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Preface

This book is meant to be a textbook for a standard one-semester introductory statistics course for

general education students. Our motivation for writing it is twofold: 1.) to provide a low-costalternative to many existing popular textbooks on the market and !.) to provide a "uality textbook

on the sub#ect with a focus on the core material of the course in a balanced presentation.

The high cost of textbooks has spiraled out of control in recent years. The high fre"uency at which

new editions of popular texts appear puts a tremendous burden on students and faculty alike$ as well

as the natural environment. %gainst this background we set out to write a "uality textbook with

materials such as examples and exercises that age well with time and that would therefore notre"uire fre"uent new editions. Our vision resonates well with the publisher&s business model which

includes free digital access$ reduced paper prints$ and easy customi'ation by instructors if additional

material is desired.

Over time the core content of this course has developed into a well-defined body of material that is

substantial for a one-semester course. The authors believe that the students in this course are best

served by a focus on the core material and not by an exposure to a plethora of peripheral topics.Therefore in writing this book we have sought to present material that comprises fully a central body

of knowledge that is defined according to convention$ realistic expectation with respect to course

duration and students& maturity level$ and our professional #udgment and experience. (e believe

that certain topics$ among them oisson and geometric distributions and the normal approximation

to the binomial distribution *particularly with a continuity correction) are distracting in nature.

Other topics$ such as nonparametric methods$ while important$ do not belong in a first course in

statistics. %s a result we envision a smaller and less intimidating textbook that trades some extended

and unnecessary topics for a better focused presentation of the central material.

Textbooks for this course cover a wide range in terms of simplicity and complexity. +ome popular

textbooks emphasi'e the simplicity of individual concepts to the point of lacking the coherence of an

overall network of concepts. Other textbooks include overly detailed conceptual and computational






discussions and as a result repel students from reading them. The authors believe that a successful

book must strike a balance between the two extremes$ however difficult it may be. %s a conse"uence

the overarching guiding principle of our writing is to seek simplicity but to preserve the coherence of

the whole body of information communicated$ both conceptually and computationally. (e seek to

remind ourselves *and others) that we teach ideas$ not #ust step-by-step algorithms$ but ideas thatcan be implemented by straightforward algorithms.

,n our experience most students come to an introductory course in statistics with a calculator that

they are familiar with and with which their proficiency is more than ade"uate for the course material.

,f the instructor chooses to use technological aids$ either calculators or statistical software such as

initab or +++$ for more than mere arithmetical computations but as a significant component of

the course then effective instruction for their use will re"uire more extensive written instruction thana mere paragraph or two in the text. iven the plethora of such aids available$ to discuss a few of

them would not provide sufficiently wide or detailed coverage and to discuss many would digress

unnecessarily from the conceptual focus of the book. The overarching philosophy of this textbook is

to present the core material of an introductory course in statistics for non-ma#ors in a complete yet

streamlined way. uch room has been intentionally left for instructors to apply their own

instructional styles as they deem appropriate for their classes and educational goals. (e believe that

the whole matter of what technological aids to use$ and to what extent$ is precisely the type of

material best left to the instructor&s discretion.

%ll figures with the exception of /igure 1.1 0The rand icture of +tatistics0$/igure !.1 0+tem and

eaf 2iagram0$ /igure !.! 0Ordered +tem and eaf 2iagram0$/igure !.13 0The 4ox lot0$ /igure 15.6

0inear 7orrelation 7oefficient 0$ /igure 15.8 0The +imple inear odel 7oncept0$ and the

unnumbered figure in 9ote !.85 0xample 1;0 of 7hapter ! 02escriptive +tatistics0 were generated

using %T%4$ copyright !515.






Chapter %

&ntroduction

,n this chapter we will introduce some basic terminology and lay the groundwork for the course. (e will explain in general terms what statistics and probability are and the problems that these two

areas of study are designed to solve.

% 'asic (e)nitions and Concepts

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1 &o lear( the bas)c *e+()t)o(s ,se* )( stat)st)cs a(* so-e of )ts key co(cepts.

(e begin with a simple example. There are millions of passenger automobiles in the <nited +tates.

(hat is their average value= ,t is obviously impractical to attempt to solve this problem directly by

assessing the value of every single car in the country$ adding up all those numbers$ and then dividing

by however many numbers there are. ,nstead$ the best we can do would be to estimate the average.

One natural way to do so would be to randomly select some of the cars$ say !55 of them$ ascertain

the value of each of those cars$ and find the average of those !55 numbers. The set of all those

millions of vehicles is called the population of interest$ and the number attached to each one$ its

value$ is a measurement . The average value is a parameter: a number that describes a characteristic

of the population$ in this case monetary worth. The set of !55 cars selected from the population is

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called a sample$ and the !55 numbers$ the monetary values of the cars we selected$ are the sample

data. The average of the data is called a statistic: a number calculated from the sample data. This

example illustrates the meaning of the following definitions.

e+()t)o(

A population is any specific collection of objects of interest. A sample is any subset or subcollection of

the population, including the case that the sample consists of the whole population, in which case it is

termed a census.

e+()t)o(

A measurement is a number or attribute computed for each member of a population or of a sample.

The measurements of sample elements are collectively called the sample data.

e+()t)o(

A parameter is a number that summarizes some aspect of the population as a whole. A statistic is a

number computed from the sample data.

7ontinuing with our example$ if the average value of the cars in our sample was >?$38@$ then it seems

reasonable to conclude that the average value of all cars is about >?$38@. ,n reasoning this way we

have drawn an inference about the population based on information obtained from the sample. ,ngeneral$ statistics is a study of data: describing properties of the data$ which is called descriptive

statistics$ and drawing conclusions about a population of interest from information extracted from a

sample$ which is called inferential statistics. 7omputing the single number >?$38@ to summari'e the

data was an operation of descriptive statistics using it to make a statement about the population was

an operation of inferential statistics.






e+()t)o(

Statistics is a collection of methods for collecting, displaying, analyzing, and drawing conclusions from data.

e+()t)o(

Descriptive statistics is the branch of statistics that involves organizing, displaying, and describing

data.

e+()t)o(

Inferential statistics is the branch of statistics that involves drawing conclusions about a population

based on information contained in a sample taken from that population.

The measurement made on each element of a sample need not be numerical. ,nthe case of

automobiles$ what is noted about each car could be its color$ its make$ its body type$ and so on. +uch

data are categorical or qualitative$ as opposed to numerical or quantitative data such as value or age.

This is a general distinction.

e+()t)o(

Qualitative data are measurements for which there is no natural numerical scale, but which consist of

attributes, labels, or other nonnumerical characteristics.






e+()t)o(

Quantitative data are numerical measurements that arise from a natural numerical scale.

Aualitative data can generate numerical sample statistics. ,n the automobile example$ for instance$

we might be interested in the proportion of all cars that are less than six years old. ,n our same

sample of !55 cars we could note for each car whether it is less than six years old or not$ which is a

"ualitative measurement. ,f 1@! cars in the sample are less than six years old$ which is 5.?; or ?;B$

then we would estimate the parameter of interest$ the population proportion$ to be about the same as

the sample statistic$ the sample proportion$ that is$ about 5.?;.

The relationship between a population of interest and a sample drawn from that population isperhaps the most important concept in statistics$ since everything else rests on it. This relationship is

illustrated graphically in /igure 1.1 0The rand icture of +tatistics0. The circles in the large box

represent elements of the population. ,n the figure there was room for only a small number of them

but in actual situations$ like our automobile example$ they could very well number in the millions.

The solid black circles represent the elements of the population that are selected at random and that

together form the sample. /or each element of the sample there is a measurement of interest$

denoted by a lower case x *which we have indexed asx1,…,xnto tell them apart) these measurements

collectively form the sample data set. /rom the data we may calculate various statistics. To anticipate

the notation that will be used later$ we might compute the sample mean x−and the sample

proportion p$ and take them as approximations to the population mean *this is the lower case

reek letter mu$ the traditional symbol for this parameter) and the population proportion p$

respectively. The other symbols in the figure stand for other parameters and statistics that we will

encounter.






!igure "." The #rand $icture of %tatistics

*+, TA*+AA,S

• Stat)st)cs )s a st,*y of *ata: *escr)b)(g propert)es of *ata *escr)pt)4e

stat)st)cs5 a(* *raw)(g co(cl,s)o(s abo,t a pop,lat)o( base* o( )(for-at)o(

)( a sa-ple )(fere(t)al stat)st)cs5.

• &he *)st)(ct)o( betwee( a pop,lat)o( together w)th )ts para-eters a(* a

sa-ple together w)th )ts stat)st)cs )s a f,(*a-e(tal co(cept )( )(fere(t)al

stat)st)cs.






• (for-at)o( )( a sa-ple )s ,se* to -ake )(fere(ces abo,t the pop,lat)o( fro-

wh)ch the sa-ple was *raw(.

++/C&S+S

1 E7pla)( what )s -ea(t by the ter- population.

2 E7pla)( what )s -ea(t by the ter- sample.

3 E7pla)( how a sa-ple *)8ers fro- a pop,lat)o(.

E7pla)( what )s -ea(t by the ter- sample data.

0 E7pla)( what a parameter )s.

E7pla)( what a statistic )s.

!)4e a( e7a-ple of a pop,lat)o( a(* two *)8ere(t character)st)cs that -ay be of

)(terest.

6 escr)be the *)8ere(ce betwee( descriptive statistics a(* inferential statistics. ll,strate

w)th a( e7a-ple.

9 *e(t)fy each of the follow)(g *ata sets as e)ther a pop,lat)o( or a sa-ple:

a &he gra*e po)(t a4erages !PAs5 of all st,*e(ts at a college.

b &he !PAs of a ra(*o-ly selecte* gro,p of st,*e(ts o( a college ca-p,s.

c &he ages of the ()(e S,pre-e %o,rt $,st)ces of the U()te* States o( $a(,ary 1

162.

* &he ge(*er of e4ery seco(* c,sto-er who e(ters a -o4)e theater.

e &he le(gths of Atla(t)c croakers ca,ght o( a +sh)(g tr)p to the beach.

1; *e(t)fy the follow)(g -eas,res as e)ther <,a(t)tat)4e or <,al)tat)4e:

a &he 3; h)gh=te-perat,re rea*)(gs of the last 3; *ays.

b &he scores of ; st,*e(ts o( a( E(gl)sh test.

c &he bloo* types of 12; teachers )( a -)**le school.

* &he last fo,r *)g)ts of soc)al sec,r)ty (,-bers of all st,*e(ts )( a class.

e &he (,-bers o( the >erseys of 03 football players o( a tea-.

11 *e(t)fy the follow)(g -eas,res as e)ther <,a(t)tat)4e or <,al)tat)4e:

a &he ge(*ers of the +rst ; (ewbor(s )( a hosp)tal o(e year.

b &he (at,ral ha)r color of 2; ra(*o-ly selecte* fash)o( -o*els.






c &he ages of 2; ra(*o-ly selecte* fash)o( -o*els.

* &he f,el eco(o-y )( -)les per gallo( of 2; (ew cars p,rchase* last -o(th.

e &he pol)t)cal a?l)at)o( of 0;; ra(*o-ly selecte* 4oters.

12 A researcher w)shes to est)-ate the a4erage a-o,(t spe(t per perso( by 4)s)tors to a

the-e park. @e takes a ra(*o- sa-ple of forty 4)s)tors a(* obta)(s a( a4erage of 26

per perso(.

a Bhat )s the pop,lat)o( of )(terestC

b Bhat )s the para-eter of )(terestC

c #ase* o( th)s sa-ple *o we k(ow the a4erage a-o,(t spe(t per perso( by

4)s)tors to the parkC E7pla)( f,lly.

13 A researcher w)shes to est)-ate the a4erage we)ght of (ewbor(s )( So,th A-er)ca )(

the last +4e years. @e takes a ra(*o- sa-ple of 230 (ewbor(s a(* obta)(s a( a4erage

of 3.2 k)logra-s.



c #ase* o( th)s sa-ple *o we k(ow the a4erage we)ght of (ewbor(s )( So,th

A-er)caC E7pla)( f,lly.

1 A researcher w)shes to est)-ate the proport)o( of all a*,lts who ow( a cell pho(e. @e

takes a ra(*o- sa-ple of 102 a*,ltsD 1296 of the- ow( a cell pho(e he(ce

1296102 F .63 or abo,t 63G ow( a cell pho(e.



c Bhat )s the stat)st)c )(4ol4e*C

* #ase* o( th)s sa-ple *o we k(ow the proport)o( of all a*,lts who ow( a cell

pho(eC E7pla)( f,lly.

10 A soc)olog)st w)shes to est)-ate the proport)o( of all a*,lts )( a certa)( reg)o( who ha4e

(e4er -arr)e*. ( a ra(*o- sa-ple of 132; a*,lts 10 ha4e (e4er -arr)e* he(ce

10132; F .11 or abo,t 11G ha4e (e4er -arr)e*.


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c Bhat )s the stat)st)c )(4ol4e*C

* #ase* o( th)s sa-ple *o we k(ow the proport)o( of all a*,lts who ha4e (e4er

-arr)e*C E7pla)( f,lly.

1 a. Bhat -,st be tr,e of a sa-ple )f )t )s to g)4e a rel)able est)-ate of the 4al,e

of a part)c,lar

pop,lat)o( para-eterC

b. Bhat -,st be tr,e of a sa-ple )f )t )s to g)4e certain k(owle*ge of the 4al,e

of a part)c,lar

pop,lat)o( para-eterC

ANS+/S

1 A pop,lat)o( )s the total collect)o( of ob>ects that are of )(terest )( a stat)st)cal st,*y.

3 A sa-ple be)(g a s,bset )s typ)cally s-aller tha( the pop,lat)o(. ( a stat)st)cal st,*y

all ele-e(ts of a sa-ple are a4a)lable for obser4at)o( wh)ch )s (ot typ)cally the case

for a pop,lat)o(.

0 A para-eter )s a 4al,e *escr)b)(g a character)st)c of a pop,lat)o(. ( a stat)st)cal st,*y

the 4al,e of a para-eter )s typ)cally ,(k(ow(.

All c,rre(tly reg)stere* st,*e(ts at a part)c,lar college for- a pop,lat)o(. &wo

pop,lat)o( character)st)cs of )(terest co,l* be the a4erage !PA a(* the proport)o( of

st,*e(ts o4er 23 years.

9 a. Pop,lat)o(.

b. Sa-ple.

c Pop,lat)o(.

* Sa-ple.

e Sa-ple.






11 a. H,al)tat)4e.

b. H,al)tat)4e.

c H,a(t)tat)4e.

* H,a(t)tat)4e.

e H,al)tat)4e.

13 a. All (ewbor( bab)es )( So,th A-er)ca )( the last +4e years.

b. &he a4erage b)rth we)ght of all (ewbor( bab)es )( So,th A-er)ca )( the last +4e

years. c. No (ot e7actly b,t we k(ow the appro7)-ate 4al,e of

the a4erage.

10 a. All a*,lts )( the reg)o(.

b. &he proport)o( of the a*,lts )( the reg)o( who ha4e (e4er -arr)e*.

c. &he proport)o( co-p,te* fro- the sa-ple ;.1.

*. No (ot e7actly b,t we k(ow the appro7)-ate 4al,e of the proport)o(.

0 1verview

LEARNN! "#$E%&'E

1 &o obta)( a( o4er4)ew of the -ater)al )( the te7t.

The example we have given in the first section seems fairly simple$ but there are some significant

problems that it illustrates. (e have supposed that the !55 cars of the sample had an average value

of >?$38@ *a number that is precisely known)$ and concluded that the population has an average of






about the same amount$ although its precise value is still unknown. (hat would happen if someone

were to take another sample of exactly the same si'e from exactly the same population= (ould he get

the same sample average as we did$ >?$38@= %lmost surely not. ,n fact$ if the investigator who took

the second sample were to report precisely the same value$ we would immediately become suspicious

of his result. The sample average is an example of what is called a random variable: a number that varies from trial to trial of an experiment *in this case$ from sample to sample)$ and does so in a way

that cannot be predicted precisely. Candom variables will be a central ob#ect of study for us$

beginning in 7hapter 6 02iscrete Candom Dariables0.

%nother issue that arises is that different samples have different levels of reliability. (e have

supposed that our sample of si'e !55 had an average of >?$38@. ,f a sample of si'e 1$555 yielded an

average value of >@$?3!$ then we would naturally regard this latter number as likely to be a betterestimate of the average value of all cars. Eow can this be expressed= %n important idea that we will

develop in 7hapter @ 0stimation0 is that of the confidence interval : from the data we will construct

an interval of values so that the process has a certain chance$ say a F8B chance$ of generating an

interval that contains the actual population average. Thus instead of reporting a single estimate$

>?$38@$ for the population mean$ we would say that we are F8B certain that the true average is

within >155 of our sample mean$ that is$ between >?$!8@ and >?$68@$ the number >155 having been

computed from the sample data #ust like the sample mean >?$38@ was. This will automatically

indicate the reliability of the sample$ since to obtain the same chance of containing the unknown

parameter a large sample will typically produce a shorter interval than a small one will. 4ut unless

we perform a census$ we can never be completely sure of the true average value of the population the

best that we can do is to make statements of probability$ an important concept that we will begin to

study formally in 7hapter 3 04asic 7oncepts of robability0.

+ampling may be done not only to estimate a population parameter$ but to test a claim that is made

about that parameter. +uppose a food package asserts that the amount of sugar in one serving of the

product is 16 grams. % consumer group might suspect that it is more. Eow would they test the

competing claims about the amount of sugar$ 16 grams versus more than 16 grams= They might take

a random sample of perhaps !5 food packages$ measure the amount of sugar in one serving of each

one$ and average those amounts. They are not interested in the true amount of sugar in one serving

in itself their interest is simply whether the claim about the true amount is accurate. +tated another

way$ they are sampling not in order to estimate the average amount of sugar in one serving$ but to






see whether that amount$ whatever it may be$ is larger than 16 grams. %gain because one can have

certain knowledge only by taking a census$ ideas of probability enter into the analysis. (e will

examine tests of hypotheses beginning in 7hapter ? 0Testing Eypotheses0.

+everal times in this introduction we have used the term Grandom sample.H enerally the value of

our data is only as good as the sample that produced it. /or example$ suppose we wish to estimate

the proportion of all students at a large university who are females$ which we denote by p. ,f we

select 85 students at random and !@ of them are female$ then a natural estimate is p≈ p-27 50-

0.54or 86B. Eow much confidence we can place in this estimate depends not only on the si'e of the

sample$ but on its "uality$ whether or not it is truly random$ or at least truly representative of the

whole population. ,f all 85 students in our sample were drawn from a 7ollege of 9ursing$ then the

proportion of female students in the sample is likely higher than that of the entire campus. ,f all 85

students were selected from a 7ollege of ngineering +ciences$ then the proportion of students in the

entire student body who are females could be underestimated. ,n either case$ the estimate would be

distorted or biased. ,n statistical practice an unbiased sampling scheme is important but in most

cases not easy to produce. /or this introductory course we will assume that all samples are either

random or at least representative.

IEJ &AIEABAJ

• Stat)st)cs co-p,te* fro- sa-ples 4ary ra(*o-ly fro- sa-ple to sa-ple.

%o(cl,s)o(s -a*e abo,t pop,lat)o( para-eters are state-e(ts of probab)l)ty.

3 2resentation o (ata

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1 &o lear( two ways that *ata w)ll be prese(te* )( the te7t.

,n this book we will use two formats for presenting data sets. The first is a data list$ which is an

explicit listing of all the individual measurements$ either as a display with space between the

individual measurements$ or in set notation with individual measurements separated by commas.

EKAPLE 1

&he *ata obta)(e* by -eas,r)(g the age of 21 ra(*o-ly selecte* st,*e(ts e(rolle* )(

fresh-a( co,rses at a ,()4ers)ty co,l* be prese(te* as the *ata l)st

18 18 19 19 19 18 22 20 18 18 17

19 18 24 18 20 18 21 20 17 19

or )( set (otat)o( as

{18,18,19,19,19,18,22,20,18,18,17,19,18,24,18,20,18,21,20,17,19}

% data set can also be presented by means of a data frequency table$ a table in which

each distinct value x is listed in the first row and its frequency f $ which is the number of times the

value x appears in the data set$ is listed below it in the second row.

EKAPLE 2

&he *ata set of the pre4)o,s e7a-ple )s represe(te* by the *ata fre<,e(cy table

x %4 %5 %6 0 0% 00 07

0 5 8 3 % % %






The data fre"uency table is especially convenient when data sets are large and the number of distinct

values is not too large.

IEJ &AIEABAJ

• ata sets ca( be prese(te* e)ther by l)st)(g all the ele-e(ts or by g)4)(g a table

of 4al,es a(* fre<,e(c)es.

EKER%SES

1 L)st all the -eas,re-e(ts for the *ata set represe(te* by the follow)(g *ata fre<,e(cy

table.

x 0% 00 00 07 08

% 8 9 7 0

2 L)st all the -eas,re-e(ts for the *ata set represe(te* by the follow)(g *ata fre<,e(cy

table.

x 64 65 66 % %% %0 %0 %8

4 8 0 7 0 0 % %

3 %o(str,ct the *ata fre<,e(cy table for the follow)(g *ata set.

22 25 22 27 24 23






26 24 22 24 26

%o(str,ct the *ata fre<,e(cy table for the follow)(g *ata set.

{1,5,2,3,5,1,4,4,4,3,2,5,1,3,2,

1,1,1,2}

ANSBERS

1 M31323232323233333333333333333030.

3

x 00 03 07 08 09 04

3 % 3 % 0 %

Chapter 0(escriptive Statistics






%s described in 7hapter 1 0,ntroduction0$ statistics naturally divides into two branches$ descriptive

statistics and inferential statistics. Our main interest is in inferential statistics$ as shown in /igure 1.1

0The rand icture of +tatistics0 in 7hapter 1 0,ntroduction0. 9evertheless$ the starting point for

dealing with a collection of data is to organi'e$ display$ and summari'e it effectively. These are the

ob#ectives of descriptive statistics$ the topic of this chapter.

0.% Three 2opular (ata (isplays

!+A/N&N: 1';+CT&<+






1 &o lear( to )(terpret the -ea()(g of three graph)cal represe(tat)o(s of sets of

*ata: ste- a(* leaf *)agra-s fre<,e(cy h)stogra-s a(* relat)4e fre<,e(cy

h)stogra-s.

% well-known adage is that Ga picture is worth a thousand words.H This saying proves true when it

comes to presenting statistical information in a data set. There are many effective ways to present

data graphically. The three graphical tools that are introduced in this section are among the most

commonly used and are relevant to the subse"uent presentation of the material in this book.

Stem and !ea (ia$rams

+uppose 35 students in a statistics class took a test and made the following scores:

86 80 25 77 73 76 100 90 69 9390 83 70 73 73 70 90 83 71 95

40 58 68 69 100 78 87 97 92 74

Eow did the class do on the test= % "uick glance at the set of 35 numbers does not immediately give a

clear answer. Eowever the data set may be reorgani'ed and rewritten to make relevant information more

visible. One way to do so is to construct a stem and leaf diagram as shown in . The numbers in the tens

place$ from ! through F$ and additionally the number 15$ are the Gstems$H and are arranged in numerical

order from top to bottom to the left of a vertical line. The number in the units place in each measurement

is a Gleaf$H and is placed in a row to the right of the corresponding stem$ the number in the tens place ofthat measurement. Thus the three leaves F$ ?$ and F in the row headed with the stem ; correspond to the

three exam scores in the ;5s$ ;F *in the first row of data)$ ;? *in the third row)$ and ;F *also in the third

row). The display is made even more useful for some purposes by rearranging the leaves in numerical

order$ as shown in . ither way$ with the data reorgani'ed certain information of interest becomes

apparent immediately. There are two perfect scores three students made scores under ;5 most students

scored in the @5s$ ?5s and F5s and the overall average is probably in the high @5s or low ?5s.

igure &." %tem and 'eaf (iagram






!igure &.& )rdered %tem and 'eaf (iagram

,n this example the scores have a natural stem *the tens place) and leaf *the ones place). One could spread

the diagram out by splitting each tens place number into lower and upper categories. /or example$ all the

scores in the ?5s may be represented on two separate stems$ lower ?5s and upper ?5s:

5 3 3






5 9 4

The definitions of stems and leaves are flexible in practice. The general purpose of a stem and leaf

diagram is to provide a "uick display of how the data are distributed across the range of their values some

improvisation could be necessary to obtain a diagram that best meets that goal.

9ote that all of the original data can be recovered from the stem and leaf diagram. This will not be true in

the next two types of graphical displays.

Fre"uency =isto$rams

The stem and leaf diagram is not practical for large data sets$ so we need a different$ purely graphical way

to represent data. % frequency histogram is such a device. (e will illustrate it using the same data set

from the previous subsection. /or the 35 scores on the exam$ it is natural to group the scores on the

standard ten-point scale$ and count the number of scores in each group. Thus there are two 155s$ seven

scores in the F5s$ six in the ?5s$ and so on. (e then construct the diagram shown in by drawing for each

group$ or class$ a vertical bar whose length is the number of observations in that group. ,n our example$

the bar labeled 155 is ! units long$ the bar labeled F5 is @ units long$ and so on. (hile the individual data

values are lost$ we know the number in each class. This number is called the frequency of the class$

hence the name fre"uency histogram.

!igure &.* !requency +istogram






The same procedure can be applied to any collection of numerical data. Observations are grouped into

several classes and the fre"uency *the number of observations) of each class is noted. These classes are

arranged and indicated in order on the hori'ontal axis *called the x -axis)$ and for each group a vertical

bar$ whose length is the number of observations in that group$ is drawn. The resulting display is a

fre"uency histogram for the data. The similarity in and is apparent$ particularly if you imagine turning the

stem and leaf diagram on its side by rotating it a "uarter turn counterclockwise.

,n general$ the definition of the classes in the fre"uency histogram is flexible. The general purpose of a

fre"uency histogram is very much the same as that of a stem and leaf diagram$ to provide a graphical

display that gives a sense of data distribution across the range of values that appear. (e will not discuss

the process of constructing a histogram from data since in actual practice it is done automatically with

statistical software or even handheld calculators.

/elative Fre"uency =isto$rams,n our example of the exam scores in a statistics class$ five students scored in the ?5s. The number 8 is

the frequency of the group labeled G?5s.H +ince there are 35 students in the entire statistics class$ the

proportion who scored in the ?5s is 8I35. The number 8I35$ which could also be expressed as0.16≈.1667$

or as 1;.;@B$ is the relative frequency of the group labeled G?5s.H very group *the @5s$ the ?5s$ and

so on) has a relative fre"uency. (e can thus construct a diagram by drawing for each group$ or class$ a






vertical bar whose length is the relative fre"uency of that group. /or example$ the bar for the ?5s will have

length 8I35 unit$ not 8 units. The diagram is a relative frequency histogram for the data$ and is

shown in . ,t is exactly the same as the fre"uency histogram except that the vertical axis in the relative

fre"uency histogram is not fre"uency but relative fre"uency.

!igure &. -elative !requency +istogram

The same procedure can be applied to any collection of numerical data. 7lasses are selected$ the relative

fre"uency of each class is noted$ the classes are arranged and indicated in order on the hori'ontal axis$

and for each class a vertical bar$ whose length is the relative fre"uency of the class$ is drawn. The resulting

display is a relative fre"uency histogram for the data. % key point is that now if each vertical bar has width

1 unit$ then the total area of all the bars is 1 or 155B.

%lthough the histograms in and have the same appearance$ the relative fre"uency histogram is more

important for us$ and it will be relative fre"uency histograms that will be used repeatedly to

represent data in this text. To see why this is so$ reflect on what it is that you are actually seeing in

the diagrams that "uickly and effectively communicates information to you about the data. ,t is

the relative sizes of the bars. The bar labeled G@5sH in either figure takes up 1I3 of the total area of all

the bars$ and although we may not think of this consciously$ we perceive the proportion 1I3 in the

figures$ indicating that a third of the grades were in the @5s. The relative fre"uency histogram is

important because the labeling on the vertical axis reflects what is important visually: the relative

si'es of the bars.

(hen the si'e n of a sample is small only a few classes can be used in constructing a relative

fre"uency histogram. +uch a histogram might look something like the one in panel *a) of . ,f the

sample si'e n were increased$ then more classes could be used in constructing a relative fre"uency






histogram and the vertical bars of the resulting histogram would be finer$ as indicated in panel *b)

of . /or a very large sample the relative fre"uency histogram would look very fine$ like the one in *c)

of. ,f the sample si'e were to increase indefinitely then the corresponding relative fre"uency

histogram would be so fine that it would look like a smooth curve$ such as the one in panel *d) of .

