averages and Variation 1.1, 1.2, 2.1

4 Correlation and Regression 1.1, 1.2, 3.1, 3.2

5 Elementary Probability Theory 1.1, 1.2, 2.1

6 The Binomial Probability 1.1, 1.2, 2.1, 3.1, 3.2, 5.1, 5.2 Distribution and Related Topics 5.3 useful but not essential

7 Normal Curves and Sampling Distributions (omit 7.6) 1.1, 1.2, 2.1, 3.1, 3.2, 5.1, 5.2, 6.1(include 7.6) also 6.2, 6.3

8 Estimation(omit 8.3) 1.1, 1.2, 2.1, 3.1, 3.2, 5.1, 5.2, 6.1, 7.1, 7.2, 7.3, 7.4, 7.5(include 8.3) also 6.2, 6.3, 7.6

9 Hypothesis Testing (omit 9.3) 1.1, 1.2, 2.1, 3.1, 3.2, 5.1, 5.2, 6.1, 7.1, 7.2, 7.3, 7.4, 7.5(include 9.3) also 6.2, 6.3, 7.6

10 Inferences About Differences(omit 10.3) 1.1, 1.2, 2.1, 3.1, 3.2, 5.1, 5.2, 6.1, 7.1, 7.2, 7.3, 7.4, 7.5,

8.1, 8.2, 9.1, 9.2(include 10.3) also 6.2, 6.3, 7.6, 9.3

11 Additional Topics Using Inference(Part I: 11.1, 11.2, 11.3) 1.1, 1.2, 2.1, 3.1, 3.2, 5.1, 5.2, 6.1, 7.1, 7.2, 7.3, 7.4, 7.5, 9.1(Part II: 11.4) Chapter 4, 8.1, 8.2 also

Table of Prerequisite Material

1

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Understanding Basic Statistics,

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Transcript of Understanding Basic Statistics,