!igure &. %ample %ize and -elative !requency +istograms

,t is common in statistics to represent a population or a very large data set by a smooth curve. ,t is

good to keep in mind that such a curve is actually #ust a very fine relative fre"uency histogram in

which the exceedingly narrow vertical bars have disappeared. 4ecause the area of each such vertical

bar is the proportion of the data that lies in the interval of numbers over which that bar stands$ this

means that for any two numbers a and b$ the proportion of the data that lies between the two

numbers a and b is the area under the curve that is above the interval *a$b) in the hori'ontal axis.

This is the area shown in . ,n particular the total area under the curve is 1$ or 155B.

!igure &./ A 0ery !ine -elative !requency +istogram






*+, TA*+AA,S

• !raph)cal represe(tat)o(s of large *ata sets pro4)*e a <,)ck o4er4)ew of the

(at,re of the *ata.

• A pop,lat)o( or a 4ery large *ata set -ay be represe(te* by a s-ooth c,r4e. &h)s

c,r4e )s a 4ery +(e relat)4e fre<,e(cy h)stogra- )( wh)ch the e7cee*)(gly (arrow

4ert)cal bars ha4e bee( o-)tte*.

• Bhe( a c,r4e *er)4e* fro- a relat)4e fre<,e(cy h)stogra- )s ,se* to *escr)be a

*ata set the proport)o( of *ata w)th 4al,es betwee( two (,-bers a a(* b )s the

area ,(*er the c,r4e betwee( a a(* b as )ll,strate* )( )g,re 2. QA 'ery )(e

Relat)4e re<,e(cy @)stogra-Q.




































0.0 >easures o Central !ocation

LEARNN! "#$E%&'ES

1 &o lear( the co(cept of the ce(ter of a *ata set.

2 &o lear( the -ea()(g of each of three -eas,res of the ce(ter of a *ata setTthe

-ea( the -e*)a( a(* the -o*eTa(* how to co-p,te each o(e.

This section could be titled Gthree kinds of averages of a data set.H %ny kind of GaverageH is meant to

be an answer to the "uestion G(here do the data center=H ,t is thus a measure of the central location






of the data set. (e will see that the nature of the data set$ as indicated by a relative fre"uency

histogram$ will determine what constitutes a good answer. 2ifferent shapes of the histogram call for

different measures of central location.

The >ean

The first measure of central location is the usual GaverageH that is familiar to everyone. ,n the formula in

the following definition we introduce the standard summation notation J$ where J is the capital reek

letter sigma. ,n general$ the notation J followed by a second mathematical symbol means to add up all the

values that the second symbol can take in the context of the problem. Eere is an example to illustrate this.

,n the definition we follow the convention of using lowercase n to denote the number of

measurements in a sample$ which is called the sample size.





















,n the examples above the data sets were described as samples. Therefore the means were sample means$

denoted by x . ,f the data come from a census$ so that there is a measurement for every element of the

population$ then the mean is calculated by exactly the same process of summing all the measurements

and dividing by how many of them there are$ but it is now the population mean and is denoted by $ the

lower case reek letter mu.

The mean of two numbers is the number that is halfway between them. /or example$ the average of the

numbers 8 and 1@ is *8 K 1@) L ! M 11$ which is ; units above 8 and ; units below 1@. ,n this sense the

average 11 is the GcenterH of the data set N8$1@. /or larger data sets the mean can similarly be regarded as

the GcenterH of the data.

The >edian

To see why another concept of average is needed$ consider the following situation. +uppose we are

interested in the average yearly income of employees at a large corporation. (e take a random sample of

seven employees$ obtaining the sample data *rounded to the nearest hundred dollars$ and expressed in

thousands of dollars).

24.8 22.8 24. !"2.# 2#.2 !8.# 2$.%

The mean *rounded to one decimal place) is x- [email protected]$ but the statement Gthe average income of employees

at this corporation is >6@$655H is surely misleading. ,t is approximately twice what six of the seven

employees in the sample make and is nowhere near what any of them makes. ,t is easy to see what went

wrong: the presence of the one executive in the sample$ whose salary is so large compared to everyone

else&s$ caused the numerator in the formula for the sample mean to be far too large$ pulling the mean far






to the right of where we think that the average GoughtH to be$ namely around >!6$555 or >!8$555. The

number 1F!.8 in our data set is called an outlier$ a number that is far removed from most or all of the

remaining measurements. any times an outlier is the result of some sort of error$ but not always$ as is

the case here. (e would get a better measure of the GcenterH of the data if we were to arrange the data in

numerical order$

!8.# 22.8 2$.% 24. 24.8 2#.2 !"2.#

then select the middle number in the list$ in this case !6.;. The result is called the median of the data set$

and has the property that roughly half of the measurements are larger than it is$ and roughly half are

smaller. ,n this sense it locates the center of the data. ,f there are an even number of measurements in the

data set$ then there will be two middle elements when all are lined up in order$ so we take the mean of the

middle two as the median. Thus we have the following definition.

e+()t)o(

The sample median xPQ of a set of sample data for which there are an odd number of measurements is

the middle measurement when the data are arranged in numerical order. The sample median xPQ

of a

set of sample data for which there are an even number of measurements is the mean of the two middle

measurements when the data are arranged in numerical order.

The population median is defined in a similar way$ but we will not have occasion to refer to it again

in this text.

The median is a value that divides the observations in a data set so that 85B of the data are on its left

and the other 85B on its right. ,n accordance with $ therefore$ in the curve that represents the

distribution of the data$ a vertical line drawn at the median divides the area in two$ area 5.8 *85B of

the total area 1) to the left and area 5.8 *85B of the total area 1) to the right$ as shown in . ,n our

income example the median$ >!6$;55$ clearly gave a much better measure of the middle of the data

set than did the mean >6@$655. This is typical for situations in which the distribution is skewed.

*+kewness and symmetry of distributions are discussed at the end of this subsection.)






!igure &.1 The 2edian






Saylor URL: http://www.saylor.org/books Saylor.org;





The relationship between the mean and the median for several common shapes of distributions is shown

in . The distributions in panels *a) and *b) are said to be symmetric because of the symmetry that they

exhibit. The distributions in the remaining two panels are said to be skewed . ,n each distribution we have

drawn a vertical line that divides the area under the curve in half$ which in accordance with is located at

the median. The following facts are true in general:

a. (hen the distribution is symmetric$ as in panels *a) and *b) of $ the mean and the median are

e"ual.






b. (hen the distribution is as shown in panel *c) of $ it is said to be skewed right . The mean has

been pulled to the right of the median by the long Gright tailH of the distribution$ the few relatively large

data values.

c. (hen the distribution is as shown in panel *d) of $ it is said to be skewed left . The mean has been

pulled to the left of the median by the long Gleft tailH of the distribution$ the few relatively small data

values.

!igure &.3 %kewness of -elative !requency +istograms

The >ode

erhaps you have heard a statement like GThe average number of automobiles owned by households

in the <nited +tates is 1.3@$H and have been amused at the thought of a fraction of an automobile






sitting in a driveway. ,n such a context the following measure for central location might make more

sense.

e+()t)o(

The sample mode of a set of sample data is the most frequently occurring value.

The population mode is defined in a similar way$ but we will not have occasion to refer to it again in

this text.

On a relative fre"uency histogram$ the highest point of the histogram corresponds to the mode of the

data set. illustrates the mode.

!igure &.4 2ode






/or any data set there is always exactly one mean and exactly one median. This need not be true of the

mode several different values could occur with the highest fre"uency$ as we will see. ,t could even happen

that every value occurs with the same fre"uency$ in which case the concept of the mode does not make

much sense.

EKAPLE 6

)(* the -o*e of the follow)(g *ata set.

−1 0 2 0

Sol,t)o(:

&he 4al,e ; )s -ost fre<,e(tly obser4e* a(* therefore the -o*e )s ;.






EKAPLE 9

%o-p,te the sa-ple -o*e for the *ata of .

Sol,t)o(:

&he two -ost fre<,e(tly obser4e* 4al,es )( the *ata set are 1 a(* 2. &herefore -o*e

)s a set of two 4al,es: M12.

The mode is a measure of central location since most real-life data sets have moreobservations near the

center of the data range and fewer observations on the lower and upper ends. The value with the highest

fre"uency is often in the middle of the data range.

IEJ &AIEABAJ

&he -ea( the -e*)a( a(* the -o*e each a(swer the <,est)o( Bhere )s the ce(ter

of the *ata setC &he (at,re of the *ata set as )(*)cate* by a relat)4e fre<,e(cy

h)stogra- *eter-)(es wh)ch o(e g)4es the best a(swer.




































!A/:+ (ATA S+T ++/ C&S+S

26 Large ata Set 1 l)sts the SA& scores a(* !PAs of 1;;; st,*e(ts.

http://www.1.7ls

a %o-p,te the -ea( a(* -e*)a( of the 1;;; SA& scores.

b %o-p,te the -ea( a(* -e*)a( of the 1;;; !PAs.29 Large ata Set 1 l)sts the SA& scores of 1;;; st,*e(ts.

http://www.1.7ls

a Regar* the *ata as ar)s)(g fro- a ce(s,s of all st,*e(ts at a h)gh school )( wh)ch the

SA& score of e4ery st,*e(t was -eas,re*. %o-p,te the pop,lat)o( -ea( μ.

b Regar* the +rst 20 obser4at)o(s as a ra(*o- sa-ple *raw( fro- th)s pop,lat)o(.

%o-p,te the sa-ple -ea( xP− a(* co-pare )t to μ.

c Regar* the (e7t 20 obser4at)o(s as a ra(*o- sa-ple *raw( fro- th)s pop,lat)o(.


3; Large ata Set 1 l)sts the !PAs of 1;;; st,*e(ts.

http://www.1.7ls

a Regar* the *ata as ar)s)(g fro- a ce(s,s of all fresh-a( at a s-all college at the e(* of

the)r +rst aca*e-)c year of college st,*y )( wh)ch the !PA of e4ery s,ch perso( was

-eas,re*. %o-p,te the pop,lat)o( -ea( μ.





31 Large ata Sets A a(* # l)st the s,r4)4al t)-es )( *ays of 1; laboratory -)ce w)th

thy-)c le,ke-)a fro- o(set to *eath.

http://www..7ls

http://www.A.7ls

http://www.#.7ls

a %o-p,te the -ea( a(* -e*)a( s,r4)4al t)-e for all -)ce w)tho,t regar* to ge(*er.

b %o-p,te the -ea( a(* -e*)a( s,r4)4al t)-e for the 0 -ale -)ce separately recor*e*

)( Large ata Set A5.

c %o-p,te the -ea( a(* -e*)a( s,r4)4al t)-e for the 0 fe-ale -)ce separately

recor*e* )( Large ata Set #5.











0.3 >easures o <ariability

LEARNN! "#$E%&'ES

1 &o lear( the co(cept of the 4ar)ab)l)ty of a *ata set.

2 &o lear( how to co-p,te three -eas,res of the 4ar)ab)l)ty of a *ata set: the

ra(ge the 4ar)a(ce a(* the sta(*ar* *e4)at)o(.

ook at the two data sets in Table !.1 0Two 2ata +ets0 and the graphical representation of each$

called a dot plot $ in /igure !.15 02ot lots of 2ata +ets0.






Table !.1 Two 2ata +ets

ata Set : ; 36 2 ; 39 39 3 ; 39 ;

ata Set : 3 ; 33 2 3 ; 3 0

!igure &."5 (ot $lots of (ata %ets

The two sets of ten measurements each center at the same value: they both have mean$ median$ and

mode 65. 9evertheless a glance at the figure shows that they are markedly different. ,n 2ata +et , the

measurements vary only slightly from the center$ while for 2ata +et ,, the measurements vary

greatly. Rust as we have attached numbers to a data set to locate its center$ we now wish to associate

to each data set numbers that measure "uantitatively how the data either scatter away from the

center or cluster close to it. These new "uantities are called measures of variability$ and we will

discuss three of them.

The /an$e

The first measure of variability that we discuss is the simplest.

(e)nition

The range of a data set is the number - defined by the formula

R=xmax−xmin

where xmaxis the largest measurement in the data set and xminis the smallest.

+A>2!+ %

)(* the ra(ge of each *ata set )( &able 2.1 Q&wo ata SetsQ.






Sol,t)o(:

or ata Set the -a7)-,- )s 3 a(* the -)()-,- )s 36 so the ra(ge )s R=43−38=5.

or ata Set the -a7)-,- )s a(* the -)()-,- )s 33 so the ra(ge )s R=47−33=14.

The range is a measure of variability because it indicates the si'e of the interval over which the data

points are distributed. % smaller range indicates less variability *less dispersion) among the data$

whereas a larger range indicates the opposite.

The <ariance and the Standard (eviation

The other two measures of variability that we will consider are more elaborate and also depend on

whether the data set is #ust a sample drawn from a much larger population or is the whole populationitself *that is$ a census).






%lthough the first formula in each case looks less complicated than the second$ the latter is easier to

use in hand computations$ and is called a shortcut formula.






The student is encouraged to compute the ten deviations for 2ata +et , and verify that their s"uares

add up to !5$ so that the sample variance and standard deviation of 2ata +et , are the much smaller

numberss2=20/9=2.2¯ ands=√ 20/9≈1.49.






The sample variance has different units from the data. /or example$ if the units in the data set were

inches$ the new units would be inches s"uared$ or s"uare inches. ,t is thus primarily of theoretical

importance and will not be considered further in this text$ except in passing.

,f the data set comprises the whole population$ then the population standard deviation$

denoted 6 *the lower case reek letter sigma)$ and its s"uare$ the population variance 6 !

$ are definedas follows.

9ote that the denominator in the fraction is the full number of observations$ not that number

reduced by one$ as is the case with the sample standard deviation. +ince most data sets are samples$

we will always work with the sample standard deviation and variance.

/inally$ in many real-life situations the most important statistical issues have to do with comparingthe means and standard deviations of two data sets. /igure !.11 02ifference between Two 2ata

+ets0 illustrates how a difference in one or both of the sample mean and the sample standard

deviation are reflected in the appearance of the data set as shown by the curves derived from the

relative fre"uency histograms built using the data.






!igure &."" (ifference between Two (ata %ets

IEJ &AIEABAJ






&he ra(ge the sta(*ar* *e4)at)o( a(* the 4ar)a(ce each g)4e a <,a(t)tat)4e a(swer

to the <,est)o( @ow 4ar)able are the *ataC
















!A/:+ (ATA S+T ++/ C&S+S


http://www.1.7ls

a %o-p,te the ra(ge a(* sa-ple sta(*ar* *e4)at)o( of the 1;;; SA& scores.

b %o-p,te the ra(ge a(* sa-ple sta(*ar* *e4)at)o( of the 1;;; !PAs.

2; Large ata Set 1 l)sts the SA& scores of 1;;; st,*e(ts.

http://www.1.7ls

a Regar* the *ata as ar)s)(g fro- a ce(s,s of all st,*e(ts at a h)gh school )( wh)ch the

SA& score of e4ery st,*e(t was -eas,re*. %o-p,te the pop,lat)o( ra(ge a(*

pop,lat)o( sta(*ar* *e4)at)o( σ .


%o-p,te the sa-ple ra(ge a(* sa-ple sta(*ar* *e4)at)o( s a(* co-pare the- to the

pop,lat)o( ra(ge a(* σ .




21 Large ata Set 1 l)sts the !PAs of 1;;; st,*e(ts.

http://www.1.7ls

a Regar* the *ata as ar)s)(g fro- a ce(s,s of all fresh-a( at a s-all college at the e(* of

the)r +rst aca*e-)c year of college st,*y )( wh)ch the !PA of e4ery s,ch perso( was

-eas,re*. %o-p,te the pop,lat)o( ra(ge a(* pop,lat)o( sta(*ar* *e4)at)o( σ .









http://www..7lshttp://www.A.7ls

http://www.#.7ls

a %o-p,te the ra(ge a(* sa-ple sta(*ar* *e4)at)o( of s,r4)4al t)-e for all -)ce w)tho,t

regar* to ge(*er.






b %o-p,te the ra(ge a(* sa-ple sta(*ar* *e4)at)o( of s,r4)4al t)-e for the 0 -ale -)ce

separately recor*e* )( Large ata Set A5.

c %o-p,te the ra(ge a(* sa-ple sta(*ar* *e4)at)o( of s,r4)4al t)-e for the 0 fe-ale

-)ce separately recor*e* )( Large ata Set #5. o yo, see a *)8ere(ce )( the res,lts

for -ale a(* fe-ale -)ceC oes )t appear to be s)g()+ca(tC






0.7 /elative 2osition o (ata

!+A/N&N: 1';+CT&<+S

1 &o lear( the co(cept of the relat)4e pos)t)o( of a( ele-e(t of a *ata set.






2 &o lear( the -ea()(g of each of two -eas,res the perce(t)le ra(k a(* the z =

score of the relat)4e pos)t)o( of a -eas,re-e(t a(* how to co-p,te each o(e.

3 &o lear( the -ea()(g of the three <,art)les assoc)ate* to a *ata set a(* how to

co-p,te the-.

&o lear( the -ea()(g of the +4e=(,-ber s,--ary of a *ata set how to co(str,ctthe bo7 plot assoc)ate* to )t a(* how to )(terpret the bo7 plot.

(hen you take an exam$ what is often as important as your actual score on the exam is the way your

score compares to other students& performance. ,f you made a @5 but the average score *whether the

mean$ median$ or mode) was ?8$ you did relatively poorly. ,f you made a @5 but the average score

was only 88 then you did relatively well. ,n general$ the significance of one observed value in a data

set strongly depends on how that value compares to the other observed values in a data set.

Therefore we wish to attach to each observed value a number that measures its relative position.

2ercentiles and ?uartiles

%nyone who has taken a national standardi'ed test is familiar with the idea of being given both a score on

the exam and a Gpercentile rankingH of that score. Sou may be told that your score was ;!8 and that it is

the ?8th percentile. The first number tells how you actually did on the exam the second says that ?8B of

the scores on the exam were less than or e"ual to your score$ ;!8.

(e)nition

#iven an observed value x in a data set $ x is the &th percentile of the data if the percentage of the data

that are less than or equal to x is $. The number $ is the percentile ran' of x .

+A>2!+ %3

Bhat perce(t)le )s the 4al,e 1.39 )( the *ata set of te( !PAs co(s)*ere* )( Note 2.12

QE7a-ple 3Q )( Sect)o( 2.2 Qeas,res of %e(tral Locat)o(QC Bhat perce(t)le )s the

4al,e 3.33C

Sol,t)o(:

&he *ata wr)tte( )( )(creas)(g or*er are

1.39 1.76 1.90 2.12 2.53 2.71 3.00 3.33 3.71 4.00

&he o(ly *ata 4al,e that )s less tha( or e<,al to 1.39 )s 1.39 )tself. S)(ce 1 )s 11; .

1; or 1;G of 1; the 4al,e 1.39 )s the 1;th perce(t)le. E)ght *ata 4al,es are less tha(






or e<,al to 3.33. S)(ce 6 )s 61; .6; or 6;G of 1; the 4al,e 3.33 )s the 6;th

perce(t)le.

The $ th percentile cuts the data set in two so that approximately $ B of the data lie below it

and(100−P)B of the data lie above it. ,n particular$ the three percentiles that cut the data into

fourths$ as shown in /igure !.1! 02ata 2ivision by Auartiles0$ are called the quartiles. The following

simple computational definition of the three "uartiles works well in practice.

!igure &."& (ata (ivision by 7uartiles

(e)nition

!or any data set8

1 The second quartile Q2of the data set is its median.

! (efine two subsets8

1 the lower set8 all observations that are strictly less than Q2






! the upper set8 all observations that are strictly greater than Q2.

3 The first quartile Q1of the data set is the median of the lower set.

6 The third quartile Q3of the data set is the median of the upper set.

+A>2!+ %7

)(* the <,art)les of the *ata set of !PAs of Note 2.12 QE7a-ple 3Q )( Sect)o( 2.2

Qeas,res of %e(tral Locat)o(Q.

Sol,t)o(:

As )( the pre4)o,s e7a-ple we +rst l)st the *ata )( (,-er)cal or*er:

1.39 1.76 1.90 2.12 2.53 2.71 3.00 3.33 3.71 4.00

&h)s *ata set has n 1; obser4at)o(s. S)(ce 1; )s a( e4e( (,-ber the -e*)a( )s the

-ea( of the two -)**le obser4at)o(s: x=(2.53 + 2.71)/2=2.62. &h,s the seco(* <,art)le

)s Q2=2.62. &he lower a(* ,pper s,bsets are

Lower:L={1.39,1.76,1.90,2.12,2.53}

Upper:U={2.71,3.00,3.33,3.71,4.00}

Each has a( o** (,-ber of ele-e(ts so the -e*)a( of each )s )ts -)**le

obser4at)o(. &h,s the +rst <,art)le )s Q1=1.90 the -e*)a( of L a(* the th)r* <,art)le

)s Q3=3.33 the -e*)a( of U.

EKAPLE 10

A*>o)( the obser4at)o( 3.66 to the *ata set of the pre4)o,s e7a-ple a(* +(* the

<,art)les of the (ew set of *ata.

Sol,t)o(:

As )( the pre4)o,s e7a-ple we +rst l)st the *ata )( (,-er)cal or*er:






1.39 1.76 1.90 2.12 2.53 2.71 3.00 3.33 3.71 3.88 4.00

&h)s *ata set has 11 obser4at)o(s. &he seco(* <,art)le )s )ts -e*)a( the -)**le 4al,e

2.1. &h,s Q2=2.71. &he lower a(* ,pper s,bsets are (ow

Lower:L={1.39,1.76,1.90,2.12,2.53}

Upper: U= {3.00,3.33,3.71,3.88,4.00}

&he lower set L has -e*)a( the -)**le 4al,e 1.9; so Q1=1.90. &he ,pper set has

-e*)a( the -)**le 4al,e 3.1 so Q3=3.71.

,n addition to the three "uartiles$ the two extreme values$ the minimum x min and the maximum x max are

also useful in describing the entire data set. Together these five numbers are called the five(

number summary of the data set:

N x min$ 71$ 7!$ 73$ x max

The five-number summary is used to construct a bo) plot as in /igure !.13 0The 4ox lot0. ach of the

five numbers is represented by a vertical line segment$ a box is formed using the line segments

at Q! and Q$ as its two vertical sides$ and two hori'ontal line segments are extended from the vertical

segments marking Q! and Q$ to the ad#acent extreme values. *The two hori'ontal line segments are

referred to as Gwhiskers$H and the diagram is sometimes called a Gbox and whisker plot.H) (e caution the

reader that there are other types of box plots that differ somewhat from the ones we are constructing$

although all are based on the three "uartiles.

!igure &."* The 9ox $lot

9ote that the distance fromQ1toQ3is the length of the interval over which the middle half of the

data range. Thus it has the following special name.






e+()t)o(

The inter"uartile range *,AC) is the quantity

IQR=Q3−Q1

EKAPLE 1

%o(str,ct a bo7 plot a(* +(* the HR for the *ata )( Note 2. QE7a-ple 1Q.

Sol,t)o(:

ro- o,r work )( Note 2. QE7a-ple 1Q we k(ow that the +4e=(,-ber s,--ary )s

xmin=1.39 Q1=1.90 Q2=2.62 Q3=3.33 xmax=4.00

&he bo7 plot )s

&he )(ter<,art)le ra(ge )s IQR=3.33−1.90=1.43.

z-scores

%nother way to locate a particular observation x in a data set is to compute its distance from the mean in

units of standard deviation.






The formulas in the definition allow us to compute the z -score when x is known. ,f the z -score is

known then x can be recovered using the corresponding inverse formulas

x=(x −)+sz or x=µ+σz

The z -score indicates how many standard deviations an individual observation x is from the center of

the data set$ its mean. ,f z is negative then x is below average. ,f z is 5 then x is e"ual to the average.

,f z is positive then x is above average. +ee /igure !.16.

!igure &." x :%cale versus z :%core











+A>2!+ %5

S,ppose the -ea( a(* sta(*ar* *e4)at)o( of the !PAs of all c,rre(tly reg)stere*

st,*e(ts at a college are μ 2.; a(* σ ;.0;. &he z =scores of the !PAs of two

st,*e(ts A(to()o a(* #eatr)ce are z=−0.62a(* z 1.26 respect)4ely. Bhat are the)r

!PAsC






Sol,t)o(:

Us)(g the seco(* for-,la r)ght after the *e+()t)o( of z =scores we co-p,te the !PAs

as

Antonio:x=µ+z σ=2.70+(−0.62)(0.50)=2.39

Beatrice:x=µ+z σ=2.70+(1.28)(0.50)=3.34

*+, TA*+AA,S

• &he perce(t)le ra(k a(* z =score of a -eas,re-e(t )(*)cate )ts relat)4e pos)t)o(

w)th regar* to the other -eas,re-e(ts )( a *ata set.

• &he three <,art)les *)4)*e a *ata set )(to fo,rths.

• &he +4e=(,-ber s,--ary a(* )ts assoc)ate* bo7 plot s,--ar)Ve the locat)o(

a(* *)str)b,t)o( of the *ata.














































30

E-)l)a a(* er*)(a(* took the sa-e fresh-a( che-)stry co,rse E-)l)a )( the fall

er*)(a(* )( the spr)(g. E-)l)a -a*e a( 63 o( the co--o( +(al e7a- that she took

o( wh)ch the -ea( was a(* the sta(*ar* *e4)at)o( 6. er*)(a(* -a*e a 9 o( the

co--o( +(al e7a- that he took wh)ch was -ore *)?c,lt s)(ce the -ea( was 0

a(* the sta(*ar* *e4)at)o( 12. &he o(e who has a h)gher z =score *)* relat)4ely better.

Bas )t E-)l)a or er*)(a(*C

3 Refer to the pre4)o,s e7erc)se. "( the +(al e7a- )( the sa-e co,rse the follow)(g

se-ester the -ea( )s 6 a(* the sta(*ar* *e4)at)o( )s 9. Bhat gra*e o( the e7a-

-atches E-)l)aWs perfor-a(ceC er*)(a(*WsC

3 Rose(cra(tV a(* !,)l*e(ster( are o( a we)ght=re*,c)(g *)et. Rose(cra(tV who we)ghs

16 lb belo(gs to a( age a(* bo*y=type gro,p for wh)ch the -ea( we)ght )s 10 lb a(*

the sta(*ar* *e4)at)o( )s 10 lb. !,)l*e(ster( who we)ghs 2; lb belo(gs to a( age a(*

bo*y=type gro,p for wh)ch the -ea( we)ght )s 10 lb a(* the sta(*ar* *e4)at)o( )s 2;

lb. Ass,-)(g z =scores are goo* -eas,res for co-par)so( )( th)s co(te7t who )s -ore

o4erwe)ght for h)s age a(* bo*y typeC

LAR!E A&A SE& EKER%SES


http://www.1.7ls

a %o-p,te the three <,art)les a(* the )(ter<,art)le ra(ge of the 1;;; SA& scores.

b %o-p,te the three <,art)les a(* the )(ter<,art)le ra(ge of the 1;;; !PAs.

39 Large ata Set 1; recor*s the scores of 2 st,*e(ts o( a stat)st)cs e7a-.

http://www.1;.7ls

a %o-p,te the +4e=(,-ber s,--ary of the *ata.






b escr)be )( wor*s the perfor-a(ce of the class o( the e7a- )( the l)ght of the res,lt )(

part a5.

; Large ata Sets 3 a(* 3A l)st the he)ghts of 1 c,sto-ers e(ter)(g a shoe store.

http://www.3.7ls

http://www.3A.7ls

a %o-p,te the +4e=(,-ber s,--ary of the he)ghts w)tho,t regar* to ge(*er.

b %o-p,te the +4e=(,-ber s,--ary of the he)ghts of the -e( )( the sa-ple.

c %o-p,te the +4e=(,-ber s,--ary of the he)ghts of the wo-e( )( the sa-ple.



http://www..7ls

http://www.A.7ls

http://www.#.7ls

a %o-p,te the three <,art)les a(* the )(ter<,art)le ra(ge of the s,r4)4al t)-es for all

-)ce w)tho,t regar* to ge(*er.

b %o-p,te the three <,art)les a(* the )(ter<,art)le ra(ge of the s,r4)4al t)-es for the 0

-ale -)ce separately recor*e* )( Large ata Set A5.

c %o-p,te the three <,art)les a(* the )(ter<,art)le ra(ge of the s,r4)4al t)-es for the 0

fe-ale -)ce separately recor*e* )( Large ata Set #5.
















0.8 The +mpirical /ule and Chebyshev#sTheorem

!+A/N&N: 1';+CT&<+S

1 &o lear( what the 4al,e of the sta(*ar* *e4)at)o( of a *ata set )-pl)es abo,t how

the *ata scatter away fro- the -ea( as *escr)be* by the E-p)r)cal R,le a(*

%hebyshe4Ws &heore-.

2 &o ,se the E-p)r)cal R,le a(* %hebyshe4Ws &heore- to *raw co(cl,s)o(s abo,t a

*ata set.

Sou probably have a good intuitive grasp of what the average of a data set says about that data set. ,n

this section we begin to learn what the standard deviation has to tell us about the nature of the data

set.

The +mpirical /ule

(e start by examining a specific set of data. Table !.! 0Eeights of en0 shows the heights in inches of 155

randomly selected adult men. % relative fre"uency histogram for the data is shown in /igure !.18 0Eeights

of %dult en0. The mean and standard deviation of the data are$ rounded to two decimal

places$ x−=69.92 and s M 1.@5. ,f we go through the data and count the number of observations that are

within one standard deviation of the mean$ that is$ that are

between69.92−1.70=68.22and69.92+1.70=71.62inches$ there are ;F of them. ,f we count the number

of observations that are within two standard deviations of the mean$ that is$ that are

bet*een69.92−2(1.70)=66.52and69.92+2(1.70)=73.32inches$ there are F8 of them. %ll of the

measurements are within three standard deviations of the mean$ that is$

between69.92−3(1.70)=64.822and69.92+3(1.70)=75.02inches. These tallies are not coincidences$ but

are in agreement with the following result that has been found to be widely applicable.

Table !.! Eeights of en

68.7 72.3 71.3 72.5 70.6 68.2 70.1 68.4 68.6 70.6

73.7 70.5 71.0 70.9 69.3 69.4 69.7 69.1 71.5 68.6

70.9 70.0 70.4 68.9 69.4 69.4 69.2 70.7 70.5 69.9

69.8 69.8 68.6 69.5 71.6 66.2 72.4 70.7 67.7 69.1






68.8 69.3 68.9 74.8 68.0 71.2 68.3 70.2 71.9 70.4

71.9 72.2 70.0 68.7 67.9 71.1 69.0 70.8 67.3 71.8

70.3 68.8 67.2 73.0 70.4 67.8 70.0 69.5 70.1 72.0

72.2 67.6 67.0 70.3 71.2 65.6 68.1 70.8 71.4 70.2

70.1 67.5 71.3 71.5 71.0 69.1 69.5 71.1 66.8 71.8

69.6 72.7 72.8 69.6 65.9 68.0 69.7 68.7 69.8 69.7

!igure &." +eights of Adult 2en

&he E-p)r)cal R,le

,f a data set has an approximately bell-shaped relative fre"uency histogram$ then *see /igure !.1; 0The

mpirical Cule0)

1 approximately ;?B of the data lie within one standard deviation of the mean$ that is$ in the interval

with endpoints x −±sfor samples and with endpointsµ±σfor populations

! approximately F8B of the data lie within two standard deviations of the mean$ that is$ in the interval

with endpoints x −±2sfor samples and with endpointsµ±2σfor populations and






3 approximately FF.@B of the data lies within three standard deviations of the mean$ that is$ in the

interval with endpoints x −±3sfor samples and with endpointsµ±3σfor populations.

!igure &."/ The ;mpirical -ule






Two key points in regard to the mpirical Cule are that the data distribution must be approximately bell:

shaped and that the percentages are only approximately true. The mpirical Cule does not apply to data

sets with severely asymmetric distributions$ and the actual percentage of observations in any of the

intervals specified by the rule could be either greater or less than those given in the rule. (e see this with

the example of the heights of the men: the mpirical Cule suggested ;? observations between ;?.!! and

@1.;! inches but we counted ;F.











Figure 2.17Distribution of eig!ts

EKAPLE 2;

Scores o( H tests ha4e a bell=shape* *)str)b,t)o( w)th -ea( μ 1;; a(* sta(*ar*

*e4)at)o( σ 1;. )sc,ss what the E-p)r)cal R,le )-pl)es co(cer()(g )(*)4)*,als w)th

H scores of 11; 12; a(* 13;.

Sol,t)o(:

A sketch of the H *)str)b,t)o( )s g)4e( )( )g,re 2.16 Q)str)b,t)o( of H ScoresQ. &he

E-p)r)cal R,le states that

1 appro7)-ately 6G of the H scores )( the pop,lat)o( l)e betwee( 9; a(* 11;

2 appro7)-ately 90G of the H scores )( the pop,lat)o( l)e betwee( 6; a(* 12;

a(*

3 appro7)-ately 99.G of the H scores )( the pop,lat)o( l)e betwee( ; a(* 13;.






Figure 2.1"Distribution of #$ %cores

S)(ce 6G of the H scores l)e &it!in the )(ter4al fro- 9; to 11; )t -,st be the case

that 32G l)e outside that )(ter4al. #y sy--etry appro7)-ately half of that 32G or

1G of all H scores w)ll l)e abo4e 11;. f 1G l)e abo4e 11; the( 6G l)e below. Be

co(cl,*e that the H score 11; )s the 6th perce(t)le.

&he sa-e a(alys)s appl)es to the score 12;. S)(ce appro7)-ately 90G of all H scores

l)e w)th)( the )(ter4al for- 6; to 12; o(ly 0G l)e o,ts)*e )t a(* half of the- or 2.0G

of all scores are abo4e 12;. &he H score 12; )s th,s h)gher tha( 9.0G of all H

scores a(* )s <,)te a h)gh score.

#y a s)-)lar arg,-e(t o(ly 10/1;; of 1G of all a*,lts or abo,t o(e or two )( e4ery

tho,sa(* wo,l* ha4e a( H score abo4e 13;. &h)s fact -akes the score 13;

e7tre-ely h)gh.






Chebyshev#s Theorem

The mpirical Cule does not apply to all data sets$ only to those that are bell-shaped$ and even then is

stated in terms of approximations. % result that applies to every data set is known as 7hebyshev&s

Theorem.

%hebyshe4Ws &heore-

/or any numerical data set$

1 at least 3I6 of the data lie within two standard deviations of the mean$ that is$ in the interval with

endpoints x −±2sfor samples and with endpointsµ±2σfor populations

! at least ?IF of the data lie within three standard deviations of the mean$ that is$ in the interval with

endpoints x −±3sfor samples and with endpointsµ±3σfor populations

3 at least1−1/k2 of the data lie within k standard deviations of the mean$ that is$ in the interval with

endpoints x −±ksfor samples and with endpointsµ±kσfor populations$ where k is any positive whole

number that is greater than 1.

/igure !.1F 07hebyshev&s Theorem0 gives a visual illustration of 7hebyshev&s Theorem.

igure &."4 <hebyshev=s Theorem






,t is important to pay careful attention to the words Gat leastH at the beginning of each of the three parts.

The theorem gives the minimum proportion of the data which must lie within a given number of standard

deviations of the mean the true proportions found within the indicated regions could be greater than

what the theorem guarantees.











+A>2!+ 00

&he (,-ber of 4eh)cles pass)(g thro,gh a b,sy )(tersect)o( betwee( 6:;; a.-.

a(* 1;:;; a.-. was obser4e* a(* recor*e* o( e4ery week*ay -or()(g of the last

year. &he *ata set co(ta)(s n 201 (,-bers. &he sa-ple -ea( )s x −=725a(* the

sa-ple sta(*ar* *e4)at)o( )s s 20. *e(t)fy wh)ch of the follow)(g

state-e(ts must be tr,e.

1 "( appro7)-ately 90G of the week*ay -or()(gs last year the (,-ber of 4eh)cles

pass)(g thro,gh the )(tersect)o( fro- 6:;; a.-. to 1;:;; a.-. was betwee( 0

a(* 0.

2 "( at least 0G of the week*ay -or()(gs last year the (,-ber of 4eh)cles

pass)(g thro,gh the )(tersect)o( fro- 6:;; a.-. to 1;:;; a.-. was betwee( 0

a(* 0.

3 "( at least 169 week*ay -or()(gs last year the (,-ber of 4eh)cles pass)(g

thro,gh the )(tersect)o( fro- 6:;; a.-. to 1;:;; a.-. was betwee( 0 a(* 0.

"( at -ost 20G of the week*ay -or()(gs last year the (,-ber of 4eh)cles

pass)(g thro,gh the )(tersect)o( fro- 6:;; a.-. to 1;:;; a.-. was e)ther less

tha( 0 or greater tha( 0.

0 "( at -ost 12.0G of the week*ay -or()(gs last year the (,-ber of 4eh)cles

pass)(g thro,gh the )(tersect)o( fro- 6:;; a.-. to 1;:;; a.-. was less tha( 0.

"( at -ost 20G of the week*ay -or()(gs last year the (,-ber of 4eh)cles

pass)(g thro,gh the )(tersect)o( fro- 6:;; a.-. to 1;:;; a.-. was less tha( 0.

Sol,t)o(:

1 S)(ce )t )s (ot state* that the relat)4e fre<,e(cy h)stogra- of the *ata )s bell=

shape* the E-p)r)cal R,le *oes (ot apply. State-e(t 15 )s base* o( the

E-p)r)cal R,le a(* therefore )t -)ght (ot be correct.

2 State-e(t 25 )s a *)rect appl)cat)o( of part 15 of %hebyshe4Ws &heore-

beca,se (x −−2s,x −+2s)=(675,775).t -,st be correct.

3 State-e(t 35 says the sa-e th)(g as state-e(t 25 beca,se 0G of 201 )s

166.20 so the -)()-,- whole (,-ber of obser4at)o(s )( th)s )(ter4al )s 169.

&h,s state-e(t 35 )s *e+()tely correct.

State-e(t 5 says the sa-e th)(g as state-e(t 25 b,t )( *)8ere(t wor*s a(*

therefore )s *e+()tely correct.






0 State-e(t 5 wh)ch )s *e+()tely correct states that at -ost 20G of the t)-e

e)ther fewer tha( 0 or -ore tha( 0 4eh)cles passe* thro,gh the )(tersect)o(.

State-e(t 05 says that half of that 20G correspo(*s to *ays of l)ght tra?c. &h)s

wo,l* be correct )f the relat)4e fre<,e(cy h)stogra- of the *ata were k(ow( to be

sy--etr)c. #,t th)s )s (ot state*D perhaps all of the obser4at)o(s o,ts)*e the

)(ter4al 005 are less tha( 0. &h,s state-e(t 05 -)ght (ot be correct

State-e(t 5 )s *e+()tely correct a(* state-e(t 5 )-pl)es state-e(t 5: e4e(

)f e4ery -eas,re-e(t that )s o,ts)*e the )(ter4al 005 )s less tha( 0

wh)ch )s co(ce)4able s)(ce sy--etry )s (ot k(ow( to hol*5 e4e( so at -ost

20G of all obser4at)o(s are less tha( 0. &h,s state-e(t 5 -,st *e+()tely be

correct.

*+, TA*+AA,S

• &he E-p)r)cal R,le )s a( appro7)-at)o( that appl)es o(ly to *ata sets w)th a bell=

shape* relat)4e fre<,e(cy h)stogra-. t est)-ates the proport)o( of the

-eas,re-e(ts that l)e w)th)( o(e two a(* three sta(*ar* *e4)at)o(s of the

-ea(.

• %hebyshe4Ws &heore- )s a fact that appl)es to all poss)ble *ata sets. t *escr)bes

the -)()-,- proport)o( of the -eas,re-e(ts that l)e -,st w)th)( o(e two or

-ore sta(*ar* *e4)at)o(s of the -ea(.

++/C&S+S

'AS&C

1 State the E-p)r)cal R,le.

2 escr)be the co(*)t)o(s ,(*er wh)ch the E-p)r)cal R,le -ay be appl)e*.

3 State %hebyshe4Ws &heore-.

escr)be the co(*)t)o(s ,(*er wh)ch %hebyshe4Ws &heore- -ay be appl)e*.

0 A sa-ple *ata set w)th a bell=shape* *)str)b,t)o( has -ea( x −=6a(* sta(*ar*

*e4)at)o( s 2. )(* the appro7)-ate proport)o( of obser4at)o(s )( the *ata set that l)e:a betwee( a(* 6D

b betwee( 2 a(* 1;D

c betwee( ; a(* 12.

A pop,lat)o( *ata set w)th a bell=shape* *)str)b,t)o( has -ea( μ a(* sta(*ar*

*e4)at)o( σ 2. )(* the appro7)-ate proport)o( of obser4at)o(s )( the *ata set that l)e:

a betwee( a(* 6D






b betwee( 2 a(* 1;D

c betwee( ; a(* 12.

A pop,lat)o( *ata set w)th a bell=shape* *)str)b,t)o( has -ea( μ 2 a(* sta(*ar*

*e4)at)o( σ 1.1. )(* the appro7)-ate proport)o( of obser4at)o(s )( the *ata set that

l)e:

a abo4e 2D

b abo4e 3.1D

c betwee( 2 a(* 3.1.6 A sa-ple *ata set w)th a bell=shape* *)str)b,t)o( has -ea( x−=2a(* sta(*ar*

*e4)at)o( s 1.1. )(* the appro7)-ate proport)o( of obser4at)o(s )( the *ata set

that l)e:

a below X;.2D

b below 3.1Dc betwee( X1.3 a(* ;.9.

9 A pop,lat)o( *ata set w)th a bell=shape* *)str)b,t)o( a(* s)Ve ' 0;; has -ea( μ

2 a(* sta(*ar* *e4)at)o( σ 1.1. )(* the appro7)-ate (,-ber of obser4at)o(s )(

the *ata set that l)e:

a abo4e 2D

b abo4e 3.1D

c betwee( 2 a(* 3.1.

1; A sa-ple *ata set w)th a bell=shape* *)str)b,t)o( a(* s)Ve n 126 has -ea( x −=2a(*

sta(*ar* *e4)at)o( s 1.1. )(* the appro7)-ate (,-ber of obser4at)o(s )( the *ata

set that l)e:

a below X;.2D

b below 3.1D

c betwee( X1.3 a(* ;.9.

11 A sa-ple *ata set has -ea( x −=6a(* sta(*ar* *e4)at)o( s 2. )(* the -)()-,-

proport)o( of obser4at)o(s )( the *ata set that -,st l)e:

a betwee( 2 a(* 1;D

b betwee( ; a(* 12D

c betwee( a(* 6.

12 A pop,lat)o( *ata set has -ea( μ 2 a(* sta(*ar* *e4)at)o( σ 1.1. )(* the

-)()-,- proport)o( of obser4at)o(s )( the *ata set that -,st l)e:

a betwee( X;.2 a(* .2D

b betwee( X1.3 a(* 0.3.

Saylor URL: http://www.saylor.org/books Saylor.org1;;





13 A pop,lat)o( *ata set of s)Ve ' 0;; has -ea( μ 0.2 a(* sta(*ar* *e4)at)o( σ

1.1. )(* the -)()-,- (,-ber of obser4at)o(s )( the *ata set that -,st l)e:

a betwee( 3 a(* .D

b betwee( 1.9 a(* 6.0.

1 A sa-ple *ata set of s)Ve n 126 has -ea(x −=2a(* sta(*ar* *e4)at)o( s 2. )(*

the -)()-,- (,-ber of obser4at)o(s )( the *ata set that -,st l)e:

a betwee( X2 a(* )(cl,*)(g X2 a(* 5D

b betwee( X a(* 6 )(cl,*)(g X a(* 65.

10 A sa-ple *ata set of s)Ve n 3; has -ea( x −=6a(* sta(*ar* *e4)at)o( s 2.

a Bhat )s the -a7)-,- proport)o( of obser4at)o(s )( the *ata set that ca( l)e

o,ts)*e the )(ter4al 21;5C

b Bhat ca( be sa)* abo,t the proport)o( of obser4at)o(s )( the *ata set that are

below 2C

c Bhat ca( be sa)* abo,t the proport)o( of obser4at)o(s )( the *ata set that are

abo4e 1;C

* Bhat ca( be sa)* abo,t the (,-ber of obser4at)o(s )( the *ata set that are

abo4e 1;C

1 A pop,lat)o( *ata set has -ea( μ 2 a(* sta(*ar* *e4)at)o( σ 1.1.

a Bhat )s the -a7)-,- proport)o( of obser4at)o(s )( the *ata set that ca( l)e

o,ts)*e the )(ter4al (−1.3,5.3)C

b Bhat ca( be sa)* abo,t the proport)o( of obser4at)o(s )( the *ata set that are

below X1.3C

c Bhat ca( be sa)* abo,t the proport)o( of obser4at)o(s )( the *ata set that are

abo4e 0.3C

A22!&CAT&1NS

1 Scores o( a +(al e7a- take( by 12;; st,*e(ts ha4e a bell=shape* *)str)b,t)o( w)th

-ea( 2 a(* sta(*ar* *e4)at)o( 9.

a Bhat )s the -e*)a( score o( the e7a-C

b Abo,t how -a(y st,*e(ts score* betwee( 3 a(* 61C

c Abo,t how -a(y st,*e(ts score* betwee( 2 a(* 9;C

* Abo,t how -a(y st,*e(ts score* below 0C

Saylor URL: http://www.saylor.org/books Saylor.org1;1





16 Le(gths of +sh ca,ght by a co--erc)al +sh)(g boat ha4e a bell=shape* *)str)b,t)o( w)th

-ea( 23 )(ches a(* sta(*ar* *e4)at)o( 1.0 )(ches.

a Abo,t what proport)o( of all +sh ca,ght are betwee( 2; )(ches a(* 2 )(ches lo(gC

b Abo,t what proport)o( of all +sh ca,ght are betwee( 2; )(ches a(* 23 )(ches lo(gC

c Abo,t how lo(g )s the lo(gest +sh ca,ght o(ly a s-all fract)o( of a perce(t are lo(ger5C

19 @ockey p,cks ,se* )( profess)o(al hockey ga-es -,st we)gh betwee( 0.0 a(*

o,(ces. f the we)ght of p,cks -a(,fact,re* by a part)c,lar process )s bell=shape* has

-ea( 0.0 o,(ces a(* sta(*ar* *e4)at)o( ;.120 o,(ce what proport)o( of the p,cks

w)ll be ,sable )( profess)o(al ga-esC

2; @ockey p,cks ,se* )( profess)o(al hockey ga-es -,st we)gh betwee( 0.0 a(*

o,(ces. f the we)ght of p,cks -a(,fact,re* by a part)c,lar process )s bell=shape* a(*

has -ea( 0.0 o,(ces how large ca( the sta(*ar* *e4)at)o( be )f 99.G of the p,cks

are to be ,sable )( profess)o(al ga-esC

21 Spee*s of 4eh)cles o( a sect)o( of h)ghway ha4e a bell=shape* *)str)b,t)o( w)th -ea(

; -ph a(* sta(*ar* *e4)at)o( 2.0 -ph.

a f the spee* l)-)t )s 00 -ph abo,t what proport)o( of 4eh)cles are spee*)(gC

b Bhat )s the -e*)a( spee* for 4eh)cles o( th)s h)ghwayC

c Bhat )s the perce(t)le ra(k of the spee* 0 -phC

* Bhat spee* correspo(*s to the 1th perce(t)leC

22 S,ppose that as )( the pre4)o,s e7erc)se spee*s of 4eh)cles o( a sect)o( of h)ghway

ha4e -ea( ; -ph a(* sta(*ar* *e4)at)o( 2.0 -ph b,t (ow the *)str)b,t)o( of

spee*s )s ,(k(ow(.

a f the spee* l)-)t )s 00 -ph at least what proport)o( of 4eh)cles -,st

spee*)(gC

b Bhat ca( be sa)* abo,t the proport)o( of 4eh)cles go)(g 0 -ph or fasterC

23 A( )(str,ctor a((o,(ces to the class that the scores o( a rece(t e7a- ha* a bell=

shape* *)str)b,t)o( w)th -ea( 0 a(* sta(*ar* *e4)at)o( 0.

a Bhat )s the -e*)a( scoreC






b Appro7)-ately what proport)o( of st,*e(ts )( the class score* betwee( ; a(*

6;C

c Appro7)-ately what proport)o( of st,*e(ts )( the class score* abo4e 60C

* Bhat )s the perce(t)le ra(k of the score 60C

2 &he !PAs of all c,rre(tly reg)stere* st,*e(ts at a large ,()4ers)ty ha4e a bell=shape*

*)str)b,t)o( w)th -ea( 2. a(* sta(*ar* *e4)at)o( ;.. St,*e(ts w)th a !PA below 1.0

are place* o( aca*e-)c probat)o(. Appro7)-ately what perce(tage of c,rre(tly

reg)stere* st,*e(ts at the ,()4ers)ty are o( aca*e-)c probat)o(C

20 &h)rty=s)7 st,*e(ts took a( e7a- o( wh)ch the a4erage was 6; a(* the sta(*ar*

*e4)at)o( was . A r,-or says that +4e st,*e(ts ha* scores 1 or below. %a( the

r,-or be tr,eC Bhy or why (otC


























Chapter 3

'asic Concepts o 2robability

+uppose a polling organi'ation "uestions 1$!55 voters in order to estimate the proportion of all

voters who favor a particular bond issue. (e would expect the proportion of the 1$!55 voters in the

survey who are in favor to be close to the proportion of all voters who are in favor$ but this need not

be true. There is a degree of randomness associated with the survey result. ,f the survey result is

highly likely to be close to the true proportion$ then we have confidence in the survey result. ,f it is

not particularly likely to be close to the population proportion$ then we would perhaps not take the

survey result too seriously. The likelihood that the survey proportion is close to the population

proportion determines our confidence in the survey result. /or that reason$ we would like to be able

to compute that likelihood. The task of computing it belongs to the realm of probability$ which we

study in this chapter.

3.% Sample Spaces@ +vents@ and Their 2robabilities

LEARNN! "#$E%&'ES

1 &o lear( the co(cept of the sa-ple space assoc)ate* w)th a ra(*o- e7per)-e(t.

2 &o lear( the co(cept of a( e4e(t assoc)ate* w)th a ra(*o- e7per)-e(t.

3 &o lear( the co(cept of the probab)l)ty of a( e4e(t.

Sample Spaces and +vents

Colling an ordinary six-sided die is a familiar example of a random experiment $ an action for which all

possible outcomes can be listed$ but for which the actual outcome on any given trial of the experiment

cannot be predicted with certainty. ,n such a situation we wish to assign to each outcome$ such as rolling a

two$ a number$ called the probability of the outcome$ that indicates how likely it is that the outcome will

occur. +imilarly$ we would like to assign a probability to any event $ or collection of outcomes$ such as

rolling an even number$ which indicates how likely it is that the event will occur if the experiment is






performed. This section provides a framework for discussing probability problems$ using the terms #ust

mentioned.

e+()t)o(

A random e)periment is a mechanism that produces a definite outcome that cannot be predicted

with certainty. The sample space associated with a random experiment is the set of all possible

outcomes. An event is a subset of the sample space.

e+()t)o(

An event ; is said to occur on a particular trial of the experiment if the outcome observed is an element

of the set ; .

+A>2!+ %

%o(str,ct a sa-ple space for the e7per)-e(t that co(s)sts of toss)(g a s)(gle co)(.

Sol,t)o(:

&he o,tco-es co,l* be labele* ! for hea*s a(* t for ta)ls. &he( the sa-ple space )s

the set S={h,t}.

+A>2!+ 0

%o(str,ct a sa-ple space for the e7per)-e(t that co(s)sts of roll)(g a s)(gle *)e. )(*

the e4e(ts that correspo(* to the phrases a( e4e( (,-ber )s rolle* a(* a (,-ber

greater tha( two )s rolle*.

Sol,t)o(:

&he o,tco-es co,l* be labele* accor*)(g to the (,-ber of *ots o( the top face of the

*)e. &he( the sa-ple space )s the set S={1,2,3,4,5,6}.






&he o,tco-es that are e4e( are 2 a(* so the e4e(t that correspo(*s to the

phrase a( e4e( (,-ber )s rolle* )s the set M2 wh)ch )t )s (at,ral to *e(ote by

the letter (. Be wr)te E={2,4,6}.

S)-)larly the e4e(t that correspo(*s to the phrase a (,-ber greater tha( two )s

rolle* )s the set T={3,4,5,6}@ wh)ch we ha4e *e(ote* ) .

% graphical representation of a sample space and events is a +enn diagram$ as shown in /igure

3.1 0Denn 2iagrams for Two +ample +paces0 for 9ote 3.; 0xample 10 and 9ote 3.@ 0xample !0.

,n general the sample space % is represented by a rectangle$ outcomes by points within the

rectangle$ and events by ovals that enclose the outcomes that compose them.

ure *." 0enn (iagrams for Two %ample %paces

+A>2!+ 3

A ra(*o- e7per)-e(t co(s)sts of toss)(g two co)(s.

a. %o(str,ct a sa-ple space for the s)t,at)o( that the co)(s are )(*)st)(g,)shable

s,ch as two bra(* (ew pe(()es.

b. %o(str,ct a sa-ple space for the s)t,at)o( that the co)(s are *)st)(g,)shable s,ch as

o(e a pe((y a(* the other a ()ckel.

Sol,t)o(:

a. After the co)(s are tosse* o(e sees e)ther two hea*s wh)ch co,l* be labele* 2htwo ta)ls wh)ch co,l* be labele* 2t or co)(s that *)8er wh)ch co,l* be labele* d. &h,s a

sa-ple space )s S={2h,2t,d}.

b. S)(ce we ca( tell the co)(s apart there are (ow two ways for the co)(s to *)8er: the

pe((y hea*s a(* the ()ckel ta)ls or the pe((y ta)ls a(* the ()ckel hea*s. Be ca(

label each o,tco-e as a pa)r of letters the +rst of wh)ch )(*)cates how the pe((y






la(*e* a(* the seco(* of wh)ch )(*)cates how the ()ckel la(*e*. A sa-ple space )s

the( S′={hh,ht,th,tt}.

% device that can be helpful in identifying all possible outcomes of a random experiment$ particularly one

that can be viewed as proceeding in stages$ is what is called a tree diagram. ,t is described in the

following example.

EKAPLE

%o(str,ct a sa-ple space that *escr)bes all three=ch)l* fa-)l)es accor*)(g to the

ge(*ers of the ch)l*re( w)th respect to b)rth or*er.

Sol,t)o(:

&wo of the o,tco-es are two boys the( a g)rl wh)ch we -)ght *e(ote bbg a(* a

g)rl the( two boys wh)ch we wo,l* *e(ote gbb.%learly there are -a(y o,tco-es

a(* whe( we try to l)st all of the- )t co,l* be *)?c,lt to be s,re that we ha4e

fo,(* the- all ,(less we procee* syste-at)cally. &he tree *)agra- show(

)()g,re 3.2 Q&ree )agra- or &hree=%h)l* a-)l)esQ g)4es a syste-at)c

approach.

Figure *.2)ree Diagram For )!ree+,!ild Families






&he *)agra- was co(str,cte* as follows. &here are two poss)b)l)t)es for the +rst

ch)l* boy or g)rl so we *raw two l)(e seg-e(ts co-)(g o,t of a start)(g po)(t

o(e e(*)(g )( a b for boy a(* the other e(*)(g )( a g for g)rl. or each of

these two poss)b)l)t)es for the +rst ch)l* there are two poss)b)l)t)es for the seco(*

ch)l* boy or g)rl so fro- each of the b a(* g we *raw two l)(e seg-e(ts o(e

seg-e(t e(*)(g )( a b a(* o(e )( a g. or each of the fo,r e(*)(g po)(ts (ow )(

the *)agra- there are two poss)b)l)t)es for the th)r* ch)l* so we repeat the

process o(ce -ore.

&he l)(e seg-e(ts are calle* branches of the tree. &he r)ght e(*)(g po)(t of each

bra(ch )s calle* a node. &he (o*es o( the e7tre-e r)ght are the )nal nodesD to

each o(e there correspo(*s a( o,tco-e as show( )( the +g,re.

ro- the tree )t )s easy to rea* o8 the e)ght o,tco-es of the e7per)-e(t so the

sa-ple space )s rea*)(g fro- the top to the botto- of the +(al (o*es )( the tree






S={bbb,bbg,bgb,bgg,gbb,gbg,ggb,ggg}

2robability

e+()t)o(

The probability of an outcome e in a sample space % is a number p between 5 and " that measures

the likelihood that e will occur on a single trial of the corresponding random experiment. The value p M

5 corresponds to the outcome e being impossible and the value p M 1 corresponds to the outcome e being

certain.

e+()t)o(

The probability of an event A is the sum of the probabilities of the individual outcomes of which it is

composed. >t is denoted P(A).

The following formula expresses the content of the definition of the probability of an event:

,f an event ; isE={e1,e2,…,ek}$ then

$ * ; )M $ *e1)K $ *e!)K Y Y Y K $ *ek)

/igure 3.3 0+ample +paces and robability0 graphically illustrates the definitions.

!igure *.* %ample %paces and $robability






+ince the whole sample space % is an event that is certain to occur$ the sum of the probabilities of all

the outcomes must be the number 1.

,n ordinary language probabilities are fre"uently expressed as percentages. /or example$ we would

say that there is a @5B chance of rain tomorrow$ meaning that the probability of rain is 5.@5. (e will

use this practice here$ but in all the computational formulas that follow we will use the form 5.@5 and

not @5B.

+A>2!+ 8

A co)( )s calle* bala(ce* or fa)r )f each s)*e )s e<,ally l)kely to la(* ,p. Ass)g( a

probab)l)ty to each o,tco-e )( the sa-ple space for the e7per)-e(t that co(s)sts of

toss)(g a s)(gle fa)r co)(.

Sol,t)o(:

B)th the o,tco-es labele* ! for hea*s a(* t for ta)ls the sa-ple space )s the

set S={h,t}.S)(ce the o,tco-es ha4e the sa-e probab)l)t)es wh)ch -,st a** ,p to 1

each o,tco-e )s ass)g(e* probab)l)ty 1/2.

+A>2!+ 9

A *)e )s calle* bala(ce* or fa)r )f each s)*e )s e<,ally l)kely to la(* o( top. Ass)g( a

probab)l)ty to each o,tco-e )( the sa-ple space for the e7per)-e(t that co(s)sts of

toss)(g a s)(gle fa)r *)e. )(* the probab)l)t)es of the e4e(ts (: a( e4e( (,-ber )s

rolle* a(* ) : a (,-ber greater tha( two )s rolle*.

Sol,t)o(:






B)th o,tco-es labele* accor*)(g to the (,-ber of *ots o( the top face of the *)e the

sa-ple space )s the set S={1,2,3,4,5,6}.S)(ce there are s)7 e<,ally l)kely o,tco-es

wh)ch -,st a** ,p to 1 each )s ass)g(e* probab)l)ty 1/.

+A>2!+ 4

&wo fa)r co)(s are tosse*. )(* the probab)l)ty that the co)(s -atch ).e. e)ther

both la(* hea*s or both la(* ta)ls.

Sol,t)o(:

( Note 3.6 QE7a-ple 3Q we co(str,cte* the sa-ple space S={2h,2t,d}for the

s)t,at)o( )( wh)ch the co)(s are )*e(t)cal a(* the sa-ple space S′={hh,ht,th,tt}for the

s)t,at)o( )( wh)ch the two co)(s ca( be tol* apart.

&he theory of probab)l)ty *oes (ot tell ,s !o& to ass)g( probab)l)t)es to the

o,tco-es o(ly what to *o w)th the- o(ce they are ass)g(e*. Spec)+cally ,s)(g

sa-ple space % -atch)(g co)(s )s the e4e(t M ={2h,2t} wh)ch has probab)l)ty P(2h)

+P(2t).Us)(g sa-ple space S′ -atch)(g co)(s )s the e4e(t M ′={hh,tt} wh)ch has

probab)l)ty P(hh)+P(tt).( the phys)cal worl* )t sho,l* -ake (o *)8ere(ce whether

the co)(s are )*e(t)cal or (ot a(* so we wo,l* l)ke to ass)g( probab)l)t)es to the

o,tco-es so that the (,-bers P(M )a(* P(M ′)are the sa-e a(* best -atch what

we obser4e whe( act,al phys)cal e7per)-e(ts are perfor-e* w)th co)(s that

see- to be fa)r. Act,al e7per)e(ce s,ggests that the o,tco-es )( S′ are e<,ally

l)kely so we ass)g( to each probab)l)ty 1 a(* the(

P(M ′)=P(hh)+P(tt)=1/4+1/4=1/2

S)-)larly fro- e7per)e(ce appropr)ate cho)ces for the o,tco-es )( % are:

P(2h)=1/4 P(2t)=1/4 P(d)=1/2

wh)ch g)4e the sa-e +(al a(swer

P(M )=P(2h)+P(2t)=1/4+1/4=1/2

The previous three examples illustrate how probabilities can be computed simply by counting

when the sample space consists of a finite number of e"ually likely outcomes. ,n some situations

the individual outcomes of any sample space that represents the experiment are unavoidably






une"ually likely$ in which case probabilities cannot be computed merely by counting$ but the

computational formula given in the definition of the probability of an event must be used.

+A>2!+ 5

&he break*ow( of the st,*e(t bo*y )( a local h)gh school accor*)(g to race a(*

eth()c)ty )s 01G wh)te 2G black 11G @)spa()c G As)a( a(* 0G for all others.

A st,*e(t )s ra(*o-ly selecte* fro- th)s h)gh school. &o select ra(*o-ly

-ea(s that e4ery st,*e(t has the sa-e cha(ce of be)(g selecte*.5 )(* the

probab)l)t)es of the follow)(g e4e(ts:

a -: the st,*e(t )s black

a. : the st,*e(t )s -)(or)ty that )s (ot wh)te5

b. ': the st,*e(t )s (ot black.

Sol,t)o(:

&he e7per)-e(t )s the act)o( of ra(*o-ly select)(g a st,*e(t fro- the st,*e(t

pop,lat)o( of the h)gh school. A( ob4)o,s sa-ple space )s S={w,b,h,a,o}.S)(ce 01G

of the st,*e(ts are wh)te a(* all st,*e(ts ha4e the sa-e cha(ce of be)(g

selecte* P(w)=0.51 a(* s)-)larly for the other o,tco-es. &h)s )(for-at)o( )s

s,--ar)Ve* )( the follow)(g table:

EKAPLE 9

&he st,*e(t bo*y )( the h)gh school co(s)*ere* )( Note 3.16 QE7a-ple 6Q -ay be

broke( *ow( )(to te( categor)es as follows: 20G wh)te -ale 2G wh)te fe-ale

12G black -ale 10G black fe-ale G @)spa()c -ale 0G @)spa()c fe-ale 3G






As)a( -ale 3G As)a( fe-ale 1G -ale of other -)(or)t)es co-b)(e* a(* G

fe-ale of other -)(or)t)es co-b)(e*. A st,*e(t )s ra(*o-ly selecte* fro- th)s

h)gh school. )(* the probab)l)t)es of the follow)(g e4e(ts:

a. -: the st,*e(t )s black

b. MF: the st,*e(t )s -)(or)ty fe-ale

c. FN : the st,*e(t )s fe-ale a(* )s (ot black.

Sol,t)o(:

Now the sa-ple space )s S={wm ,bm ,hm ,am ,om ,wf,bf,hf,af,of}. &he )(for-at)o( g)4e( )(

the e7a-ple ca( be s,--ar)Ve* )( the follow)(g table calle* a t&o+&a/

contingenc/ table:

:ender

/ace +thnicity

hite 'lack =ispanic Asian 1thers

ale ;.20 ;.12 ;.; ;.;3 ;.;1

e-ale ;.2 ;.10 ;.;0 ;.;3 ;.;

a. S)(ce B={bm ,bf}@ P(B)=P(bm )+P(bf)=0.12+0.15=0.27.

b. S)(ce MF={bf,hf,af,of}@

P(M )=P(bf)+P(hf)+P(af)+P(of)=0.15+0.05+0.03+0.04=0.27

c. S)(ce FN ={wf,hf,af,of}@

P(FN )=P(wf)+P(hf)+P(af)+P(of)=0.26+0.05+0.03+0.04=0.38

*+, TA*+AA,S

• &he sa-ple space of a ra(*o- e7per)-e(t )s the collect)o( of all poss)bleo,tco-es.

• A( e4e(t assoc)ate* w)th a ra(*o- e7per)-e(t )s a s,bset of the sa-ple space.

• &he probab)l)ty of a(y o,tco-e )s a (,-ber betwee( ; a(* 1. &he probab)l)t)es of

all the o,tco-es a** ,p to 1.






• &he probab)l)ty of a(y e4e(t 0 )s the s,- of the probab)l)t)es of the o,tco-es

)( 0.

EKER%SES

#AS%

1 A bo7 co(ta)(s 1; wh)te a(* 1; black -arbles. %o(str,ct a sa-ple space for the

e7per)-e(t of ra(*o-ly *raw)(g o,t w)th replace-e(t two -arbles )( s,ccess)o( a(*

(ot)(g the color each t)-e. &o *raw w)th replace-e(t -ea(s that the +rst -arble )s

p,t back before the seco(* -arble )s *raw(.5

2 A bo7 co(ta)(s 1 wh)te a(* 1 black -arbles. %o(str,ct a sa-ple space for the

e7per)-e(t of ra(*o-ly *raw)(g o,t w)th replace-e(t three -arbles )( s,ccess)o(

a(* (ot)(g the color each t)-e. &o *raw w)th replace-e(t -ea(s that each -arble )s

p,t back before the (e7t -arble )s *raw(.5

3 A bo7 co(ta)(s 6 re* 6 yellow a(* 6 gree( -arbles. %o(str,ct a sa-ple space for the

e7per)-e(t of ra(*o-ly *raw)(g o,t w)th replace-e(t two -arbles )( s,ccess)o( a(*

(ot)(g the color each t)-e.

A bo7 co(ta)(s re* yellow a(* gree( -arbles. %o(str,ct a sa-ple space for the

e7per)-e(t of ra(*o-ly *raw)(g o,t w)th replace-e(t three -arbles )( s,ccess)o(

a(* (ot)(g the color each t)-e.

0 ( the s)t,at)o( of E7erc)se 1 l)st the o,tco-es that co-pr)se each of the follow)(g

e4e(ts.

a At least o(e -arble of each color )s *raw(.






b No wh)te -arble )s *raw(.

( the s)t,at)o( of E7erc)se 2 l)st the o,tco-es that co-pr)se each of the follow)(g

e4e(ts.

a At least o(e -arble of each color )s *raw(.

b No wh)te -arble )s *raw(.

c ore black tha( wh)te -arbles are *raw(.

( the s)t,at)o( of E7erc)se 3 l)st the o,tco-es that co-pr)se each of the follow)(g

e4e(ts.

a No yellow -arble )s *raw(.

b &he two -arbles *raw( ha4e the sa-e color.

c At least o(e -arble of each color )s *raw(.

6 ( the s)t,at)o( of E7erc)se l)st the o,tco-es that co-pr)se each of the follow)(g

e4e(ts.

a No yellow -arble )s *raw(.

b &he three -arbles *raw( ha4e the sa-e color.

c At least o(e -arble of each color )s *raw(.

9 Ass,-)(g that each o,tco-e )s e<,ally l)kely +(* the probab)l)ty of each e4e(t )(

E7erc)se 0.

1; Ass,-)(g that each o,tco-e )s e<,ally l)kely +(* the probab)l)ty of each e4e(t )(

E7erc)se .


E7erc)se .


E7erc)se 6.

13 A sa-ple space )s S={a,b,c,d,e}.*e(t)fy two e4e(ts

as U={a,b,d}a(* V={b,c,d}.S,ppose P(a)a(* P(b)are each ;.2 a(* P(c)a(* P(d)are each ;.1.

a eter-)(e what P(e)-,st be.






b )(* P(U).

c )(* P(V).

1 A sa-ple space )s S={u,v,w,x}.*e(t)fy two e4e(ts

as A={v,w}a(* B={u,w,x}.S,ppose P(u)=0.22@ P(w)=0.36 a(* P(x)=0.27.

a eter-)(e what P(v)-,st be.

b )(* P(A).

c )(* P(B).

APPL%A&"NS

1 &he sa-ple space that *escr)bes all three=ch)l* fa-)l)es accor*)(g to the ge(*ers of the

ch)l*re( w)th respect to b)rth or*er was co(str,cte* )( Note 3.9 QE7a-ple Q. *e(t)fy

the o,tco-es that co-pr)se each of the follow)(g e4e(ts )( the e7per)-e(t of select)(g

a three=ch)l* fa-)ly at ra(*o-.






a At least o(e ch)l* )s a g)rl.

b At -ost o(e ch)l* )s a g)rl.

c All of the ch)l*re( are g)rls.

* E7actly two of the ch)l*re( are g)rls.

e &he +rst bor( )s a g)rl.

16 &he sa-ple space that *escr)bes three tosses of a co)( )s the sa-e as the o(e

co(str,cte* )( Note 3.9 QE7a-ple Q w)th boy replace* by hea*s a(* g)rl replace*

by ta)ls. *e(t)fy the o,tco-es that co-pr)se each of the follow)(g e4e(ts )( the

e7per)-e(t of toss)(g a co)( three t)-es.

a &he co)( la(*s hea*s -ore ofte( tha( ta)ls.

b &he co)( la(*s hea*s the sa-e (,-ber of t)-es as )t la(*s ta)ls.

c &he co)( la(*s hea*s at least tw)ce.

* &he co)( la(*s hea*s o( the last toss.

19 Ass,-)(g that the o,tco-es are e<,ally l)kely +(* the probab)l)ty of each e4e(t )(

E7erc)se 1.

2; Ass,-)(g that the o,tco-es are e<,ally l)kely +(* the probab)l)ty of each e4e(t )(

E7erc)se 16.

A((&T&1NA! ++/C&S+S

21 &he follow)(g two=way co(t)(ge(cy table g)4es the break*ow( of the pop,lat)o( )( a

part)c,lar locale accor*)(g to age a(* tobacco ,sage:

Age

Tobacco Use

Smoker Non-smoker

Under 30 0.05 0.20

Over 30 0.20 0.55

A perso( )s selecte* at ra(*o-. )(* the probab)l)ty of each of the follow)(g e4e(ts.

a &he perso( )s a s-oker.b &he perso( )s ,(*er 3;.

c &he perso( )s a s-oker who )s ,(*er 3;.


part)c,lar locale accor*)(g to party a?l)at)o( 0 - , or 'one5 a(* op)()o( o( a bo(*

)ss,e:






Affiliation

Opinion

Favors Opposes Undecided

A 0.12 0.09 0.07

B 0.16 0.12 0.14C 0.04 0.03 0.06

None 0.08 0.06 0.03

A perso( )s selecte* at ra(*o-. )(* the probab)l)ty of each of the follow)(g e4e(ts.

a &he perso( )s a?l)ate* w)th party -.

b &he perso( )s a?l)ate* w)th so-e party.

c &he perso( )s )( fa4or of the bo(* )ss,e.

* &he perso( has (o party a?l)at)o( a(* )s ,(*ec)*e* abo,t the bo(* )ss,e.

23 &he follow)(g two=way co(t)(ge(cy table g)4es the break*ow( of the pop,lat)o( of

-arr)e* or pre4)o,sly -arr)e* wo-e( beyo(* ch)l*=bear)(g age )( a part)c,lar locale

accor*)(g to age at +rst -arr)age a(* (,-ber of ch)l*re(:

Age

Number of Children

0 or ! " or #ore

Under 20 0.02 0.14 0.08

20–29 0.07 0.37 0.11

30 and above 0.10 0.10 0.01

A wo-a( )s selecte* at ra(*o-. )(* the probab)l)ty of each of the follow)(g e4e(ts.

a &he wo-a( was )( her twe(t)es at her +rst -arr)age.

b &he wo-a( was 2; or ol*er at her +rst -arr)age.

c &he wo-a( ha* (o ch)l*re(.

* &he wo-a( was )( her twe(t)es at her +rst -arr)age a(* ha* at least three

ch)l*re(.

e


a*,lts )( a part)c,lar locale accor*)(g to h)ghest le4el of e*,cat)o( a(* whether or (ot

the )(*)4)*,al reg,larly takes *)etary s,pple-e(ts:






$ducation

Use of Supplements

Takes %oes Not Take

No High !hoo" #i$"o%a 0.04 0.06

High !hoo" #i$"o%a 0.06 0.44Undergrad&a'e #egree 0.09 0.28

(rad&a'e #egree 0.01 0.02

A( a*,lt )s selecte* at ra(*o-. )(* the probab)l)ty of each of the follow)(g e4e(ts.

a &he perso( has a h)gh school *)plo-a a(* takes *)etary s,pple-e(ts

reg,larly.

b &he perso( has a( ,(*ergra*,ate *egree a(* takes *)etary s,pple-e(ts

reg,larly.

c &he perso( takes *)etary s,pple-e(ts reg,larly.

* &he perso( *oes (ot take *)etary s,pple-e(ts reg,larly.

LAR!E A&A SE& EKER%SES

20 Large ata Sets a(* A recor* the res,lts of 0;; tosses of a co)(. )(* the relat)4e

fre<,e(cy of each o,tco-e 1 2 3 0 a(* . oes the co)( appear to be bala(ce*

or fa)rC

http://www..7ls

http://www.A.7ls

2 Large ata Sets A a(* # recor* res,lts of a ra(*o- s,r4ey of 2;; 4oters )( each of

two reg)o(s )( wh)ch they were aske* to e7press whether they prefer %a(*)*ate 0for a

U.S. Se(ate seat or prefer so-e other ca(*)*ate.

a )(* the probab)l)ty that a ra(*o-ly selecte* 4oter a-o(g these ;; prefers

%a(*)*ate 0.

b )(* the probab)l)ty that a ra(*o-ly selecte* 4oter a-o(g the 2;; who l)4e )(

Reg)o( 1 prefers %a(*)*ate 0 separately recor*e* )( Large ata Set A5.






c )(* the probab)l)ty that a ra(*o-ly selecte* 4oter a-o(g the 2;; who l)4e )(

Reg)o( 2 prefers %a(*)*ate 0 separately recor*e* )( Large ata Set #5.

http://www..7ls

http://www.A.7ls

http://www.#.7ls











3.0 Complements@ &ntersections@ and Bnions

LEARNN! "#$E%&'ES

1 &o lear( how so-e e4e(ts are (at,rally e7press)ble )( ter-s of other e4e(ts.

2 &o lear( how to ,se spec)al for-,las for the probab)l)ty of a( e4e(t that )s

e7presse* )( ter-s of o(e or -ore other e4e(ts.






+ome events can be naturally expressed in terms of other$ sometimes simpler$ events.

Complements

e+()t)o(

The complement of an event A in a sample space % $ denoted Ac$ is the collection of all outcomes

in % that are not elements of the set A. >t corresponds to negating any description in words of the

event A.

EKAPLE 1;

&wo e4e(ts co((ecte* w)th the e7per)-e(t of roll)(g a s)(gle *)e are (: the (,-ber

rolle* )s e4e( a(* ) : the (,-ber rolle* )s greater tha( two. )(* the co-ple-e(t of each.

Sol,t)o(:

( the sa-ple space S={1,2,3,4,5,6}the correspo(*)(g sets of o,tco-es

are E={2,4,6}and T={3,4,5,6}. &he co-ple-e(ts are Ec={1,3,5}a(* Tc={1,2}.

( wor*s the co-ple-e(ts are *escr)be* by the (,-ber rolle* )s (ot e4e( a(* the

(,-ber rolle* )s (ot greater tha( two. "f co,rse eas)er *escr)pt)o(s wo,l* be the

(,-ber rolle* )s o** a(* the (,-ber rolle* )s less tha( three.

,f there is a ;5B chance of rain tomorrow$ what is the probability of fair weather= The obvious

answer$ 65B$ is an instance of the following general rule.

2robability /ule or Complements

P Ac)=1−P(A)

This formula is particularly useful when finding the probability of an event

EKAPLE 11

)(* the probab)l)ty that at least o(e hea*s w)ll appear )( +4e tosses of a fa)r co)(.






Sol,t)o(:

*e(t)fy o,tco-es by l)sts of +4e !s a(* t s s,ch as tthtta(* hhttt.Altho,gh )t )s

te*)o,s to l)st the- all )t )s (ot *)?c,lt to co,(t the-. &h)(k of ,s)(g a tree

*)agra- to *o so. &here are two cho)ces for the +rst toss. or each of these there

are two cho)ces for the seco(* toss he(ce 2×2=4o,tco-es for two tosses. or

each of these fo,r o,tco-es there are two poss)b)l)t)es for the th)r* toss

he(ce 4×2=8o,tco-es for three tosses. S)-)larly there are 8×2=16o,tco-es for

fo,r tosses a(* +(ally 16×2=32o,tco-es for +4e tosses.

Let *e(ote the e4e(t at least o(e hea*s. &here are -a(y ways to obta)( at least

o(e hea*s b,t o(ly o(e way to fa)l to *o so: all ta)ls. &h,s altho,gh )t )s *)?c,lt to

l)st all the o,tco-es that for- )t )s easy to wr)te Oc={ttttt}.S)(ce there are 32

e<,ally l)kely o,tco-es each has probab)l)ty 1/32 so P(Oc)=1/32

he(ce P(O)=1−1/32≈0.97or abo,t a 9G cha(ce.

&ntersection o +vents

(e)nitionThe intersection of events A and 9$ denoted A ) 9$ is the collection of all outcomes that are elements

of both of the sets A and 9. >t corresponds to combining descriptions of the two events using the word

?and.@

To say that the event A ) 9 occurred means that on a particular trial of the experiment

both A and 9 occurred. % visual representation of the intersection of events A and 9 in a sample

space % is given in /igure 3.6 0The ,ntersection of vents 0. The intersection corresponds to the

shaded lens-shaped region that lies within both ovals.






!igure *. The >ntersection of ;vents Aand 9











e+()t)o(

;vents A and 9 are mutually e)clusive if they have no elements in common.

/or A and 9 to have no outcomes in common means precisely that it is impossible for both A and 9 to

occur on a single trial of the random experiment. This gives the following rule.

Probab)l)ty R,le for ,t,ally E7cl,s)4e E4e(ts

vents A and 9 are mutually exclusive if and only if

P(A∩B)=0

%ny event A and its complement Ac are mutually exclusive$ but A and 9 can be mutually exclusive without

being complements.

EKAPLE 1






( the e7per)-e(t of roll)(g a s)(gle *)e +(* three cho)ces for a( e4e(t 0 so that the

e4e(ts 0 a(* (: the (,-ber rolle* )s e4e( are -,t,ally e7cl,s)4e.

Sol,t)o(:

S)(ce E={2,4,6}a(* we wa(t 0 to ha4e (o ele-e(ts )( co--o( w)th ( a(y e4e(t that

*oes (ot co(ta)( a(y e4e( (,-ber w)ll *o. &hree cho)ces are M130 the

co-ple-e(t (c the o**s5 M13 a(* M0.

Bnion o +vents

e+()t)o(

The union of events A and 9$ denoted A Z 9$ is the collection of all outcomes that are elements of one

or the other of the sets A and 9$ or of both of them. >t corresponds to combining descriptions of the two

events using the word ?or.@

To say that the event A Z 9 occurred means that on a particular trial of the experiment

either A or 9 occurred *or both did). % visual representation of the union of events A and 9 in a sample

space % is given in /igure 3.8 0The <nion of vents 0. The union corresponds to the shaded region.

!igure *. The nion of ;vents A and 9






EKAPLE 10

( the e7per)-e(t of roll)(g a s)(gle *)e +(* the ,()o( of the e4e(ts (: the (,-ber

rolle* )s e4e( a(* ) : the (,-ber rolle* )s greater tha( two.

Sol,t)o(:

S)(ce the o,tco-es that are )( e)ther E={2,4,6}or T={3,4,5,6}or both5 are 2 3 0 a(*

EZT={2,3,4,5,6}.Note that a( o,tco-e s,ch as that )s )( both sets )s st)ll l)ste* o(ly

o(ce altho,gh str)ctly speak)(g )t )s (ot )(correct to l)st )t tw)ce5.

( wor*s the ,()o( )s *escr)be* by the (,-ber rolle* )s e4e( or )s greater tha( two.

E4ery (,-ber betwee( o(e a(* s)7 e7cept the (,-ber o(e )s e)ther e4e( or )s greater

tha( two correspo(*)(g to ( Z ) g)4e( abo4e.

EKAPLE 1

A two=ch)l* fa-)ly )s selecte* at ra(*o-. Let - *e(ote the e4e(t that at least o(e

ch)l* )s a boy let D *e(ote the e4e(t that the ge(*ers of the two ch)l*re( *)8er a(*

let *e(ote the e4e(t that the ge(*ers of the two ch)l*re( -atch. )(* - Z D a(* B

M .

Sol,t)o(:






A sa-ple space for th)s e7per)-e(t )s S={bb,bg,gb,gg} where the +rst letter *e(otes

the ge(*er of the +rstbor( ch)l* a(* the seco(* letter *e(otes the ge(*er of the

seco(* ch)l*. &he e4e(ts - D a(* are

B={bb,bg,gb} D={bg,gb} M ={bb,gg}

Each o,tco-e )( D )s alrea*y )( - so the o,tco-es that are )( at least o(e or the

other of the sets - a(* D )s >,st the set - )tself: BD={bb,bg,gb}=B.

E4ery o,tco-e )( the whole sa-ple space % )s )( at least o(e or the other of the

sets - a(* so B M ={bb,bg,gb,gg}=S.

The following ,dditive -ule of &robability is a useful formula for calculating the probability

ofAZB.

Additive /ule o 2robability

P(A B)=P(A)+P(B)−P(A∩B)

The next example$ in which we compute the probability of a union both by counting and by using the

formula$ shows why the last term in the formula is needed.











EKAPLE 16

A t,tor)(g ser4)ce spec)al)Ves )( prepar)(g a*,lts for h)gh school e<,)4ale(ce tests.

A-o(g all the st,*e(ts seek)(g help fro- the ser4)ce 3G (ee* help )( -athe-at)cs

3G (ee* help )( E(gl)sh a(* 2G (ee* help )( both -athe-at)cs a(* E(gl)sh. Bhat

)s the perce(tage of st,*e(ts who (ee* help )( e)ther -athe-at)cs or E(gl)shC

Sol,t)o(:

-ag)(e select)(g a st,*e(t at ra(*o- that )s )( s,ch a way that e4ery st,*e(t has

the sa-e cha(ce of be)(g selecte*. Let *e(ote the e4e(t the st,*e(t (ee*s help

)( -athe-at)cs a(* let ( *e(ote the e4e(t the st,*e(t (ee*s help )( E(gl)sh. &he

)(for-at)o( g)4e( )s that P(M )=0.63@ P(E)=0.34@ a(* P(M ∩E)=0.27. &he A**)t)4e R,le of

Probab)l)ty g)4es






P(M E)=P(M )+P(E)−P(M ∩E)=0.63+0.34−0.27=0.70

9ote how the nave reasoning that if ;3B need help in mathematics and 36B need help in nglish

then ;3 plus 36 or F@B need help in one or the other gives a number that is too large. The percentage

that need help in both sub#ects must be subtracted off$ else the people needing help in both are

counted twice$ once for needing help in mathematics and once again for needing help in nglish. The

simple sum of the probabilities would work if the events in "uestion were mutually exclusive$ for

thenP(A∩B)is 'ero$ and makes no difference.
















*+, TA*+AA,

•

&he probab)l)ty of a( e4e(t that )s a co-ple-e(t or ,()o( of e4e(ts of k(ow(probab)l)ty ca( be co-p,te* ,s)(g for-,las.


























R S T

M 0.09 0.25 0.19

N 0.31 0.16 0.00

a P(R)@ P(S)@ P(R∩S).

b P(M )@ P(N )@ P(M ∩N ).

c P(RS).

d P(Rc).

e eter-)(e whether or (ot the e4e(ts ' a(* % are -,t,ally e7cl,s)4eD the e4e(ts ' a(* ) .A22!&CAT&1NS

11 ake a state-e(t )( or*)(ary E(gl)sh that *escr)bes the co-ple-e(t of each e4e(t *o

(ot s)-ply )(sert the wor* (ot5.

a ( the roll of a *)e: +4e or -ore.

b ( a roll of a *)e: a( e4e( (,-ber.

c ( two tosses of a co)(: at least o(e hea*s.

* ( the ra(*o- select)o( of a college st,*e(t: Not a fresh-a(.

12 ake a state-e(t )( or*)(ary E(gl)sh that *escr)bes the co-ple-e(t of each e4e(t *o

(ot s)-ply )(sert the wor* (ot5.

a ( the roll of a *)e: two or less.

b ( the roll of a *)e: o(e three or fo,r.

c ( two tosses of a co)(: at -ost o(e hea*s.

* ( the ra(*o- select)o( of a college st,*e(t: Ne)ther a fresh-a( (or a se()or.

%3 &he sa-ple space that *escr)bes all three=ch)l* fa-)l)es accor*)(g to the ge(*ers of the

ch)l*re( w)th respect to b)rth or*er )s

S={bbb,bbg,bgb,bgg,gbb,gbg,ggb,ggg}.

or each of the follow)(g e4e(ts )( the e7per)-e(t of select)(g a three=ch)l* fa-)ly at

ra(*o- state the co-ple-e(t of the e4e(t )( the s)-plest poss)ble ter-s the( +(* the

o,tco-es that co-pr)se the e4e(t a(* )ts co-ple-e(t.






a At least o(e ch)l* )s a g)rl.

b At -ost o(e ch)l* )s a g)rl.

c All of the ch)l*re( are g)rls.

* E7actly two of the ch)l*re( are g)rls.

e &he +rst bor( )s a g)rl.

1 &he sa-ple space that *escr)bes the two=way class)+cat)o( of c)t)Ve(s accor*)(g to

ge(*er a(* op)()o( o( a pol)t)cal )ss,e )s

S={mf,ma,mn,ff,fa,fn},

where the +rst letter *e(otes ge(*er m: -ale f : fe-ale5 a(* the seco(* op)()o( f :

for a: aga)(st n: (e,tral5. or each of the follow)(g e4e(ts )( the e7per)-e(t of

select)(g a c)t)Ve( at ra(*o- state the co-ple-e(t of the e4e(t )( the s)-plest

poss)ble ter-s the( +(* the o,tco-es that co-pr)se the e4e(t a(* )ts co-ple-e(t.

a &he perso( )s -ale.

b &he perso( )s (ot )( fa4or.

c &he perso( )s e)ther -ale or )( fa4or.

* &he perso( )s fe-ale a(* (e,tral.

10 A to,r)st who speaks E(gl)sh a(* !er-a( b,t (o other la(g,age 4)s)ts a reg)o( of

Slo4e()a. f 30G of the res)*e(ts speak E(gl)sh 10G speak !er-a( a(* 3G speak both

E(gl)sh a(* !er-a( what )s the probab)l)ty that the to,r)st w)ll be able to talk w)th a

ra(*o-ly e(co,(tere* res)*e(t of the reg)o(C

1 ( a certa)( co,(try 3G of all a,to-ob)les ha4e a)rbags 2G ha4e a(t)=lock brakes

a(* 13G ha4e both. Bhat )s the probab)l)ty that a ra(*o-ly selecte* 4eh)cle w)ll ha4e

both a)rbags a(* a(t)=lock brakesC

1 A -a(,fact,rer e7a-)(es )ts recor*s o4er the last year o( a co-po(e(t part

rece)4e* fro- o,ts)*e s,ppl)ers. &he break*ow( o( so,rce s,ppl)er 0 s,ppl)er -5

a(* <,al)ty : h)gh U: ,sable D: *efect)4e5 )s show( )( the two=way co(t)(ge(cy

table.






H U D

A 0.6937 0.0049 0.0014

B 0.2982 0.0009 0.0009

&he recor* of a part )s selecte* at ra(*o-. )(* the probab)l)ty of each of the follow)(g

e4e(ts.

a &he part was *efect)4e.

b &he part was e)ther of h)gh <,al)ty or was at least ,sable )( two ways: )5 by a**)(g

(,-bers )( the table a(* ))5 ,s)(g the a(swer to a5 a(* the Probab)l)ty R,le for

%o-ple-e(ts.

c &he part was *efect)4e a(* ca-e fro- s,ppl)er -.

* &he part was *efect)4e or ca-e fro- s,ppl)er - )( two ways: by +(*)(g the cells )( the

table that correspo(* to th)s e4e(t a(* a**)(g the)r probab)l)t)es a(* ))5 ,s)(g the

A**)t)4e R,le of Probab)l)ty.

16. (*)4)*,als w)th a part)c,lar -e*)cal co(*)t)o( were class)+e* accor*)(g to the prese(ce

) 5 or abse(ce '5 of a pote(t)al to7)( )( the)r bloo* a(* the o(set of the co(*)t)o( (:

early : -)*ra(ge L: late5. &he break*ow( accor*)(g to th)s class)+cat)o( )s show( )( the

two=way co(t)(ge(cy table.

E M L

T 0.012 0.124 0.013

N 0.170 0.638 0.043

"(e of these )(*)4)*,als )s selecte* at ra(*o-. )(* the probab)l)ty of each of the

follow)(g e4e(ts.

a &he perso( e7per)e(ce* early o(set of the co(*)t)o(.

b &he o(set of the co(*)t)o( was e)ther -)*ra(ge or late )( two ways: )5 by

a**)(g (,-bers )( the table a(* ))5 ,s)(g the a(swer to a5 a(* the

Probab)l)ty R,le for %o-ple-e(ts.

c &he to7)( )s prese(t )( the perso(Ws bloo*.* &he perso( e7per)e(ce* early o(set of the co(*)t)o( a(* the to7)( )s prese(t

)( the perso(Ws bloo*.

e &he perso( e7per)e(ce* early o(set of the co(*)t)o( or the to7)( )s prese(t )(

the perso(Ws bloo* )( two ways: )5 by +(*)(g the cells )( the table that

correspo(* to th)s e4e(t a(* a**)(g the)r probab)l)t)es a(* ))5 ,s)(g the

A**)t)4e R,le of Probab)l)ty.






19 &he break*ow( of the st,*e(ts e(rolle* )( a ,()4ers)ty co,rse by class F :

fresh-a( So: sopho-ore : >,()or Se: se()or5 a(* aca*e-)c -a>or %: sc)e(ce

-athe-at)cs or e(g)(eer)(g L: l)beral arts : other5 )s show( )( the two=way

class)+cat)o( table.

#a&or

Class

F So J Se

S 92 42 20 13

L 368 167 80 53

O 460 209 100 67

A st,*e(t e(rolle* )( the co,rse )s selecte* at ra(*o-. A*>o)( the row a(* col,-(

totals to the table a(* ,se the e7pa(*e* table to +(* the probab)l)ty of each of the

follow)(g e4e(ts.

a &he st,*e(t )s a fresh-a(.

b &he st,*e(t )s a l)beral arts -a>or.

c &he st,*e(t )s a fresh-a( l)beral arts -a>or.

* &he st,*e(t )s e)ther a fresh-a( or a l)beral arts -a>or.

e &he st,*e(t )s (ot a l)beral arts -a>or.

2; &he table relates the respo(se to a f,(*=ra)s)(g appeal by a college to )ts al,-() to

the (,-ber of years s)(ce gra*,at)o(.

'esponse

(ears Since )raduation

0*+ ,*!0 !*"+ Over "+

*o+i'ive 120 440 210 90

None 1380 3560 3290 910

A( al,-(,s )s selecte* at ra(*o-. A*>o)( the row a(* col,-( totals to the table a(*

,se the e7pa(*e* table to +(* the probab)l)ty of each of the follow)(g e4e(ts.

a &he al,-(,s respo(*e*.

b &he al,-(,s *)* (ot respo(*.

c &he al,-(,s gra*,ate* at least 21 years ago.

* &he al,-(,s gra*,ate* at least 21 years ago a(* respo(*e*.

A((&T&1NA! ++/C&S+S

21 &he sa-ple space for toss)(g three co)(s )s

S={hhh,hht,hth,htt,thh,tht,tth,ttt}

a L)st the o,tco-es that correspo(* to the state-e(t All the co)(s are hea*s.






b L)st the o,tco-es that correspo(* to the state-e(t Not all the co)(s are hea*s.

c L)st the o,tco-es that correspo(* to the state-e(t All the co)(s are (ot hea*s.











3.3 Conditional 2robability and &ndependent +vents

LEARNN! "#$E%&'ES

1 &o lear( the co(cept of a co(*)t)o(al probab)l)ty a(* how to co-p,te )t.

2 &o lear( the co(cept of )(*epe(*e(ce of e4e(ts a(* how to apply )t.






Conditional 2robability

+uppose a fair die has been rolled and you are asked to give the probability that it was a five. There are six

e"ually likely outcomes$ so your answer is 1I;. 4ut suppose that before you give your answer you are given

the extra information that the number rolled was odd. +ince there are only three odd numbers that arepossible$ one of which is five$ you would certainly revise your estimate of the likelihood that a five was

rolled from 1I; to 1I3. ,n general$ the revised probability that an event A has occurred$ taking into

account the additional information that another event 9 has definitely occurred on this trial of the

experiment$ is called the conditional probability of A given 9 and is denoted byP(A|B).The reasoning

employed in this example can be generali'ed to yield the computational formula in the following

definition.

(e)nition

The conditional probability of A given 9$ denoted P(A|B) is the probability that event A has

occurred in a trial of a random experiment for which it is known that event 9 has definitely occurred. >t

may be computed by means of the following formula8

Cule for 7onditional robability

P(A|B)=P(A∩B)/P(B)

+A>2!+ 0

A fa)r *)e )s rolle*.

a. )(* the probab)l)ty that the (,-ber rolle* )s a +4e g)4e( that )t )s o**.

b. )(* the probab)l)ty that the (,-ber rolle* )s o** g)4e( that )t )s a +4e.

Sol,t)o(:

&he sa-ple space for th)s e7per)-e(t )s the set S={1,2,3,4,5,6}co(s)st)(g of s)7 e<,ally

l)kely o,tco-es. Let F *e(ote the e4e(t a +4e )s rolle* a(* let *e(ote the e4e(t

a( o** (,-ber )s rolle* so that

F={5} and O={1,3,5}











Rust as we did not need the computational formula in this example$ we do not need it when the

information is presented in a two-way classification table$ as in the next example.

+A>2!+ 0%

( a sa-ple of 9;2 )(*)4)*,als ,(*er ; who were or ha* pre4)o,sly bee( -arr)e*

each perso( was class)+e* accor*)(g to ge(*er a(* age at +rst -arr)age. &he res,lts

are s,--ar)Ve* )( the follow)(g two=way class)+cat)o( table where the -ea()(g of

the labels )s:

• : -ale

• F : fe-ale

• (: a tee(ager whe( +rst -arr)e*

• 3 : )( o(eWs twe(t)es whe( +rst -arr)e*

• : )( o(eWs th)rt)es whe( +rst -arr)e*

E W H Total

M 43 293 114 450

F 82 299 71 452

,o'a" 125 592 185 902

&he (,-bers )( the +rst row -ea( that 3 people )( the sa-ple were -e( who were

+rst -arr)e* )( the)r tee(s 293 were -e( who were +rst -arr)e* )( the)r twe(t)es

11 -e( who were +rst -arr)e* )( the)r th)rt)es a(* a total of 0; people )( the

sa-ple were -e(. S)-)larly for the (,-bers )( the seco(* row. &he (,-bers )( the

last row -ea( that )rrespect)4e of ge(*er 120 people )( the sa-ple were -arr)e* )(






the)r tee(s 092 )( the)r twe(t)es 160 )( the)r th)rt)es a(* that there were 9;2 people

)( the sa-ple )( all. S,ppose that the proport)o(s )( the sa-ple acc,rately re[ect

those )( the pop,lat)o( of all )(*)4)*,als )( the pop,lat)o( who are ,(*er ; a(* who

are or ha4e pre4)o,sly bee( -arr)e*. S,ppose s,ch a perso( )s selecte* at ra(*o-.

a )(* the probab)l)ty that the )(*)4)*,al selecte* was a tee(ager at +rst

-arr)age.

b )(* the probab)l)ty that the )(*)4)*,al selecte* was a tee(ager at +rst

-arr)age g)4e( that the perso( )s -ale.

Sol,t)o(:

t )s (at,ral to let ( also *e(ote the e4e(t that the perso( selecte* was a tee(ager at

+rst -arr)age a(* to let *e(ote the e4e(t that the perso( selecte* )s -ale.

a. Accor*)(g to the table the proport)o( of )(*)4)*,als )( the sa-ple who were )(

the)r tee(s at the)r +rst -arr)age )s 120/9;2. &h)s )s the relat)4e fre<,e(cy of s,ch people

)( the pop,lat)o( he(ce P(E)=125/902≈0.139or abo,t 1G.

S)(ce )t )s k(ow( that the perso( selecte* )s -ale all the fe-ales -ay be

re-o4e* fro- co(s)*erat)o( so that o(ly the row )( the table correspo(*)(g

to -e( )( the sa-ple appl)es:

E W H Total

M 43 293 114 450

&he proport)o( of -ales )( the sa-ple who were )( the)r tee(s at the)r +rst -arr)age

)s 3/0;. &h)s )s the relat)4e fre<,e(cy of s,ch people )( the pop,lat)o( of -ales

he(ce P(E|M )=43/450≈0.096or abo,t 1;G.

,n the next example$ the computational formula in the definition must be used.

+A>2!+ 00






S,ppose that )( a( a*,lt pop,lat)o( the proport)o( of people who are both

o4erwe)ght a(* s,8er hyperte(s)o( )s ;.;9D the proport)o( of people who are (ot

o4erwe)ght b,t s,8er hyperte(s)o( )s ;.11D the proport)o( of people who are

o4erwe)ght b,t *o (ot s,8er hyperte(s)o( )s ;.;2D a(* the proport)o( of people

who are (e)ther o4erwe)ght (or s,8er hyperte(s)o( )s ;.6. A( a*,lt )s ra(*o-ly

selecte* fro- th)s pop,lat)o(.

a. )(* the probab)l)ty that the perso( selecte* s,8ers hyperte(s)o( g)4e( that he )s

o4erwe)ght.

b. )(* the probab)l)ty that the selecte* perso( s,8ers hyperte(s)o( g)4e( that he )s (ot

o4erwe)ght.

c. %o-pare the two probab)l)t)es >,st fo,(* to g)4e a( a(swer to the <,est)o( as to

whether o4erwe)ght people te(* to s,8er fro- hyperte(s)o(.

Sol,t)o(:

Let *e(ote the e4e(t the perso( selecte* s,8ers hyperte(s)o(. Let *e(ote

the e4e(t the perso( selecte* )s o4erwe)ght. &he probab)l)ty )(for-at)o( g)4e(

)( the proble- -ay be orga()Ve* )(to the follow)(g co(t)(ge(cy table:

O Oc

H 0.09 0.11

H c 0.02 0.78






&ndependent +vents

%lthough typically we expect the conditional probabilityP(A|B)to be different from the

probabilityP(A)of A$ it does not have to be different from P(A). (henP(A|B)=P(A) the occurrence

of 9 has no effect on the likelihood of A. (hether or not the event A has occurred is independent of

the event 9.

<sing algebra it can be shown that the e"ualityP(A|B)=P(A)holds if and only if the e"ualityP(A∩

B)=P(A)DP(B)holds$ which in turn is true if and only if P(B|A)=P(B).This is the basis for the

following definition.

(e)nition






;vents A and 9 are independent if

P(A∩B)=P(A)DP(B)

>f A and 9 are not independent then they are dependent.

The formula in the definition has two practical but exactly opposite uses:

1 ,n a situation in which we can compute all three probabilitiesP(A)P(B) andP(A∩B) it is used to

check whether or not the events A and 9 are independent:

o ,fP(A∩B)=P(A)DP(B) then A and 9 are independent.

o ,fP(A∩B)≠P(A)DP(B) then A and 9 are not independent.

! ,n a situation in which each ofP(A)andP(B)can be computed and it is known that A and 9 are

independent$ then we can computeP(A∩B) by multiplying

togetherP(A)andP(B)/P(A∩B)=P(A)DP(B).

+A>2!+ 03

A s)(gle fa)r *)e )s rolle*. Let A={3}a(* B={1,3,5}.Are 0 a(* - )(*epe(*e(tC

Sol,t)o(:

( th)s e7a-ple we ca( co-p,te all three probab)l)t)es P(A)=1/6@ P(B)=1/2 a(* P(A∩

B)=P({3})=1/6.S)(ce the pro*,ct P(A)DP(B)=(1/6)(1/2)=1/12)s (ot the sa-e (,-ber as P(A∩

B)=1/6 the e4e(ts 0 a(* - are (ot )(*epe(*e(t.

+A>2!+ 07

&he two=way class)+cat)o( of -arr)e* or pre4)o,sly -arr)e* a*,lts ,(*er ; accor*)(g

to ge(*er a(* age at +rst -arr)age )( Note 3.6 QE7a-ple 21Q pro*,ce* the table

E W H Total

M 43 293 114 450






E W H Total

F 82 299 71 452

,o'a" 125 592 185 902

eter-)(e whether or (ot the e4e(ts F : fe-ale a(* (: was a tee(ager at +rst

-arr)age are )(*epe(*e(t.

+A>2!+ 08a(y *)ag(ost)c tests for *etect)(g *)seases *o (ot test for the *)sease *)rectly

b,t for a che-)cal or b)olog)cal pro*,ct of the *)sease he(ce are (ot perfectly

rel)able. &he sensitivit/ of a test )s the probab)l)ty that the test w)ll be pos)t)4e

whe( a*-)()stere* to a perso( who has the *)sease. &he h)gher the se(s)t)4)ty

the greater the *etect)o( rate a(* the lower the false (egat)4e rate.






S,ppose the se(s)t)4)ty of a *)ag(ost)c proce*,re to test whether a perso( has a

part)c,lar *)sease )s 92G. A perso( who act,ally has the *)sease )s teste* for )t

,s)(g th)s proce*,re by two )(*epe(*e(t laborator)es.

a. Bhat )s the probab)l)ty that both test res,lts w)ll be pos)t)4eC

b. Bhat )s the probab)l)ty that at least o(e of the two test res,lts w)ll be pos)t)4eC

Sol,t)o(:

a. Let 01 *e(ote the e4e(t the test by the +rst laboratory )s pos)t)4e a(*

let 02 *e(ote the e4e(t the test by the seco(* laboratory )s pos)t)4e.

S)(ce 01 a(* 02 are )(*epe(*e(t

P(A1 ∩A2)=P(A1)DP(A2)=0.92×0.92=0.8464

b. Us)(g the A**)t)4e R,le for Probab)l)ty a(* the probab)l)ty >,st

co-p,te*

P(A1 A2)=P(A1)+P(A2)−P(A1 ∩A2)=0.92+0.92−0.8464=0.9936

+A>2!+ 09

&he speci4cit/ of a *)ag(ost)c test for a *)sease )s the probab)l)ty that the test w)ll be

(egat)4e whe( a*-)()stere* to a perso( who *oes (ot ha4e the *)sease. &he h)gher

the spec)+c)ty the lower the false pos)t)4e rate.

S,ppose the spec)+c)ty of a *)ag(ost)c proce*,re to test whether a perso( has a

part)c,lar *)sease )s 69G.

a. A perso( who *oes (ot ha4e the *)sease )s teste* for )t ,s)(g th)s proce*,re.

Bhat )s the probab)l)ty that the test res,lt w)ll be pos)t)4eC

b. A perso( who *oes (ot ha4e the *)sease )s teste* for )t by two )(*epe(*e(t

laborator)es ,s)(g th)s proce*,re. Bhat )s the probab)l)ty that both test res,lts w)ll be

pos)t)4eC

Sol,t)o(:






a. Let - *e(ote the e4e(t the test res,lt )s pos)t)4e. &he co-ple-e(t of - )s

that the test res,lt )s (egat)4e a(* has probab)l)ty the spec)+c)ty of the test ;.69.

&h,s

P(B)=1−P(Bc)=1−0.89=0.11.

b. Let -1 *e(ote the e4e(t the test by the +rst laboratory )s pos)t)4e a(*

let -2 *e(ote the e4e(t the test by the seco(* laboratory )s pos)t)4e.

S)(ce -1 a(* -2 are )(*epe(*e(t by part a5 of the e7a-ple

P(B1∩B2)=P(B1)DP(B2)=0.11×0.11=0.0121.

The concept of independence applies to any number of events. /or example$ three events A$ 9$

and < are independent if P(A∩B∩C)=P(A)DP(B)DP(C).9ote carefully that$ as is the case with

#ust two events$ this is not a formula that is always valid$ but holds precisely when the events in

"uestion are independent.











2robabilities on Tree (ia$rams

+ome probability problems are made much simpler when approached using a tree diagram. The next

example illustrates how to place probabilities on a tree diagram and use it to solve a problem.

EKAPLE 26

A >ar co(ta)(s 1; -arbles black a(* 3 wh)te. &wo -arbles are *raw( w)tho,t

replace-e(t wh)ch -ea(s that the +rst o(e )s (ot p,t back before the seco(* o(e )s

*raw(.

a. Bhat )s the probab)l)ty that both -arbles are blackC

b. Bhat )s the probab)l)ty that e7actly o(e -arble )s blackC

c. Bhat )s the probab)l)ty that at least o(e -arble )s blackC

Sol,t)o(:






A tree *)agra- for the s)t,at)o( of *raw)(g o(e -arble after the other w)tho,t

replace-e(t )s show( )( )g,re 3. Q&ree )agra- for raw)(g &wo arblesQ. &he

c)rcle a(* recta(gle w)ll be e7pla)(e* later a(* sho,l* be )g(ore* for (ow.

Figure *.5)ree Diagram for Dra&ing )&o arbles

&he (,-bers o( the two left-ost bra(ches are the probab)l)t)es of gett)(g e)ther

a black -arble o,t of 1; or a wh)te -arble 3 o,t of 1; o( the +rst *raw. &he

(,-ber o( each re-a)()(g bra(ch )s the probab)l)ty of the e4e(t correspo(*)(g to

the (o*e o( the r)ght e(* of the bra(ch occ,rr)(g g)4e( that the e4e(t

correspo(*)(g to the (o*e o( the left e(* of the bra(ch has occ,rre*. &h,s for

the top bra(ch co((ect)(g the two #s )t )s P(B2|B1)@ where -1 *e(otes the e4e(t

the +rst -arble *raw( )s black a(* -2 *e(otes the e4e(t the seco(* -arble

*raw( )s black. S)(ce after *raw)(g a black -arble o,t there are 9 -arbles left

of wh)ch are black th)s probab)l)ty )s /9.

&he (,-ber to the r)ght of each +(al (o*e )s co-p,te* as show( ,s)(g the

pr)(c)ple that )f the for-,la )( the %o(*)t)o(al R,le for Probab)l)ty )s -,lt)pl)e*

by P(B)@ the( the res,lt )s






P(B∩A)=P(B)DP(A|B)

a. &he e4e(t both -arbles are black )s B1 ∩B2 a(* correspo(*s to the top r)ght

(o*e )( the tree wh)ch has bee( c)rcle*. &h,s as )(*)cate* there )t )s ;..

b. &he e4e(t e7actly o(e -arble )s black correspo(*s to the two (o*es of the tree

e(close* by the recta(gle. &he e4e(ts that correspo(* to these two (o*es are

-,t,ally e7cl,s)4e: black followe* by wh)te )s )(co-pat)ble w)th wh)te followe* by

black. &h,s )( accor*a(ce w)th the A**)t)4e R,le for Probab)l)ty we -erely a** the

two probab)l)t)es (e7t to these (o*es s)(ce what wo,l* be s,btracte* fro- the s,-

)s Vero. &h,s the probab)l)ty of *raw)(g e7actly o(e black -arble )( two tr)es

)s 0.23+0.23=0.46.

&he e4e(t at least o(e -arble )s black correspo(*s to the three (o*es of the

tree e(close* by e)ther the c)rcle or the recta(gle. &he e4e(ts that correspo(*

to these (o*es are -,t,ally e7cl,s)4e so as )( part b5 we -erely a** the

probab)l)t)es (e7t to these (o*es. &h,s the probab)l)ty of *raw)(g at least o(e

black -arble )( two tr)es )s 0.47+0.23+0.23=0.93.

"f co,rse th)s a(swer co,l* ha4e bee( fo,(* -ore eas)ly ,s)(g the Probab)l)ty

Law for %o-ple-e(ts s)-ply s,btract)(g the probab)l)ty of the co-ple-e(tary

e4e(t two wh)te -arbles are *raw( fro- 1 to obta)( 1−0.07=0.93.

%s this example shows$ finding the probability for each branch is fairly straightforward$ since we

compute it knowing everything that has happened in the se"uence of steps so far. Two principles that

are true in general emerge from this example:

2robabilities on Tree (ia$rams

1 The probability of the event corresponding to any node on a tree is the product of the numbers on the

uni"ue path of branches that leads to that node from the start.

! ,f an event corresponds to several final nodes$ then its probability is obtained by adding the numbers

next to those nodes.






*+, TA*+AA,S

• A co(*)t)o(al probab)l)ty )s the probab)l)ty that a( e4e(t has occ,rre* tak)(g )(to

acco,(t a**)t)o(al )(for-at)o( abo,t the res,lt of the e7per)-e(t.

• A co(*)t)o(al probab)l)ty ca( always be co-p,te* ,s)(g the for-,la )( the

*e+()t)o(. So-et)-es )t ca( be co-p,te* by *)scar*)(g part of the sa-ple space.

• &wo e4e(ts 0 a(* - are )(*epe(*e(t )f the probab)l)ty P(A∩B)of the)r

)(tersect)o( 0 \ - )s e<,al to the pro*,ct P(A)DP(B)of the)r )(*)4)*,al probab)l)t)es.











a &he probab)l)ty that the car* *raw( )s re*.

b &he probab)l)ty that the car* )s re* g)4e( that )t )s (ot gree(.

c &he probab)l)ty that the car* )s re* g)4e( that )t )s (e)ther re* (or yellow.

* &he probab)l)ty that the car* )s re* g)4e( that )t )s (ot a fo,r.

1; A spec)al *eck of 1 car*s has that are bl,e yellow gree( a(* re*. &he fo,r

car*s of each color are (,-bere* fro- o(e to fo,r. A s)(gle car* )s *raw( at ra(*o-.

)(* the follow)(g probab)l)t)es.






a &he probab)l)ty that the car* *raw( )s a two or a fo,r.

b &he probab)l)ty that the car* )s a two or a fo,r g)4e( that )t )s (ot a o(e.

c &he probab)l)ty that the car* )s a two or a fo,r g)4e( that )t )s e)ther a two or a

three.

* &he probab)l)ty that the car* )s a two or a fo,r g)4e( that )t )s re* or gree(.

11 A ra(*o- e7per)-e(t ga4e r)se to the two=way co(t)(ge(cy table show(. Use )t to

co-p,te the probab)l)t)es )(*)cate*.

R S

A 0.12 0.18

B 0.28 0.42

a P(A)@ P(R)@ P(A∩R).

b #ase* o( the a(swer to a5 *eter-)(e whether or (ot the e4e(ts 0 a(* 6 are

)(*epe(*e(t.

c #ase* o( the a(swer to b5 *eter-)(e whether or (otP(A|R)ca( be pre*)cte* w)tho,t

a(y co-p,tat)o(. f so -ake the pre*)ct)o(. ( a(y case co-p,te P(A|R),s)(g the R,le

for %o(*)t)o(al Probab)l)ty.

12 A ra(*o- e7per)-e(t ga4e r)se to the two=way co(t)(ge(cy table show(. Use )t to

co-p,te the probab)l)t)es )(*)cate*.

R S A 0.13 0.07

B 0.61 0.19

a P(A)@ P(R)@ P(A∩R).

b #ase* o( the a(swer to a5 *eter-)(e whether or (ot the e4e(ts 0 a(* 6 are

)(*epe(*e(t.

c #ase* o( the a(swer to b5 *eter-)(e whether or (ot P(A|R)ca( be pre*)cte*

w)tho,t a(y co-p,tat)o(. f so -ake the pre*)ct)o(. ( a(y case co-p,te P(A|

R),s)(g the R,le for %o(*)t)o(al Probab)l)ty.

13 S,ppose for e4e(ts 0 a(* - )( a ra(*o- e7per)-e(t P(A)=0.70a(* P(B)=0.30.%o-p,te

the )(*)cate* probab)l)ty or e7pla)( why there )s (ot e(o,gh )(for-at)o( to *o so.

a P(A∩B).

b P(A∩B)@ w)th the e7tra )(for-at)o( that 0 a(* - are )(*epe(*e(t.

c P(A∩B)@ w)th the e7tra )(for-at)o( that 0 a(* - are -,t,ally e7cl,s)4e.






1 S,ppose for e4e(ts 0 a(* - co((ecte* to so-e ra(*o-

e7per)-e(t P(A)=0.50a(* P(B)=0.50.%o-p,te the )(*)cate* probab)l)ty or e7pla)( why

there )s (ot e(o,gh )(for-at)o( to *o so.

a P(A∩B).

b P(A∩B)@ w)th the e7tra )(for-at)o( that 0 a(* - are )(*epe(*e(t.

c P(A∩B)@ w)th the e7tra )(for-at)o( that 0 a(* - are -,t,ally e7cl,s)4e.

10 S,ppose for e4e(ts 0 - a(* , co((ecte* to so-e ra(*o- e7per)-e(t 0 - a(* , are

)(*epe(*e(t a(* P(A)=0.88@ P(B)=0.65@ and P(C)=0.44.%o-p,te the )(*)cate* probab)l)ty or

e7pla)( why there )s (ot e(o,gh )(for-at)o( to *o so.

a P(A∩B∩C)

b P Ac∩Bc∩Cc)

1 S,ppose for e4e(ts 0 - a(* , co((ecte* to so-e ra(*o- e7per)-e(t 0 - a(* , are

)(*epe(*e(t a(* P(A)=0.95@ P(B)=0.73 a(* P(C)=0.62.%o-p,te the )(*)cate* probab)l)ty or

e7pla)( why there )s (ot e(o,gh )(for-at)o( to *o so.

a P(A∩B∩C)

b P Ac ∩Bc

∩Cc)

A22!&CAT&1NS

1 &he sa-ple space that *escr)bes all three=ch)l* fa-)l)es accor*)(g to the ge(*ers of the

ch)l*re( w)th respect to b)rth or*er )s

S={bbb,bbg,bgb,bgg,gbb,gbg,ggb,ggg}

( the e7per)-e(t of select)(g a three=ch)l* fa-)ly at ra(*o- co-p,te each of the

follow)(g probab)l)t)es ass,-)(g all o,tco-es are e<,ally l)kely.

a &he probab)l)ty that the fa-)ly has at least two boys.

b &he probab)l)ty that the fa-)ly has at least two boys g)4e( that (ot all of the

ch)l*re( are g)rls.

c &he probab)l)ty that at least o(e ch)l* )s a boy.

* &he probab)l)ty that at least o(e ch)l* )s a boy g)4e( that the +rst bor( )s a

g)rl.


part)c,lar locale accor*)(g to age a(* (,-ber of 4eh)c,lar -o4)(g 4)olat)o(s )( the past

three years:






Age

iolations

0 !.

Under 21 0.04 0.06 0.02

21–40 0.25 0.16 0.0141–60 0.23 0.10 0.02

60 0.08 0.03 0.00

A perso( )s selecte* at ra(*o-. )(* the follow)(g probab)l)t)es.

a &he perso( )s ,(*er 21.

b &he perso( has ha* at least two 4)olat)o(s )( the past three years.

c &he perso( has ha* at least two 4)olat)o(s )( the past three years g)4e( that

he )s ,(*er 21.

* &he perso( )s ,(*er 21 g)4e( that he has ha* at least two 4)olat)o(s )( the

past three years.

e eter-)(e whether the e4e(ts the perso( )s ,(*er 21 a(* the perso( has

ha* at least two 4)olat)o(s )( the past three years are )(*epe(*e(t or (ot.


part)c,lar locale accor*)(g to party a?l)at)o( 0 - , or 'one5 a(* op)()o( o( a bo(*

)ss,e:

Affiliation

Opinion

Favors Opposes Undecided

A 0.12 0.09 0.07

B 0.16 0.12 0.14

C 0.04 0.03 0.06

None 0.08 0.06 0.03

A perso( )s selecte* at ra(*o-. )(* each of the follow)(g probab)l)t)es.

a &he perso( )s )( fa4or of the bo(* )ss,e.

b &he perso( )s )( fa4or of the bo(* )ss,e g)4e( that he )s a?l)ate* w)thparty 0.

c &he perso( )s )( fa4or of the bo(* )ss,e g)4e( that he )s a?l)ate* w)th

party -.

2; &he follow)(g two=way co(t)(ge(cy table g)4es the break*ow( of the pop,lat)o( of

patro(s at a grocery store accor*)(g to the (,-ber of )te-s p,rchase* a(* whether

or (ot the patro( -a*e a( )-p,lse p,rchase at the checko,t co,(ter:






Number of /tems

/mpulse urchase

#ade Not #ade

e/ 0.01 0.19

an 0.04 0.76A patro( )s selecte* at ra(*o-. )(* each of the follow)(g probab)l)t)es.

a &he patro( -a*e a( )-p,lse p,rchase.

b &he patro( -a*e a( )-p,lse p,rchase g)4e( that the total (,-ber of )te-s

p,rchase* was -a(y.

c eter-)(e whether or (ot the e4e(ts few p,rchases a(* -a*e a( )-p,lse

p,rchase at the checko,t co,(ter are )(*epe(*e(t.


a*,lts )( a part)c,lar locale accor*)(g to e-ploy-e(t type a(* le4el of l)fe )(s,ra(ce:

$mplo1ment T1pe

2evel of /nsurance

2o3 #edium 4igh

Un+i""ed 0.07 0.19 0.00

e%i-+i""ed 0.04 0.28 0.08

i""ed 0.03 0.18 0.05

*roe++iona" 0.01 0.05 0.02

A( a*,lt )s selecte* at ra(*o-. )(* each of the follow)(g probab)l)t)es.

a &he perso( has a h)gh le4el of l)fe )(s,ra(ce.b &he perso( has a h)gh le4el of l)fe )(s,ra(ce g)4e( that he *oes (ot ha4e a

profess)o(al pos)t)o(.

c &he perso( has a h)gh le4el of l)fe )(s,ra(ce g)4e( that he has a profess)o(al

pos)t)o(.

* eter-)(e whether or (ot the e4e(ts has a h)gh le4el of l)fe )(s,ra(ce a(*

has a profess)o(al pos)t)o( are )(*epe(*e(t.






2 A -a( has two l)ghts )( h)s well ho,se to keep the p)pes fro- freeV)(g )( w)(ter. @e

checks the l)ghts *a)ly. Each l)ght has probab)l)ty ;.;;2 of b,r()(g o,t before )t )s

checke* the (e7t *ay )(*epe(*e(tly of the other l)ght5.

a f the l)ghts are w)re* )( parallel o(e w)ll co(t)(,e to sh)(e e4e( )f the other

b,r(s o,t. ( th)s s)t,at)o( co-p,te the probab)l)ty that at least o(e l)ght w)ll

co(t)(,e to sh)(e for the f,ll 2 ho,rs. Note the greatly )(crease* rel)ab)l)ty of

the syste- of two b,lbs o4er that of a s)(gle b,lb.

b f the l)ghts are w)re* )( ser)es (e)ther o(e w)ll co(t)(,e to sh)(e e4e( )f o(ly

o(e of the- b,r(s o,t. ( th)s s)t,at)o( co-p,te the probab)l)ty that at least

o(e l)ght w)ll co(t)(,e to sh)(e for the f,ll 2 ho,rs. Note the sl)ghtly

*ecrease* rel)ab)l)ty of the syste- of two b,lbs o4er that of a s)(gle b,lb.






20 A( acco,(ta(t has obser4e* that 0G of all cop)es of a part)c,lar two=part for- ha4e

a( error )( Part a(* 2G ha4e a( error )( Part . f the errors occ,r )(*epe(*e(tly

+(* the probab)l)ty that a ra(*o-ly selecte* for- w)ll be error=free.

2 A bo7 co(ta)(s 2; screws wh)ch are )*e(t)cal )( s)Ve b,t 12 of wh)ch are V)(c coate* a(*

6 of wh)ch are (ot. &wo screws are selecte* at ra(*o- w)tho,t replace-e(t.

a )(* the probab)l)ty that both are V)(c coate*.b )(* the probab)l)ty that at least o(e )s V)(c coate*.

A((&T&1NA! ++/C&S+S

2 E4e(ts 0 a(* - are -,t,ally e7cl,s)4e. )(* P(A|B).

26 &he c)ty co,(c)l of a part)c,lar c)ty )s co-pose* of +4e -e-bers of party 0 fo,r

-e-bers of party - a(* three )(*epe(*e(ts. &wo co,(c)l -e-bers are ra(*o-ly

selecte* to for- a( )(4est)gat)4e co--)ttee.

a )(* the probab)l)ty that both are fro- party 0.

b )(* the probab)l)ty that at least o(e )s a( )(*epe(*e(t.

c )(* the probab)l)ty that the two ha4e *)8ere(t party a?l)at)o(s that )s (ot

both 0 (ot both - a(* (ot both )(*epe(*e(t5.

29 A basketball player -akes ;G of the free throws that he atte-pts e7cept that )f he

has >,st tr)e* a(* -)sse* a free throw the( h)s cha(ces of -ak)(g a seco(* o(e go

*ow( to o(ly 3;G. S,ppose he has >,st bee( awar*e* two free throws.

a )(* the probab)l)ty that he -akes both.

b )(* the probab)l)ty that he -akes at least o(e. A tree *)agra- co,l* help.5

3; A( eco(o-)st w)shes to ascerta)( the proport)o( p of the pop,lat)o( of )(*)4)*,al

ta7payers who ha4e p,rposely s,b-)tte* fra,*,le(t )(for-at)o( o( a( )(co-e ta7

ret,r(. &o tr,ly g,ara(tee a(o(y-)ty of the ta7payers )( a ra(*o- s,r4ey ta7payers

<,est)o(e* are g)4e( the follow)(g )(str,ct)o(s.

1 l)p a co)(.

2 f the co)( la(*s hea*s a(swer Jes to the <,est)o( @a4e yo, e4er

s,b-)tte* fra,*,le(t )(for-at)o( o( a ta7 ret,r(C e4e( )f yo, ha4e

(ot.

3 f the co)( la(*s ta)ls g)4e a tr,thf,l Jes or No a(swer to the

<,est)o( @a4e yo, e4er s,b-)tte* fra,*,le(t )(for-at)o( o( a ta7

ret,r(C






&he <,est)o(er )s (ot tol* how the co)( la(*e* so he *oes (ot k(ow )f a Jes a(swer )s

the tr,th or )s g)4e( o(ly beca,se of the co)( toss.

a Us)(g the Probab)l)ty R,le for %o-ple-e(ts a(* the )(*epe(*e(ce of the co)(

toss a(* the ta7payersW stat,s +ll )( the e-pty cells )( the two=way

co(t)(ge(cy table show(. Ass,-e that the co)( )s fa)r. Each cell e7cept thetwo )( the botto- row w)ll co(ta)( the ,(k(ow( proport)o( or probab)l)ty5 p.

Status

Coin

robabilit14 T

ra&d p

No ra&d

*robabi"i' 1

b &he o(ly )(for-at)o( that the eco(o-)st sees are the e(tr)es )( the follow)(g

table:





















Chapter 7

(iscrete /andom <ariables

,t is often the case that a number is naturally associated to the outcome of a random experiment: thenumber of boys in a three-child family$ the number of defective light bulbs in a case of 155 bulbs$ the

length of time until the next customer arrives at the drive-through window at a bank. +uch a number

varies from trial to trial of the corresponding experiment$ and does so in a way that cannot be

predicted with certainty hence$ it is called a random variable. ,n this chapter and the next we study

such variables.

7.% /andom <ariables

!+A/N&N: 1';+CT&<+S

1 &o lear( the co(cept of a ra(*o- 4ar)able.

2 &o lear( the *)st)(ct)o( betwee( *)screte a(* co(t)(,o,s ra(*o- 4ar)ables.

(e)nition

A random variable is a numerical quantity that is generated by a random experiment.

(e will denote random variables by capital letters$ such as B or C $ and the actual values that they can

take by lowercase letters$ such as x and z .

Table 6.1 0/our Candom Dariables0 gives four examples of random variables. ,n the second example$ the

three dots indicates that every counting number is a possible value for B . %lthough it is highly unlikely$ for

example$ that it would take 85 tosses of the coin to observe heads for the first time$ nevertheless it is

conceivable$ hence the number 85 is a possible value. The set of possible values is infinite$ but is still at

least countable$ in the sense that all possible values can be listed one after another. ,n the last two

examples$ by way of contrast$ the possible values cannot be individually listed$ but take up a wholeinterval of numbers. ,n the fourth example$ since the light bulb could conceivably continue to shine

indefinitely$ there is no natural greatest value for its lifetime$ so we simply place the symbol ∞ for infinity

as the right endpoint of the interval of possible values.






Table 6.1 /our Candom Dariables

+xperiment Number X

2ossible <alues

o X

Roll two fa)r *)ce

S,- of the (,-ber of *ots o(

the top faces

2 3 0 6 9

1; 11 12

l)p a fa)r co)( repeate*ly

N,-ber of tosses ,(t)l the co)(

la(*s hea*s 1 2 3 ]

eas,re the 4oltage at a(

electr)cal o,tlet 'oltage -eas,re* 116 ^ 122

"perate a l)ght b,lb ,(t)l )t

b,r(s o,t &)-e ,(t)l the b,lb b,r(s o,t ; _ `

e+()t)o(

A random variable is called discrete if it has either a finite or a countable number of possible values. A

random variable is called continuous if its possible values contain a whole interval of numbers.

The examples in the table are typical in that discrete random variables typically arise from a counting

process$ whereas continuous random variables typically arise from a measurement.

*+, TA*+AA,S

• A ra(*o- 4ar)able )s a (,-ber ge(erate* by a ra(*o- e7per)-e(t.

• A ra(*o- 4ar)able )s calle* discrete )f )ts poss)ble 4al,es for- a +()te or

co,(table set.

• A ra(*o- 4ar)able )s calle* continuous )f )ts poss)ble 4al,es co(ta)( a whole

)(ter4al of (,-bers.

++/C&S+S

'AS&C






1 %lass)fy each ra(*o- 4ar)able as e)ther *)screte or co(t)(,o,s.

a &he (,-ber of arr)4als at a( e-erge(cy roo- betwee( -)*()ght a(* :;;

a.-.

b &he we)ght of a bo7 of cereal labele* 16 o,(ces.

c &he *,rat)o( of the (e7t o,tgo)(g telepho(e call fro- a b,s)(ess o?ce.

* &he (,-ber of ker(els of popcor( )( a 1=po,(* co(ta)(er.

e &he (,-ber of appl)ca(ts for a >ob.


a &he t)-e betwee( c,sto-ers e(ter)(g a checko,t la(e at a reta)l store.

b &he we)ght of ref,se o( a tr,ck arr)4)(g at a la(*+ll.

c &he (,-ber of passe(gers )( a passe(ger 4eh)cle o( a h)ghway at r,sh ho,r.

* &he (,-ber of cler)cal errors o( a -e*)cal chart.

e &he (,-ber of acc)*e(t=free *ays )( o(e -o(th at a factory.


a &he (,-ber of boys )( a ra(*o-ly selecte* three=ch)l* fa-)ly.

b &he te-perat,re of a c,p of co8ee ser4e* at a resta,ra(t.

c &he (,-ber of (o=shows for e4ery 1;; reser4at)o(s -a*e w)th a co--erc)al

a)rl)(e.

* &he (,-ber of 4eh)cles ow(e* by a ra(*o-ly selecte* ho,sehol*.

e &he a4erage a-o,(t spe(t o( electr)c)ty each $,ly by a ra(*o-ly selecte*

ho,sehol* )( a certa)( state.

%lass)fy each ra(*o- 4ar)able as e)ther *)screte or co(t)(,o,s.

a &he (,-ber of patro(s arr)4)(g at a resta,ra(t betwee( 0:;; p.-. a(* :;;

p.-.

b &he (,-ber of (ew cases of )([,e(Va )( a part)c,lar co,(ty )( a co-)(g

-o(th.

c &he a)r press,re of a t)re o( a( a,to-ob)le.

* &he a-o,(t of ra)( recor*e* at a( a)rport o(e *ay.






e &he (,-ber of st,*e(ts who act,ally reg)ster for classes at a ,()4ers)ty (e7t

se-ester.

0 *e(t)fy the set of poss)ble 4al,es for each ra(*o- 4ar)able. ake a reaso(able

est)-ate base* o( e7per)e(ce where (ecessary.5

a &he (,-ber of hea*s )( two tosses of a co)(.

b &he a4erage we)ght of (ewbor( bab)es bor( )( a part)c,lar co,(ty o(e -o(th.

c &he a-o,(t of l)<,)* )( a 12=o,(ce ca( of soft *r)(k.

* &he (,-ber of ga-es )( the (e7t Borl* Ser)es best of ,p to se4e( ga-es5.

e &he (,-ber of co)(s that -atch whe( three co)(s are tosse* at o(ce.

*e(t)fy the set of poss)ble 4al,es for each ra(*o- 4ar)able. ake a reaso(able

est)-ate base* o( e7per)e(ce where (ecessary.5a &he (,-ber of hearts )( a +4e=car* ha(* *raw( fro- a *eck of 02 car*s that

co(ta)(s 13 hearts )( all.

b &he (,-ber of p)tches -a*e by a start)(g p)tcher )( a -a>or leag,e baseball

ga-e.

c &he (,-ber of break*ow(s of c)ty b,ses )( a large c)ty )( o(e week.

* &he *)sta(ce a re(tal car re(te* o( a *a)ly rate )s *r)4e( each *ay.

e &he a-o,(t of ra)(fall at a( a)rport (e7t -o(th.

ANS+/S

1 a. *)screte

a co(t)(,o,s

b co(t)(,o,s

c *)screte

* *)screte

3

a *)screte

b co(t)(,o,s

c *)screte

* *)screte

e co(t)(,o,s






0

a {0.1.2}

b a( )(ter4al (a,b)a(swers 4ary5

c a( )(ter4al (a,b)a(swers 4ary5

* M0

e M23

7.0 2robability (istributions or (iscrete/andom <ariables

!+A/N&N: 1';+CT&<+S

1 &o lear( the co(cept of the probab)l)ty *)str)b,t)o( of a *)screte ra(*o- 4ar)able.

2 &o lear( the co(cepts of the -ea( 4ar)a(ce a(* sta(*ar* *e4)at)o( of a *)screte

ra(*o- 4ar)able a(* how to co-p,te the-.

2robability (istributions %ssociated to each possible value x of a discrete random variable B is the probabilityP(x)that B will take

the value x in one trial of the experiment.

(e)nition

The probability distribution of a discrete random variable B is a list of each possible value

of B together with the probability that B takes that value in one trial of the experiment.

The probabilities in the probability distribution of a random variable B must satisfy the following two

conditions:

1 ach probabilityP(x)must be between 5 and 1:0≤P(x)≤1.

! The sum of all the probabilities is 1:ΣP(x)=1.





















Figure 8.29robabilit/ Distribution for )ossing )&o Fair Dice






The >ean and Standard (eviation o a (iscrete /andom

<ariable

e+()t)o(

The mean Dalso called the e)pected value E of a discrete random variable B is the number

µ=E(X)=Σx P(x)

The mean of a random variable may be interpreted as the average of the values assumed by the random

variable in repeated trials of the experiment.











The concept of expected value is also basic to the insurance industry$ as the following simplified

example illustrates.






EKAPLE 0

A l)fe )(s,ra(ce co-pa(y w)ll sell a 2;;;;; o(e=year ter- l)fe )(s,ra(ce pol)cy to a(

)(*)4)*,al )( a part)c,lar r)sk gro,p for a pre-),- of 190. )(* the e7pecte* 4al,e to

the co-pa(y of a s)(gle pol)cy )f a perso( )( th)s r)sk gro,p has a 99.9G cha(ce of

s,r4)4)(g o(e year.

Sol,t)o(:

Let : *e(ote the (et ga)( to the co-pa(y fro- the sale of o(e s,ch pol)cy. &here are

two poss)b)l)t)es: the )(s,re* perso( l)4es the whole year or the )(s,re* perso( *)es

before the year )s ,p. Apply)(g the )(co-e -)(,s o,tgo pr)(c)ple )( the for-er case

the 4al,e of : )s 190 X ;D )( the latter case )t )s 195−200,000=−199,805.S)(ce the

probab)l)ty )( the +rst case )s ;.999 a(* )( the seco(* case )s 1−0.9997=0.0003 the

probab)l)ty *)str)b,t)o( for : )s:
















%o-p,te each of the follow)(g <,a(t)t)es.

a. a.

b. P(0).

c. 9 : ;5.

*. 9 : ;5.

e. P(X≤−2).

f. &he -ea( μ of : .

g. &he 4ar)a(ce σ2of : .

h. &he sta(*ar* *e4)at)o( σ of : .

Sol,t)o(:

a. S)(ce all probab)l)t)es -,st a** ,p to 1 a=1−(0.2+0.5+0.1)=0.2.

b. )rectly fro- the table P(0)=0.5.

c. ro- the table P(X>0)=P(1)+P(4)=0.2+0.1=0.3.

d. ro- the table P(X≥0)=P(0)+P(1)+P(4)=0.5+0.2+0.1=0.8.

e. S)(ce (o(e of the (,-bers l)ste* as poss)ble 4al,es for : )s less tha( or e<,al to

X2 the e4e(t : ^ X2 )s )-poss)ble so 9 : ^ X25 ;.

f. Us)(g the for-,la )( the *e+()t)o( of μ

µ=Σx P(x)=(−1)D0.2+0D0.5+1D0.2+4D0.1=0.4






*+, TA*+AA,S

• &he probab)l)ty *)str)b,t)o( of a *)screte ra(*o- 4ar)able : )s a l)st)(g of each

poss)ble 4al,e take( by : alo(g w)th the probab)l)ty P(x)that : takes that 4al,e

)( o(e tr)al of the e7per)-e(t.

• &he -ea( μ of a *)screte ra(*o- 4ar)able : )s a (,-ber that )(*)cates the

a4erage 4al,e of : o4er (,-ero,s tr)als of the e7per)-e(t. t )s co-p,te* ,s)(g

the for-,la µ=Σx P(x).

• &he 4ar)a(ce σ2a(* sta(*ar* *e4)at)o( σ of a *)screte ra(*o- 4ar)able : are

(,-bers that )(*)cate the 4ar)ab)l)ty of : o4er (,-ero,s tr)als of the e7per)-e(t.

&hey -ay be co-p,te* ,s)(g the for-,la σ2= Σx2 P(x) ]−µ2 tak)(g the s<,are root

to obta)( σ .





















1; Let : *e(ote the (,-ber of t)-es a fa)r co)( la(*s hea*s )( three tosses. %o(str,ct

the probab)l)ty *)str)b,t)o( of : .

11 )4e tho,sa(* lottery t)ckets are sol* for 1 each. "(e t)cket w)ll w)( 1;;; two t)ckets

w)ll w)( 0;; each a(* te( t)ckets w)ll w)( 1;; each. Let : *e(ote the (et ga)( fro-

the p,rchase of a ra(*o-ly selecte* t)cket.

a %o(str,ct the probab)l)ty *)str)b,t)o( of : .

b %o-p,te the e7pecte* 4al,e E(X)of : . (terpret )ts -ea()(g.






c %o-p,te the sta(*ar* *e4)at)o( σ of : .

12 Se4e( tho,sa(* lottery t)ckets are sol* for 0 each. "(e t)cket w)ll w)( 2;;; two

t)ckets w)ll w)( 0; each a(* +4e t)ckets w)ll w)( 1;; each. Let : *e(ote the (et ga)(

fro- the p,rchase of a ra(*o-ly selecte* t)cket.

a %o(str,ct the probab)l)ty *)str)b,t)o( of : .b %o-p,te the e7pecte* 4al,e E(X)of : . (terpret )ts -ea()(g.

c %o-p,te the sta(*ar* *e4)at)o( σ of : .

13 A( )(s,ra(ce co-pa(y w)ll sell a 9;;;; o(e=year ter- l)fe )(s,ra(ce pol)cy to a(



s,r4)4)(g o(e year.

1 A( )(s,ra(ce co-pa(y w)ll sell a 1;;;; o(e=year ter- l)fe )(s,ra(ce pol)cy to a(



s,r4)4)(g o(e year.

10 A( )(s,ra(ce co-pa(y est)-ates that the probab)l)ty that a( )(*)4)*,al )( a part)c,lar

r)sk gro,p w)ll s,r4)4e o(e year )s ;.9620. S,ch a perso( w)shes to b,y a 10;;;; o(e=

year ter- l)fe )(s,ra(ce pol)cy. Let , *e(ote how -,ch the )(s,ra(ce co-pa(y charges

s,ch a perso( for s,ch a pol)cy.a %o(str,ct the probab)l)ty *)str)b,t)o( of : . &wo e(tr)es )( the table w)ll

co(ta)(,.5

b %o-p,te the e7pecte* 4al,e E(X)of : .

c eter-)(e the 4al,e , -,st ha4e )( or*er for the co-pa(y to break e4e( o(

all s,ch pol)c)es that )s to a4erage a (et ga)( of Vero per pol)cy o( s,ch

pol)c)es5.

* eter-)(e the 4al,e , -,st ha4e )( or*er for the co-pa(y to a4erage a (et

ga)( of 20; per pol)cy o( all s,ch pol)c)es.

1 A( )(s,ra(ce co-pa(y est)-ates that the probab)l)ty that a( )(*)4)*,al )( a part)c,lar r)sk

gro,p w)ll s,r4)4e o(e year )s ;.99. S,ch a perso( w)shes to b,y a 0;;; o(e=year ter-

l)fe )(s,ra(ce pol)cy. Let , *e(ote how -,ch the )(s,ra(ce co-pa(y charges s,ch a

perso( for s,ch a pol)cy.

a %o(str,ct the probab)l)ty *)str)b,t)o( of : . &wo e(tr)es )( the table w)ll

co(ta)(,.5






b %o-p,te the e7pecte* 4al,e E(X)of : .

c eter-)(e the 4al,e , -,st ha4e )( or*er for the co-pa(y to break e4e( o(

all s,ch pol)c)es that )s to a4erage a (et ga)( of Vero per pol)cy o( s,ch

pol)c)es5.

* eter-)(e the 4al,e , -,st ha4e )( or*er for the co-pa(y to a4erage a (etga)( of 10; per pol)cy o( all s,ch pol)c)es.

1 A ro,lette wheel has 36 slots. &h)rty=s)7 slots are (,-bere* fro- 1 to 3D half of the- are

re* a(* half are black. &he re-a)()(g two slots are (,-bere* ; a(* ;; a(* are gree(. (

a 1 bet o( re* the bettor pays 1 to play. f the ball la(*s )( a re* slot he rece)4es back

the *ollar he bet pl,s a( a**)t)o(al *ollar. f the ball *oes (ot la(* o( re* he loses h)s

*ollar. Let : *e(ote the (et ga)( to the bettor o( o(e play of the ga-e.

a %o(str,ct the probab)l)ty *)str)b,t)o( of : .

b %o-p,te the e7pecte* 4al,e E(X)of : a(* )(terpret )ts -ea()(g )( the

co(te7t of the proble-.

c %o-p,te the sta(*ar* *e4)at)o( of : .

16 A ro,lette wheel has 36 slots. &h)rty=s)7 slots are (,-bere* fro- 1 to 3D the re-a)()(g

two slots are (,-bere* ; a(* ;;. S,ppose the (,-ber ;; )s co(s)*ere* (ot to be

e4e( b,t the (,-ber ; )s st)ll e4e(. ( a 1 bet o( e4e( the bettor pays 1 to play. f

the ball la(*s )( a( e4e( (,-bere* slot he rece)4es back the *ollar he bet pl,s a(

a**)t)o(al *ollar. f the ball *oes (ot la(* o( a( e4e( (,-bere* slot he loses h)s *ollar.

Let : *e(ote the (et ga)( to the bettor o( o(e play of the ga-e.

a %o(str,ct the probab)l)ty *)str)b,t)o( of : .b %o-p,te the e7pecte* 4al,e E(X)of : a(* e7pla)( why th)s ga-e )s (ot

o8ere* )( a cas)(o where ; )s (ot co(s)*ere* e4e(5.

c %o-p,te the sta(*ar* *e4)at)o( of : .




































7.3 The 'inomial (istribution

LEARNN! "#$E%&'ES

1 &o lear( the co(cept of a b)(o-)al ra(*o- 4ar)able.

2 &o lear( how to recog()Ve a ra(*o- 4ar)able as be)(g a b)(o-)al ra(*o- 4ar)able.

The experiment of tossing a fair coin three times and the experiment of observing the genders

according to birth order of the children in a randomly selected three-child family are completely

different$ but the random variables that count the number of heads in the coin toss and the number






of boys in the family *assuming the two genders are e"ually likely) are the same random variable$ the

one with probability distribution

% histogram that graphically illustrates this probability distribution is given in/igure 6.6 0robability

2istribution for Three 7oins and Three 7hildren0. (hat is common to the two experiments is that we

perform three identical and independent trials of the same action$ each trial has only two outcomes

*heads or tails$ boy or girl)$ and the probability of success is the same number$ 5.8$ on every trial. The

random variable that is generated is called the binomial random variable *ith parameters n M

3 and p M 5.8. This is #ust one case of a general situation.

!igure . $robability (istribution for Three <oins and Three <hildren

(e)nition






%uppose a random experiment has the following characteristics.

1 There are n identical and independent trials of a common procedure.

! There are exactly two possible outcomes for each trial, one termed ?success@ and the other ?failure.@

3 The probability of success on any one trial is the same number p.

Then the discrete random variable B that counts the number of successes in the n trials is the binomial

random variable *ith parameters nand p. Fe also say that B has a binomial distribution *ith

parameters n and p.

The following four examples illustrate the definition. 9ote how in every case GsuccessH is the outcome

that is counted$ not the outcome that we prefer or think is better in some sense.

1 % random sample of 1!8 students is selected from a large college in which the proportion of students

who are females is 8@B. +uppose B denotes the number of female students in the sample. ,n this

situation there are n M 1!8 identical and independent trials of a common procedure$ selecting a

student at random there are exactly two possible outcomes for each trial$ GsuccessH *what we are

counting$ that the student be female) and GfailureH and finally the probability of success on any one

trial is the same number pM 5.8@. B is a binomial random variable with parameters n M 1!8 and p M

5.8@.

! % multiple-choice test has 18 "uestions$ each of which has five choices. %n unprepared student taking

the test answers each of the "uestions completely randomly by choosing an arbitrary answer from the

five provided. +uppose B denotes the number of answers that the student gets right. B is a binomial

random variable with parameters n M 18 and p=1/5=0.20.

3 ,n a survey of 1$555 registered voters each voter is asked if he intends to vote for a candidate Titania

Aueen in the upcoming election. +uppose B denotes the number of voters in the survey who intend to

vote for Titania Aueen. B is a binomial random variable with n M 1555 and p e"ual to the true

proportion of voters *surveyed or not) who intend to vote for Titania Aueen.

6 %n experimental medication was given to 35 patients with a certain medical condition.

+uppose B denotes the number of patients who develop severe side effects. B is a binomial random

variable with n M 35 and p e"ual to the true probability that a patient with the underlying condition

will experience severe side effects if given that medication.

2robability Formula or a 'inomial /andom <ariable

Often the most difficult aspect of working a problem that involves the binomial random variable is

recogni'ing that the random variable in "uestion has a binomial distribution. Once that is known$

probabilities can be computed using the following formula.
















Figure 8.;9robabilit/ Distribution of t!e -inomial 6andom <ariable in 'ote 8.2 >(ample

7>











Special Formulas or the >ean and Standard (eviation

o a 'inomial /andom <ariable

+ince a binomial random variable is a discrete random variable$ the formulas for its mean$ variance$

and standard deviation given in the previous section apply to it$ as we #ust saw in 9ote 6.!F

0xample @0 in the case of the mean. Eowever$ for the binomial random variable there are much

simpler formulas.

The Cumulative 2robability (istribution o a 'inomial

/andom <ariable

,n order to allow a broader range of more realistic problems 7hapter 1! 0%ppendix0 contains

probability tables for binomial random variables for various choices of the parameters n and p. These

tables are not the probability distributions that we have seen so far$ but are cumulative probability

distributions. ,n the place of the probability P(x)the table contains the probability






P(X≤x)=P(0)+P(1)+D D D +P(x)

This is illustrated in /igure 6.; 07umulative robabilities0. The probability entered in the tablecorresponds to the area of the shaded region. The reason for providing a cumulative table is that in

practical problems that involve a binomial random variable typically the probability that is sought is

of the form P(X≤x)orP(X≥x).The cumulative table is much easier to use for

computingP(X≤x)since all the individual probabilities have already been computed and added. The

one table suffices for bothP(X≤x)orP(X≥x)and can be used to readily obtain probabilities of the

formP(x) too$ because of the following formulas. The first is #ust the robability Cule for

7omplements.

!igure ./ <umulative $robabilities

,f B is a discrete random variable$ then






P(X≥x)=1−P(X≤x−1) and P(x)=P(X≤x)−P(X≤x−1)






b &he st,*e(t -,st g,ess correctly o( at least ;G of the <,est)o(s wh)ch

)s 0.60Y10=6<,est)o(s. &he probab)l)ty so,ght )s not P(6)a( easy -)stake to

-ake5 b,t

P(X≥6)=P(6)+P(7)+P(8)+P(9)+P(10)

(stea* of co-p,t)(g each of these +4e (,-bers ,s)(g the for-,la a(* a**)(g the-

we ca( ,se the table to obta)(

P(X≥6)=1−P(X≤5)=1−0.6230=0.3770

wh)ch )s -,ch less work a(* of s,?c)e(t acc,racy for the s)t,at)o( at ha(*.

EKAPLE 1;

A( appl)a(ce repa)r-a( ser4)ces +4e wash)(g -ach)(es o( s)te each *ay. "(e=

th)r* of the ser4)ce calls re<,)re )(stallat)o( of a part)c,lar part.

a. &he repa)r-a( has o(ly o(e s,ch part o( h)s tr,ck to*ay. )(* the probab)l)ty that

the o(e part w)ll be e(o,gh to*ay that )s that at -ost o(e wash)(g -ach)(e he ser4)ces

w)ll re<,)re )(stallat)o( of th)s part)c,lar part.

b. )(* the -)()-,- (,-ber of s,ch parts he sho,l* take w)th h)- each *ay )( or*er

that the probab)l)ty that he ha4e e(o,gh for the *ays ser4)ce calls )s at least 90G.

Sol,t)o(:

Let : *e(ote the (,-ber of ser4)ce calls to*ay o( wh)ch the part )s re<,)re*.

&he( : )s a b)(o-)al ra(*o- 4ar)able w)th para-eters n 0 a(* p=1 3=0.35−.

a. Note that the probab)l)ty )( <,est)o( )s (otP(1) b,t rather 9 : ^ 15. Us)(g the

c,-,lat)4e *)str)b,t)o( table )( %hapter 12 QAppe(*)7Q

P(X≤1)=0.4609

b. &he a(swer )s the s-allest (,-ber s,ch that the table e(try P(X≤x))s at

least ;.90;;. S)(ce P(X≤2)=0.7901)s less tha( ;.90 two parts are (ot e(o,gh.






S)(ce P(X≤3)=0.9547)s as large as ;.90 three parts w)ll s,?ce at least 90G of the

t)-e. &h,s the -)()-,- (ee*e* )s three.

*+, TA*+AA,S

• &he *)screte ra(*o- 4ar)able : that co,(ts the (,-ber of s,ccesses )( n

)*e(t)cal )(*epe(*e(t tr)als of a proce*,re that always res,lts )( e)ther of two

o,tco-es s,ccess or fa)l,re a(* )( wh)ch the probab)l)ty of s,ccess o( each

tr)al )s the sa-e (,-ber p )s calle* the b)(o-)al ra(*o- 4ar)able w)th

para-eters n a(* p.

• &here )s a for-,la for the probab)l)ty that the b)(o-)al ra(*o- 4ar)able w)th

para-eters n a(* p w)ll take a part)c,lar 4al,e .

• &here are spec)al for-,las for the -ea( 4ar)a(ce a(* sta(*ar* *e4)at)o( of the

b)(o-)al ra(*o- 4ar)able w)th para-eters n a(* p that are -,ch s)-pler tha(

the ge(eral for-,las that apply to all *)screte ra(*o- 4ar)ables.

• %,-,lat)4e probab)l)ty *)str)b,t)o( tables whe( a4a)lable fac)l)tate co-p,tat)o(

of probab)l)t)es e(co,(tere* )( typ)cal pract)cal s)t,at)o(s.

'AS&C

1 eter-)(e whether or (ot the ra(*o- 4ar)able : )s a b)(o-)al ra(*o- 4ar)able. f so

g)4e the 4al,es of n a(* p. f (ot e7pla)( why (ot.

a : )s the (,-ber of *ots o( the top face of fa)r *)e that )s rolle*.

b : )s the (,-ber of hearts )( a +4e=car* ha(* *raw( w)tho,t replace-e(t5

fro- a well=sh,de* or*)(ary *eck.

c : )s the (,-ber of *efect)4e parts )( a sa-ple of te( ra(*o-ly selecte* parts

co-)(g fro- a -a(,fact,r)(g process )( wh)ch ;.;2G of all parts are

*efect)4e.

* : )s the (,-ber of t)-es the (,-ber of *ots o( the top face of a fa)r *)e )s

e4e( )( s)7 rolls of the *)e.

e : )s the (,-ber of *)ce that show a( e4e( (,-ber of *ots o( the top face

whe( s)7 *)ce are rolle* at o(ce.

2 eter-)(e whether or (ot the ra(*o- 4ar)able : )s a b)(o-)al ra(*o- 4ar)able. f so

g)4e the 4al,es of n a(* p. f (ot e7pla)( why (ot.

a : )s the (,-ber of black -arbles )( a sa-ple of 0 -arbles *raw( ra(*o-ly

a(* w)tho,t replace-e(t fro- a bo7 that co(ta)(s 20 wh)te -arbles a(* 10

black -arbles.






b : )s the (,-ber of black -arbles )( a sa-ple of 0 -arbles *raw( ra(*o-ly

a(* w)th replace-e(t fro- a bo7 that co(ta)(s 20 wh)te -arbles a(* 10 black

-arbles.

c : )s the (,-ber of 4oters )( fa4or of propose* law )( a sa-ple 12;; ra(*o-ly

selecte* 4oters *raw( fro- the e(t)re electorate of a co,(try )( wh)ch 30G ofthe 4oters fa4or the law.

* : )s the (,-ber of +sh of a part)c,lar spec)es a-o(g the (e7t te( la(*e* by

a co--erc)al +sh)(g boat that are -ore tha( 13 )(ches )( le(gth whe( 1G

of all s,ch +sh e7cee* 13 )(ches )( le(gth.

e : )s the (,-ber of co)(s that -atch at least o(e other co)( whe( fo,r co)(s

are tosse* at o(ce.

3 : )s a b)(o-)al ra(*o- 4ar)able w)th para-eters n 12 a(* p ;.62. %o-p,te the

probab)l)ty )(*)cate*.

a P(11)

b P(9)

c P(0)

d P(13)

: )s a b)(o-)al ra(*o- 4ar)able w)th para-eters n 1 a(* p ;.. %o-p,te the

probab)l)ty )(*)cate*.

a P(14)

b P(4)

c P(0)

d P(20)

0 : )s a b)(o-)al ra(*o- 4ar)able w)th para-eters n 0 p ;.0. Use the tables

)(%hapter 12 QAppe(*)7Q to co-p,te the probab)l)ty )(*)cate*.

a 9 : ^ 35






b 9 : 35

c P(3)

d P(0)

e P(5)

6 : )s a b)(o-)al ra(*o- 4ar)able w)th para-eters n 0 p=0.35−.Use the table

)(%hapter 12 QAppe(*)7Q to co-p,te the probab)l)ty )(*)cate*.

a 9 : ^ 25

b 9 : 25

c P(2)

d P(0)

e P(5)

: )s a b)(o-)al ra(*o- 4ar)able w)th the para-eters show(. Use the tables )(%hapter

12 QAppe(*)7Q to co-p,te the probab)l)ty )(*)cate*.

a n 1; p ;.20 9 : ^ 5

b n 1; p ;.0 9 : ^ 5

c n 10 p ;.0 9 : ^ 5

* n 10 p ;.0 P(12)

e n 10 p=0.6−@ P(10≤X≤12)

6 : )s a b)(o-)al ra(*o- 4ar)able w)th the para-eters show(. Use the tables )(

%hapter 12 QAppe(*)7Q to co-p,te the probab)l)ty )(*)cate*.

a n 0 p ;.;0 9 : ^ 15

b n 0 p ;.0 9 : ^ 15

c n 1; p ;.0 9 : ^ 05

* n 1; p ;.0 P(12)

e n 1; p=0.6−@ P(5≤X≤8)






9 : )s a b)(o-)al ra(*o- 4ar)able w)th the para-eters show(. Use the spec)al for-,las

to co-p,te )ts -ea( μ a(* sta(*ar* *e4)at)o( σ .

a n 6 p ;.3

b n p ;.62

c n 12;; p ;.

* n 21;; p ;.2

1; : )s a b)(o-)al ra(*o- 4ar)able w)th the para-eters show(. Use the spec)al for-,las to

co-p,te )ts -ea( μ a(* sta(*ar* *e4)at)o( σ .

a n 1 p ;.00

b n 63 p ;.;0

c n 90 p ;.30

* n 10; p ;.9






1 A co)( )s be(t so that the probab)l)ty that )t la(*s hea*s ,p )s 2/3. &he co)( )s tosse* te(

t)-es.






a )(* the probab)l)ty that )t la(*s hea*s ,p at -ost +4e t)-es.

b )(* the probab)l)ty that )t la(*s hea*s ,p -ore t)-es tha( )t la(*s ta)ls ,p.

APPL%A&"NS

1 A( E(gl)sh=speak)(g to,r)st 4)s)ts a co,(try )( wh)ch 3;G of the pop,lat)o( speaks

E(gl)sh. @e (ee*s to ask so-eo(e *)rect)o(s.

a )(* the probab)l)ty that the +rst perso( he e(co,(ters w)ll be able to speak

E(gl)sh.

b &he to,r)st sees fo,r local people sta(*)(g at a b,s stop. )(* the probab)l)ty

that at least o(e of the- w)ll be able to speak E(gl)sh.

16 &he probab)l)ty that a( egg )( a reta)l package )s cracke* or broke( )s ;.;20.

a )(* the probab)l)ty that a carto( of o(e *oVe( eggs co(ta)(s (o eggs that are

e)ther cracke* or broke(.

b )(* the probab)l)ty that a carto( of o(e *oVe( eggs has )5 at least o(e that )s

e)ther cracke* or broke(D ))5 at least two that are cracke* or broke(.

c )(* the a4erage (,-ber of cracke* or broke( eggs )( o(e *oVe( carto(s.

19 A( appl)a(ce store sells 2; refr)gerators each week. &e( perce(t of all p,rchasers of a

refr)gerator b,y a( e7te(*e* warra(ty. Let : *e(ote the (,-ber of the (e7t 2;p,rchasers who *o so.

a 'er)fy that : sat)s+es the co(*)t)o(s for a b)(o-)al ra(*o- 4ar)able a(* +(* n

a(* p.

b )(* the probab)l)ty that : )s Vero.

c )(* the probab)l)ty that : )s two three or fo,r.

* )(* the probab)l)ty that : )s at least +4e.

2; A*4erse grow)(g co(*)t)o(s ha4e ca,se* 0G of grapefr,)t grow( )( a certa)( reg)o( to

be of )(fer)or <,al)ty. !rapefr,)t are sol* by the *oVe(.

a )(* the a4erage (,-ber of )(fer)or <,al)ty grapefr,)t per bo7 of a *oVe(.

b A bo7 that co(ta)(s two or -ore grapefr,)t of )(fer)or <,al)ty w)ll ca,se a

stro(g a*4erse c,sto-er react)o(. )(* the probab)l)ty that a bo7 of o(e *oVe(

grapefr,)t w)ll co(ta)( two or -ore grapefr,)t of )(fer)or <,al)ty.






21 &he probab)l)ty that a =o,(ce ske)( of a *)sco,(t worste* we)ght k()tt)(g yar( co(ta)(s

a k(ot )s ;.20. !o(er)l b,ys te( ske)(s to crochet a( afgha(.

a )(* the probab)l)ty that )5 (o(e of the te( ske)(s w)ll co(ta)( a k(otD ))5 at

-ost o(e w)ll.

b )(* the e7pecte* (,-ber of ske)(s that co(ta)( k(ots.

c )(* the -ost l)kely (,-ber of ske)(s that co(ta)( k(ots.

22 "(e=th)r* of all pat)e(ts who ,(*ergo a (o(=)(4as)4e b,t ,(pleasa(t -e*)cal test

re<,)re a se*at)4e. A laboratory perfor-s 2; s,ch tests *a)ly. Let : *e(ote the (,-ber

of pat)e(ts o( a(y g)4e( *ay who re<,)re a se*at)4e.

a 'er)fy that : sat)s+es the co(*)t)o(s for a b)(o-)al ra(*o- 4ar)able a(* +(* n

a(* p.

b )(* the probab)l)ty that o( a(y g)4e( *ay betwee( +4e a(* ()(e pat)e(ts w)ll

re<,)re a se*at)4e )(cl,*e +4e a(* ()(e5.

c )(* the a4erage (,-ber of pat)e(ts each *ay who re<,)re a se*at)4e.

* Us)(g the c,-,lat)4e probab)l)ty *)str)b,t)o( for : )( %hapter 12 QAppe(*)7Q

+(* the -)()-,- (,-ber x minof *oses of the se*at)4e that sho,l* be o( ha(*

at the start of the *ay so that there )s a 99G cha(ce that the laboratory w)ll

(ot r,( o,t.

23 Abo,t 2G of al,-() g)4e -o(ey ,po( rece)4)(g a sol)c)tat)o( fro- the college or

,()4ers)ty fro- wh)ch they gra*,ate*. )(* the a4erage (,-ber -o(etary g)fts a

college ca( e7pect fro- e4ery 2;;; sol)c)tat)o(s )t se(*s.

2 "f all college st,*e(ts who are el)g)ble to g)4e bloo* abo,t 16G *o so o( a reg,lar

bas)s. Each -o(th a local bloo* ba(k se(*s a( appeal to g)4e bloo* to 20; ra(*o-ly

selecte* st,*e(ts. )(* the a4erage (,-ber of appeals )( s,ch -a)l)(gs that are -a*e

to st,*e(ts who alrea*y g)4e bloo*.

20 Abo,t 12G of all )(*)4)*,als wr)te w)th the)r left ha(*s. A class of 13; st,*e(ts -eets )(

a classroo- w)th 13; )(*)4)*,al *esks e7actly 1 of wh)ch are co(str,cte* for people

who wr)te w)th the)r left ha(*s. )(* the probab)l)ty that e7actly 1 of the st,*e(ts

e(rolle* )( the class wr)te w)th the)r left ha(*s.

2 A tra4ell)(g sales-a( -akes a sale o( 0G of h)s calls o( reg,lar c,sto-ers. @e -akes

fo,r sales calls each *ay.






a %o(str,ct the probab)l)ty *)str)b,t)o( of : the (,-ber of sales -a*e each

*ay.

b )(* the probab)l)ty that o( a ra(*o-ly selecte* *ay the sales-a( w)ll -ake

a sale.

c Ass,-)(g that the sales-a( -akes 2; sales calls per week +(* the -ea(a(* sta(*ar* *e4)at)o( of the (,-ber of sales -a*e per &ee? .

2 A corporat)o( has a*4ert)se* hea4)ly to try to )(s,re that o4er half the a*,lt pop,lat)o(

recog()Ves the bra(* (a-e of )ts pro*,cts. ( a ra(*o- sa-ple of 2; a*,lts 1

recog()Ve* )ts bra(* (a-e. Bhat )s the probab)l)ty that 1 or -ore people )( s,ch a

sa-ple wo,l* recog()Ve )ts bra(* (a-e )f the act,al proport)o( p of all a*,lts who

recog()Ve the bra(* (a-e were o(ly ;.0;C

A((&T&1NA! ++/C&S+S

26 Bhe( *roppe* o( a har* s,rface a th,-btack la(*s w)th )ts sharp po)(t to,ch)(g the

s,rface w)th probab)l)ty 2/3D )t la(*s w)th )ts sharp po)(t *)recte* ,p )(to the a)r w)th

probab)l)ty 1/3. &he tack )s *roppe* a(* )ts la(*)(g pos)t)o( obser4e* 10 t)-es.

a )(* the probab)l)ty that )t la(*s w)th )ts po)(t )( the a)r at least t)-es.

b f the e7per)-e(t of *ropp)(g the tack 10 t)-es )s *o(e repeate*ly what )s

the a4erage (,-ber of t)-es )t la(*s w)th )ts po)(t )( the a)rC

29 A profess)o(al proofrea*er has a 96G cha(ce of *etect)(g a( error )( a p)ece of wr)tte(

work other tha( -)sspell)(gs *o,ble wor*s a(* s)-)lar errors that are -ach)(e

*etecte*5. A work co(ta)(s fo,r errors.

a )(* the probab)l)ty that the proofrea*er w)ll -)ss at least o(e of the-.

b Show that two s,ch proofrea*ers work)(g )(*epe(*e(tly ha4e a 99.9G

cha(ce of *etect)(g a( error )( a p)ece of wr)tte( work.

c )(* the probab)l)ty that two s,ch proofrea*ers work)(g )(*epe(*e(tly w)ll

-)ss at least o(e error )( a work that co(ta)(s fo,r errors.

3; A -,lt)ple cho)ce e7a- has 2; <,est)o(sD there are fo,r cho)ces for each <,est)o(.

a A st,*e(t g,esses the a(swer to e4ery <,est)o(. )(* the cha(ce that he

g,esses correctly betwee( fo,r a(* se4e( t)-es.

b )(* the -)()-,- score the )(str,ctor ca( set so that the probab)l)ty that a

st,*e(t w)ll pass >,st by g,ess)(g )s 2;G or less.

31 ( sp)te of the re<,)re-e(t that all *ogs boar*e* )( a ke((el be )(oc,late* the

cha(ce that a healthy *og boar*e* )( a clea( well=4e(t)late* ke((el w)ll *e4elop

ke((el co,gh fro- a carr)er )s ;.;;6.






a f a carr)er (ot k(ow( to be s,ch of co,rse5 )s boar*e* w)th three other *ogs

what )s the probab)l)ty that at least o(e of the three healthy *ogs w)ll *e4elop

ke((el co,ghC

b f a carr)er )s boar*e* w)th fo,r other *ogs what )s the probab)l)ty that at least

o(e of the fo,r healthy *ogs w)ll *e4elop ke((el co,ghC

c &he patter( e4)*e(t fro- parts a5 a(* b5 )s that )f K+1*ogs are boar*e*

together o(e a carr)er a(* @ healthy *ogs the( the probab)l)ty that at least

o(e of the healthy *ogs w)ll *e4elop ke((el co,gh )s P(X≥1)=1−(0.992)K

where : )s the b)(o-)al ra(*o- 4ar)able that co,(ts the (,-ber of healthy

*ogs that *e4elop the co(*)t)o(. E7per)-e(t w)th *)8ere(t 4al,es of @ )( th)s

for-,la to +(* the -a7)-,- (,-ber K+1of *ogs that a ke((el ow(er ca(

boar* together so that )f o(e of the *ogs has the co(*)t)o( the cha(ce that

a(other *og w)ll be )(fecte* )s less tha( ;.;0.

32 (4est)gators (ee* to *eter-)(e wh)ch of ;; a*,lts ha4e a -e*)cal co(*)t)o( that

a8ects 2G of the a*,lt pop,lat)o(. A bloo* sa-ple )s take( fro- each of the

)(*)4)*,als.

a Show that the e7pecte* (,-ber of *)sease* )(*)4)*,als )( the gro,p of ;; )s

12 )(*)4)*,als.

b (stea* of test)(g all ;; bloo* sa-ples to +(* the e7pecte* 12 *)sease*

)(*)4)*,als )(4est)gators gro,p the sa-ples )(to ; gro,ps of 1; each -)7 a

l)ttle of the bloo* fro- each of the 1; sa-ples )( each gro,p a(* test each of

the ; -)7t,res. Show that the probab)l)ty that a(y s,ch -)7t,re w)ll co(ta)(

the bloo* of at least o(e *)sease* perso( he(ce test pos)t)4e )s abo,t ;.16.

c #ase* o( the res,lt )( b5 show that the e7pecte* (,-ber of -)7t,res that

test pos)t)4e )s abo,t 11. S,ppos)(g that )(*ee* 11 of the ; -)7t,res test

pos)t)4e the( we k(ow that (o(e of the 9; perso(s whose bloo* was )( the

re-a)()(g 9 sa-ples that teste* (egat)4e has the *)sease. Be ha4e

el)-)(ate* 9; perso(s fro- o,r search wh)le perfor-)(g o(ly ; tests.5
















Chapter 8

Continuous /andom <ariables

%s discussed in +ection 6.1 0Candom Dariables0 in 7hapter 6 02iscrete Candom Dariables0$ a random variable is called continuous if its set of possible values contains a whole interval of decimal numbers.

,n this chapter we investigate such random variables.

8.% Continuous /andom <ariables

LEARNN! "#$E%&'ES

1 &o lear( the co(cept of the probab)l)ty *)str)b,t)o( of a co(t)(,o,s ra(*o-

4ar)able a(* how )t )s ,se* to co-p,te probab)l)t)es.

2 &o lear( bas)c facts abo,t the fa-)ly of (or-ally *)str)b,te* ra(*o- 4ar)ables.

The 2robability (istribution o a Continuous /andom

<ariable

/or a discrete random variable B the probability that B assumes one of its possible values on a single trial

of the experiment makes good sense. This is not the case for a continuous random variable. /or example$suppose B denotes the length of time a commuter #ust arriving at a bus stop has to wait for the next bus. ,f

buses run every 35 minutes without fail$ then the set of possible values of B is the interval denoted[0,30]

the set of all decimal numbers between 5 and 35. 4ut although the number @.!11F1; is a possible value

of B $ there is little or no meaning to the concept of the probability that the commuter will wait precisely

@.!11F1; minutes for the next bus. ,f anything the probability should be 'ero$ since if we could

meaningfully measure the waiting time to the nearest millionth of a minute it is practically inconceivable

that we would ever get exactly @.!11F1; minutes. ore meaningful "uestions are those of the form: (hat

is the probability that the commuterUs waiting time is less than 15 minutes$ or is between 8 and 15

minutes= ,n other words$ with continuous random variables one is concerned not with the event that the

variable assumes a single particular value$ but with the event that the random variable assumes a value in

a particular interval.

e+()t)o(






The probability distribution of a continuous random variable B is an assignment of

probabilities to intervals of decimal numbers using a function f(x)$ called a density function$ in the

following way8 the probability that B assumes a value in the interval [a,b]is equal to the area of the

region that is bounded above by the graph of the equation y=f(x)$bounded below by the x:axis, and

bounded on the left and right by the vertical lines through a and b$ as illustrated in !igure ."

G$robability #iven as Area of a -egion under a <urveG .

!igure ." $robability #iven as Area of a -egion under a <urve

This definition can be understood as a natural outgrowth of the discussion in+ection !.1.3 0Celative

/re"uency Eistograms0 in 7hapter ! 02escriptive +tatistics0. There we saw that if we have in view a

population *or a very large sample) and make measurements with greater and greater precision$ then

as the bars in the relative fre"uency histogram become exceedingly fine their vertical sides merge

and disappear$ and what is left is #ust the curve formed by their tops$ as shown in /igure !.8 0+ample

+i'e and Celative /re"uency Eistograms0 in 7hapter ! 02escriptive +tatistics0. oreover the total

area under the curve is 1$ and the proportion of the population with measurements between two

numbersa and b is the area under the curve and between a and b$ as shown in /igure !.; 0% Dery






/ine Celative /re"uency Eistogram0 in 7hapter ! 02escriptive +tatistics0. ,f we think of B as a

measurement to infinite precision arising from the selection of any one member of the population at

random$ thenP(a<X<b)is simply the proportion of the population with measurements

between a and b$ the curve in the relative fre"uency histogram is the density function for B $ and we

arrive at the definition #ust above.

very density functionf(x)must satisfy the following two conditions:

1 /or all numbers x f(x)≥0$ so that the graph of y=f(x)never drops below the x -axis.

! The area of the region under the graph ofy=f(x)and above the x -axis is 1.

4ecause the area of a line segment is 5$ the definition of the probability distribution of a continuous

random variable implies that for any particular decimal number$ say a$ the probability

that B assumes the exact value a is 5. This property implies that whether or not the endpoints of an

interval are included makes no difference concerning the probability of the interval.

/or any continuous random variable B :

P(a≤X≤b)=P(a<X≤b)=P(a≤X<b)=P(a<X<b)

EKAPLE 1

A ra(*o- 4ar)able : has the ,()for- *)str)b,t)o( o( the )(ter4al [0,1]: the *e(s)ty

f,(ct)o( )s f(x)=1)f )s betwee( ; a(* 1 a(* f(x)=0for all other 4al,es of as

show( )( )g,re 0.2 QU()for- )str)b,t)o( o( Q.

Figure ;.2Uniform Distribution on [0,1]






a. )(* 9 : ;.05 the probab)l)ty that : ass,-es a 4al,e greater tha( ;.0.

b. )(* 9 : ;.25 the probab)l)ty that : ass,-es a 4al,e less tha( or e<,al

to ;.2.

c. )(* 9;. _ : _ ;.5 the probab)l)ty that : ass,-es a 4al,e betwee( ;.

a(* ;..

Sol,t)o(:

a. 9 : ;.05 )s the area of the recta(gle of he)ght 1 a(* base le(gth 1−0.75=0.25

he(ce )s base×height=(0.25)Y(1)=0.25.See )g,re 0.3 QProbab)l)t)es fro- the U()for- )str)b,t)o(

o( Qa5.

b. 9 : ;.25 )s the area of the recta(gle of he)ght 1 a(* base le(gth 0.2−0=0.2 he(ce

)s base×height=(0.2)Y(1)=0.2.See )g,re 0.3 QProbab)l)t)es fro- the U()for- )str)b,t)o( o(

Qb5.

c. 9;. _ : _ ;.5 )s the area of the recta(gle of he)ght 1 a(* le(gth 0.7−0.4=0.3 he(ce

)s base×height=(0.3)Y(1)=0.3.See )g,re 0.3 QProbab)l)t)es fro- the U()for- )str)b,t)o( o(Qc5.

Figure ;.*9robabilities from t!e Uniform Distribution on [0,1]






EKAPLE 2

A -a( arr)4es at a b,s stop at a ra(*o- t)-e that )s w)th (o regar* for the

sche*,le* ser4)ce5 to catch the (e7t b,s. #,ses r,( e4ery 3; -)(,tes w)tho,t fa)l

he(ce the (e7t b,s w)ll co-e a(y t)-e *,r)(g the (e7t 3; -)(,tes w)th e4e(ly

*)str)b,te* probab)l)ty a ,()for- *)str)b,t)o(5. )(* the probab)l)ty that a b,s w)ll

co-e w)th)( the (e7t 1; -)(,tes.

Sol,t)o(:

&he graph of the *e(s)ty f,(ct)o( )s a hor)Vo(tal l)(e abo4e the )(ter4al fro- ; to

3; a(* )s the =a7)s e4erywhere else. S)(ce the total area ,(*er the c,r4e -,st

be 1 the he)ght of the hor)Vo(tal l)(e )s 1/3;. See )g,re 0. QProbab)l)ty of

Ba)t)(g At ost 1; )(,tes for a #,sQ. &he probab)l)ty so,ght )s P(0≤X≤10).#y

*e+()t)o( th)s probab)l)ty )s the area of the recta(g,lar reg)o( bo,(*e* abo4e by

the hor)Vo(tal l)(e f(x)=1/30 bo,(*e* below by the =a7)s bo,(*e* o( the left by

the 4ert)cal l)(e at ; the / =a7)s5 a(* bo,(*e* o( the r)ght by the 4ert)cal l)(e at

1;. &h)s )s the sha*e* reg)o( )( )g,re 0. QProbab)l)ty of Ba)t)(g At ost 1;

)(,tes for a #,sQ. ts area )s the base of the recta(gle t)-es )ts

he)ght 10D(1/30)=1/3. &h,s P(0≤X≤10)=1/3.






Figure ;.89robabilit/ of 3aiting 0t ost 1A inutes for a -us






!igure . 9ell <urves with 6 H 5.& and (ifferent 0alues of

The value of 6 determines whether the bell curve is tall and thin or short and s"uat$ sub#ect always to

the condition that the total area under the curve be e"ual to 1. This is shown in /igure 8.; 04ell

7urves with 0$ where we have arbitrarily chosen to center the curves at M ;.

!igure ./ 9ell <urves with H / and (ifferent 0alues of 6






e+()t)o(

The probability distribution corresponding to the density function for the bell curve with

parameters and 6 is called the normal distribution with mean and standard deviation 6 .

e+()t)o(

A continuous random variable whose probabilities are described by the normal distribution with

mean and standard deviation 6 is called a normally distributed random variable or a

normal random variable for short, with mean and standard deviation 6 .

/igure 8.@ 02ensity /unction for a 9ormally 2istributed Candom Dariable with ean 0 shows the

density function that determines the normal distribution with mean and standard deviation 6 . (e

repeat an important fact about this curve:






The density curve for the normal distribution is symmetric about the mean.

!igure .1 (ensity !unction for a Iormally (istributed -andom 0ariable with 2ean and %tandard

(eviation 6

EKAPLE 3

@e)ghts of 20=year=ol* -e( )( a certa)( reg)o( ha4e -ea( 9.0 )(ches a(*

sta(*ar* *e4)at)o( 2.09 )(ches. &hese he)ghts are appro7)-ately (or-ally

*)str)b,te*. &h,s the he)ght : of a ra(*o-ly selecte* 20=year=ol* -a( )s a (or-al

ra(*o- 4ar)able w)th -ea( μ 9.0 a(* sta(*ar* *e4)at)o( σ 2.09. Sketch a

<,al)tat)4ely acc,rate graph of the *e(s)ty f,(ct)o( for : . )(* the probab)l)ty that

a ra(*o-ly selecte* 20=year=ol* -a( )s -ore tha( 9.0 )(ches tall.

Sol,t)o(:

&he *)str)b,t)o( of he)ghts looks l)ke the bell c,r4e )( )g,re 0.6 Qe(s)ty ,(ct)o(

for @e)ghts of 20=Jear="l* e(Q. &he )-porta(t po)(t )s that )t )s ce(tere* at )ts

-ea( 9.0 a(* )s sy--etr)c abo,t the -ea(.






Figure ;."Densit/ Function for eig!ts of 2;+Bear+ld en

S)(ce the total area ,(*er the c,r4e )s 1 by sy--etry the area to the r)ght of

9.0 )s half the total or ;.0. #,t th)s area )s prec)sely the probab)l)ty 9 :

9.05 the probab)l)ty that a ra(*o-ly selecte* 20=year=ol* -a( )s -ore tha(

9.0 )(ches tall.

Be w)ll lear( how to co-p,te other probab)l)t)es )( the (e7t two sect)o(s.

*+, TA*+AA,S

• or a co(t)(,o,s ra(*o- 4ar)able : the o(ly probab)l)t)es that are co-p,te* are

those of : tak)(g a 4al,e )( a spec)+e* )(ter4al.

• &he probab)l)ty that : take a 4al,e )( a part)c,lar )(ter4al )s the sa-e whether or

(ot the e(*po)(ts of the )(ter4al are )(cl,*e*.

• &he probab)l)ty P(a<X<b)@ that : take a 4al,e )( the )(ter4al fro- a to b )s the area

of the reg)o( betwee( the 4ert)cal l)(es thro,gh a a(* b abo4e the =a7)s a(*

below the graph of a f,(ct)o( f(x)calle* the *e(s)ty f,(ct)o(.

• A (or-ally *)str)b,te* ra(*o- 4ar)able )s o(e whose *e(s)ty f,(ct)o( )s a bell

c,r4e.






• E4ery bell c,r4e )s sy--etr)c abo,t )ts -ea( a(* l)es e4erywhere abo4e the =

a7)s wh)ch )t approaches asy-ptot)cally arb)trar)ly closely w)tho,t to,ch)(g5.

EKER%SES

#AS%

1 A co(t)(,o,s ra(*o- 4ar)able : has a ,()for- *)str)b,t)o( o( the )(ter4al [5,12].Sketch the

graph of )ts *e(s)ty f,(ct)o(.

2 A co(t)(,o,s ra(*o- 4ar)able : has a ,()for- *)str)b,t)o( o( the )(ter4al [−3,3].Sketch

the graph of )ts *e(s)ty f,(ct)o(.

3 A co(t)(,o,s ra(*o- 4ar)able : has a (or-al *)str)b,t)o( w)th -ea( 1;; a(* sta(*ar*

*e4)at)o( 1;. Sketch a <,al)tat)4ely acc,rate graph of )ts *e(s)ty f,(ct)o(.

A co(t)(,o,s ra(*o- 4ar)able : has a (or-al *)str)b,t)o( w)th -ea( 3 a(* sta(*ar*

*e4)at)o( 2.0. Sketch a <,al)tat)4ely acc,rate graph of )ts *e(s)ty f,(ct)o(.

0 A co(t)(,o,s ra(*o- 4ar)able : has a (or-al *)str)b,t)o( w)th -ea( 3. &he probab)l)ty

that : takes a 4al,e greater tha( 6; )s ;.212. Use th)s )(for-at)o( a(* the sy--etry of

the *e(s)ty f,(ct)o( to +(* the probab)l)ty that : takes a 4al,e less tha( . Sketch the

*e(s)ty c,r4e w)th rele4a(t reg)o(s sha*e* to )ll,strate the co-p,tat)o(.

A co(t)(,o,s ra(*o- 4ar)able : has a (or-al *)str)b,t)o( w)th -ea( 19. &he

probab)l)ty that : takes a 4al,e greater tha( 16; )s ;.1. Use th)s )(for-at)o( a(* the

sy--etry of the *e(s)ty f,(ct)o( to +(* the probab)l)ty that : takes a 4al,e less tha(

106. Sketch the *e(s)ty c,r4e w)th rele4a(t reg)o(s sha*e* to )ll,strate the

co-p,tat)o(.

A co(t)(,o,s ra(*o- 4ar)able : has a (or-al *)str)b,t)o( w)th -ea( 0;.0. &he

probab)l)ty that : takes a 4al,e less tha( 0 )s ;.. Use th)s )(for-at)o( a(* the

sy--etry of the *e(s)ty f,(ct)o( to +(* the probab)l)ty that : takes a 4al,e greater

tha( . Sketch the *e(s)ty c,r4e w)th rele4a(t reg)o(s sha*e* to )ll,strate the

co-p,tat)o(.






6 A co(t)(,o,s ra(*o- 4ar)able : has a (or-al *)str)b,t)o( w)th -ea( 12.20. &he

probab)l)ty that : takes a 4al,e less tha( 13 )s ;.62. Use th)s )(for-at)o( a(* the

sy--etry of the *e(s)ty f,(ct)o( to +(* the probab)l)ty that : takes a 4al,e greater

tha( 11.0;. Sketch the *e(s)ty c,r4e w)th rele4a(t reg)o(s sha*e* to )ll,strate the

co-p,tat)o(.

9 &he +g,re pro4)*e* shows the *e(s)ty c,r4es of three (or-ally *)str)b,te* ra(*o-

4ar)ables : 0 : - a(* : ,. &he)r sta(*ar* *e4)at)o(s )( (o part)c,lar or*er5 are 10

a(* 2;. Use the +g,re to )*e(t)fy the 4al,es of the -ea(s µA µB a(* µC a(* sta(*ar*

*e4)at)o(s σA σB a(* σC of the three ra(*o- 4ar)ables.

1; &he +g,re pro4)*e* shows the *e(s)ty c,r4es of three (or-ally *)str)b,te* ra(*o-

4ar)ables : 0 : - a(* : ,. &he)r sta(*ar* *e4)at)o(s )( (o part)c,lar or*er5 are 2; 0

a(* 1;. Use the +g,re to )*e(t)fy the 4al,es of the -ea(s µA µB a(* µC a(* sta(*ar*

*e4)at)o(s σA σB a(* σC of the three ra(*o- 4ar)ables.






A22!&CAT&1NS

11 ogberrys alar- clock )s battery operate*. &he battery co,l* fa)l w)th e<,al probab)l)ty

at a(y t)-e of the *ay or ()ght. E4ery *ay ogberry sets h)s alar- for :3; a.-. a(*

goes to be* at 1;:;; p.-. )(* the probab)l)ty that whe( the clock battery +(ally *)es )t

w)ll *o so at the -ost )(co(4e()e(t t)-e betwee( 1;:;; p.-. a(* :3; a.-.

12 #,ses r,(()(g a b,s l)(e (ear es*e-o(as ho,se r,( e4ery 10 -)(,tes. B)tho,t

pay)(g atte(t)o( to the sche*,le she walks to the (earest stop to take the b,s to tow(.

)(* the probab)l)ty that she wa)ts -ore tha( 1; -)(,tes.

13 &he a-o,(t : of ora(ge >,)ce )( a ra(*o-ly selecte* half=gallo( co(ta)(er 4ar)es

accor*)(g to a (or-al *)str)b,t)o( w)th -ea( o,(ces a(* sta(*ar* *e4)at)o( ;.20

o,(ce.

a Sketch the graph of the *e(s)ty f,(ct)o( for : .

b Bhat proport)o( of all co(ta)(ers co(ta)( less tha( a half gallo( o,(ces5C

E7pla)(.

c Bhat )s the -e*)a( a-o,(t of ora(ge >,)ce )( s,ch co(ta)(ersC E7pla)(.

1 &he we)ght : of grass see* )( bags -arke* 0; lb 4ar)es accor*)(g to a (or-al

*)str)b,t)o( w)th -ea( 0; lb a(* sta(*ar* *e4)at)o( 1 o,(ce ;.;20 lb5.a Sketch the graph of the *e(s)ty f,(ct)o( for : .

b Bhat proport)o( of all bags we)gh less tha( 0; po,(*sC E7pla)(.

c Bhat )s the -e*)a( we)ght of s,ch bagsC E7pla)(.






8.0 The Standard Normal (istribution

LEARNN! "#$E%&'ES

1 &o lear( what a sta(*ar* (or-al ra(*o- 4ar)able )s.

2 &o lear( how to ,se )g,re 12.2 Q%,-,lat)4e Nor-al Probab)l)tyQ to co-p,te

probab)l)t)es relate* to a sta(*ar* (or-al ra(*o- 4ar)able.

e+()t)o(

A standard normal random variable is a normally distributed random variable with mean M

5 and standard deviation 6 M 1. >t will always be denoted by the letter C .






The density function for a standard normal random variable is shown in /igure 8.F 02ensity 7urve

for a +tandard 9ormal Candom Dariable0.

!igure .4 (ensity <urve for a %tandard Iormal -andom 0ariable

To compute probabilities for C we will not work with its density function directly but instead read

probabilities out of /igure 1!.! 07umulative 9ormal robability0 in 7hapter 1! 0%ppendix0. The

tables are tables of cumulative probabilities their entries are probabilities of the form P(Z<z).The

use of the tables will be explained by the following series of examples.

+A>2!+ 7

)(* the probab)l)t)es )(*)cate* where as always C *e(otes a sta(*ar* (or-al

ra(*o- 4ar)able.a. 9 C _ 1.65.

b. 9 C _ X;.205.

Sol,t)o(:

a. )g,re 0.1; Q%o-p,t)(g Probab)l)t)es Us)(g the %,-,lat)4e &ableQ shows how th)s

probab)l)ty )s rea* *)rectly fro- the table w)tho,t a(y co-p,tat)o( re<,)re*. &he *)g)ts )(






the o(es a(* te(ths places of 1.6 (a-ely 1. are ,se* to select the appropr)ate row of

the tableD the h,(*re*ths part of 1.6 (a-ely ;.;6 )s ,se* to select the appropr)ate

col,-( of the table. &he fo,r *ec)-al place (,-ber )( the )(ter)or of the table that l)es )(

the )(tersect)o( of the row a(* col,-( selecte* ;.93; )s the probab)l)ty

so,ght: P(Z<1.48)=0.9306.

Figure ;.1A,omputing 9robabilities Using t!e ,umulative )able

b. &he -)(,s s)g( )( X;.20 -akes (o *)8ere(ce )( the proce*,reD the table )s ,se*

)( e7actly the sa-e way as )( part a5: the probab)l)ty so,ght )s the (,-ber that )s )( the

)(tersect)o( of the row w)th hea*)(g X;.2 a(* the col,-( w)th hea*)(g ;.;0 the (,-ber

;.;13. &h,s 9 C _ X;.205 ;.;13.

EKAPLE 0

)(* the probab)l)t)es )(*)cate*.

a. 9 C 1.;5.

b. 9 C X1.;25.

Sol,t)o(:

a. #eca,se the e4e(ts C 1.; a(* C ^ 1.; are co-ple-e(ts the Probab)l)ty

R,le for %o-ple-e(ts )-pl)es that

P(Z>1.60)=1−P(Z≤1.60)

S)(ce )(cl,s)o( of the e(*po)(t -akes (o *)8ere(ce for the co(t)(,o,s ra(*o-

4ar)able C P(Z≤1.60)=P(Z<1.60)@ wh)ch we k(ow how to +(* fro- the table. &he






(,-ber )( the row w)th hea*)(g 1. a(* )( the col,-( w)th hea*)(g ;.;; )s

;.902. &h,s P(Z<1.60)=0.9452so

P(Z>1.60)=1−P(Z≤1.60)=1−0.9452=0.0548

)g,re 0.11 Q%o-p,t)(g a Probab)l)ty for a R)ght @alf=L)(eQ)ll,strates the )*eas

geo-etr)cally. S)(ce the total area ,(*er the c,r4e )s 1 a(* the area of the

reg)o( to the left of 1.; )s fro- the table5 ;.902 the area of the reg)o( to

the r)ght of 1.; -,st be 1−0.9452=0.0548.

Figure ;.11,omputing a 9robabilit/ for a 6ig!t alf+Line

b &he -)(,s s)g( )( X1.;2 -akes (o *)8ere(ce )( the proce*,reD the table )s

,se* )( e7actly the sa-e way as )( part a5. &he (,-ber )( the )(tersect)o( of

the row w)th hea*)(g X1.; a(* the col,-( w)th hea*)(g ;.;2 )s ;.1039. &h)s

-ea(s that P(Z<−1.02)=P(Z≤−1.02)=0.1539 he(ce

P(Z>−1.02)=1−P(Z≤−1.02)=1−0.1539=0.8461






Figure ;.12 ,omputing a 9robabilit/ for an #nterval of Finite Lengt!

b &he proce*,re for +(*)(g the probab)l)ty that C takes a 4al,e )( a +()te

)(ter4al whose e(*po)(ts ha4e oppos)te s)g(s )s e7actly the sa-e proce*,re

,se* )( part a5 a(* )s )ll,strate* )( )g,re 0.13 Q%o-p,t)(g a Probab)l)ty for

a( (ter4al of )()te Le(gthQ. ( sy-bols the co-p,tat)o( )s

P(−2.55<Z<0.09)==P(Z<0.09)−P(Z<−2.55)

=0.5359−0.0054=0.5305






Figure ;.1* ,omputing a 9robabilit/ for an #nterval of Finite Lengt!

The next example shows what to do if the value of C that we want to look up in the table is not

present there.

EKAPLE


a. P(1.13<Z<4.16).

b. P(−5.22<Z<2.15).

Sol,t)o(:

a. Be atte-pt to co-p,te the probab)l)ty e7actly as )( Note 0.2; QE7a-ple

Q by look)(g ,p the (,-bers 1.13 a(* .1 )( the table. Be obta)( the 4al,e ;.6;6

for the area of the reg)o( ,(*er the *e(s)ty c,r4e to left of 1.13 w)tho,t a(y proble-

b,t whe( we go to look ,p the (,-ber .1 )( the table )t )s (ot there. Be ca( see

fro- the last row of (,-bers )( the table that the area to the left of .1 -,st be so

close to 1 that to fo,r *ec)-al places )t ro,(*s to 1.;;;;. &herefore

P(1.13<Z<4.16)=1.0000−0.8708=0.1292

b. S)-)larly here we ca( rea* *)rectly fro- the table that the area ,(*er the

*e(s)ty c,r4e a(* to the left of 2.10 )s ;.962 b,t X0.22 )s too far to the left o(

the (,-ber l)(e to be )( the table. Be ca( see fro- the +rst l)(e of the table that






the area to the left of X0.22 -,st be so close to ; that to fo,r *ec)-al places )t

ro,(*s to ;.;;;;. &herefore

P(−5.22<Z<2.15)=0.9842−0.0000=0.9842

The final example of this section explains the origin of the proportions given in the mpirical Cule.

+A>2!+ 5


a. P(−1<Z<1).

b. P(−2<Z<2).

c. P(−3<Z<3).

Sol,t)o(:

a. Us)(g the table as was *o(e )( Note 0.2; QE7a-ple Qb5 we obta)(

P(−1<Z<1)=0.8413−0.1587=0.6826

S)(ce C has -ea( ; a(* sta(*ar* *e4)at)o( 1 for C to take a 4al,e betwee( X1

a(* 1 -ea(s that C takes a 4al,e that )s w)th)( o(e sta(*ar* *e4)at)o( of the-ea(. ",r co-p,tat)o( shows that the probab)l)ty that th)s happe(s )s abo,t

;.6 the proport)o( g)4e( by the E-p)r)cal R,le for h)stogra-s that are -o,(*

shape* a(* sy--etr)cal l)ke the bell c,r4e.

b. Us)(g the table )( the sa-e way

P(−2<Z<2)=0.9772−0.0228=0.9544

&h)s correspo(*s to the proport)o( ;.90 for *ata w)th)( two sta(*ar* *e4)at)o(s

of the -ea(.

c. S)-)larly






P(−3<Z<3)=0.9987−0.0013=0.9974

wh)ch correspo(*s to the proport)o( ;.99 for *ata w)th)( three sta(*ar*

*e4)at)o(s of the -ea(.

IEJ &AIEABAJS

• A sta(*ar* (or-al ra(*o- 4ar)able C )s a (or-ally *)str)b,te* ra(*o- 4ar)able

w)th -ea( μ ; a(* sta(*ar* *e4)at)o( σ 1.

• Probab)l)t)es for a sta(*ar* (or-al ra(*o- 4ar)able are co-p,te* ,s)(g)g,re

12.2 Q%,-,lat)4e Nor-al Probab)l)tyQ.


























8.3 2robability Computations or :eneral Normal /andom

<ariables

LEARNN! "#$E%&'E

1 &o lear( how to co-p,te probab)l)t)es relate* to a(y (or-al ra(*o- 4ar)able.

,f B is any normally distributed normal random variable then /igure 1!.! 07umulative 9ormal

robability0 can also be used to compute a probability of the form P(a<X<b) by means of the

following e"uality.






The new endpoints(a−µ)/σand(b−µ)/σare the z -scores of a and b as defined in +ection

!.6.! in 7hapter ! 02escriptive +tatistics0.

/igure 8.16 0robability for an ,nterval of /inite ength0 illustrates the meaning of the e"uality

geometrically: the two shaded regions$ one under the density curve for B and the other under the

density curve for C $ have the same area. ,nstead of drawing both bell curves$ though$ we will always

draw a single generic bell-shaped curve with both an x -axis and a z -axis below it.

!igure ." $robability for an >nterval of !inite 'ength






+A>2!+ 6

Let : be a (or-al ra(*o- 4ar)able w)th -ea( μ 1; a(* sta(*ar* *e4)at)o( σ

2.0. %o-p,te the follow)(g probab)l)t)es.

a. 9 : _ 15.

b. P(8<X<14).

Sol,t)o(:











EKAPLE 1;

&he l)fet)-es of the trea* of a certa)( a,to-ob)le t)re are (or-ally *)str)b,te*

w)th -ea( 30;; -)les a(* sta(*ar* *e4)at)o( 0;; -)les. )(* the probab)l)ty

that the trea* l)fe of a ra(*o-ly selecte* t)re w)ll be betwee( 3;;;; a(* ;;;;

-)les.






Sol,t)o(:

Let : *e(ote the trea* l)fe of a ra(*o-ly selecte* t)re. &o -ake the (,-bers

eas)er to work w)th we w)ll choose tho,sa(*s of -)les as the ,()ts. &h,s μ

3.0 σ .0 a(* the proble- )s to co-p,te P(30<X<40).)g,re 0.1 QProbab)l)ty%o-p,tat)o( for &)re &rea* BearQ )ll,strates the follow)(g co-p,tat)o(:






EKAPLE 11

Scores o( a sta(*ar*)Ve* college e(tra(ce e7a-)(at)o( ,((5 are (or-ally

*)str)b,te* w)th -ea( 01; a(* sta(*ar* *e4)at)o( ;. A select)4e ,()4ers)ty

co(s)*ers for a*-)ss)o( o(ly appl)ca(ts w)th ,(( scores o4er 0;. )(*

perce(tage of all )(*)4)*,als who took the ,(( who -eet the ,()4ers)tys ,((

re<,)re-e(t for co(s)*erat)o( for a*-)ss)o(.

Sol,t)o(:

Let : *e(ote the score -a*e o( the ,(( by a ra(*o-ly selecte* )(*)4)*,al.

&he( : )s (or-ally *)str)b,te* w)th -ea( 01; a(* sta(*ar* *e4)at)o( ;. &he

probab)l)ty that : l)e )( a part)c,lar )(ter4al )s the sa-e as the proport)o( of all

e7a- scores that l)e )( that )(ter4al. &h,s the sol,t)o( to the proble- )s 9 :

0;5 e7presse* as a perce(tage. )g,re 0.16 QProbab)l)ty %o-p,tat)o( for E7a-

ScoresQ )ll,strates the follow)(g co-p,tat)o(:






*+, TA*+AA, • Probab)l)t)es for a ge(eral (or-al ra(*o- 4ar)able are co-p,te* ,s)(g)g,re 12.2

Q%,-,lat)4e Nor-al Probab)l)tyQ after co(4ert)(g =4al,es to z =scores.

++/C&S+S

'AS&C

1 : )s a (or-ally *)str)b,te* ra(*o- 4ar)able w)th -ea( 0 a(* sta(*ar* *e4)at)o( . )(*

the probab)l)ty )(*)cate*.

a 9 : _ 09.05

b 9 : _ .25

c 9 : 02.25

* 9 : ;5

2 : )s a (or-ally *)str)b,te* ra(*o- 4ar)able w)th -ea( X20 a(* sta(*ar* *e4)at)o( .

)(* the probab)l)ty )(*)cate*.

a 9 : _ X2.25






b 9 : _ X1.65

c 9 : X33.15

* 9 : X1.05

3 : )s a (or-ally *)str)b,te* ra(*o- 4ar)able w)th -ea( 112 a(* sta(*ar* *e4)at)o(

10. )(* the probab)l)ty )(*)cate*.

a P(100<X<125)

b P(91<X<107)

c P(118<X<160)

: )s a (or-ally *)str)b,te* ra(*o- 4ar)able w)th -ea( 2 a(* sta(*ar* *e4)at)o( 22.


a P(78<X<127)

b P(60<X<90)

c P(49<X<71)

0 : )s a (or-ally *)str)b,te* ra(*o- 4ar)able w)th -ea( 0;; a(* sta(*ar* *e4)at)o(

20. )(* the probab)l)ty )(*)cate*.

a 9 : _ ;;5

b P(466<X<625)

: )s a (or-ally *)str)b,te* ra(*o- 4ar)able w)th -ea( ; a(* sta(*ar* *e4)at)o( ;.0.


a 9X.;2 _ : _ 3.625

b 9 : .115

: )s a (or-ally *)str)b,te* ra(*o- 4ar)able w)th -ea( 10 a(* sta(*ar* *e4)at)o( 1.

Use )g,re 12.2 Q%,-,lat)4e Nor-al Probab)l)tyQ to +(* the +rst probab)l)ty l)ste*.

)(* the seco(* probab)l)ty ,s)(g the sy--etry of the *e(s)ty c,r4e. Sketch the


a 9 : _ 125 9 : 165

b 9 : _ 15 9 : 15

c 9 : _ 11.205 9 : 16.05

* 9 : _ 12.5 9 : 1.335

6 : )s a (or-ally *)str)b,te* ra(*o- 4ar)able w)th -ea( 1;; a(* sta(*ar* *e4)at)o(

1;. Use )g,re 12.2 Q%,-,lat)4e Nor-al Probab)l)tyQ to +(* the +rst probab)l)ty l)ste*.






)(* the seco(* probab)l)ty ,s)(g the sy--etry of the *e(s)ty c,r4e. Sketch the


a 9 : _ 6;5 9 : 12;5

b 9 : _ 05 9 : 1205

c 9 : _ 6.005 9 : 110.05

* 9 : _ .25 9 : 122.065

9 : )s a (or-ally *)str)b,te* ra(*o- 4ar)able w)th -ea( a(* sta(*ar* *e4)at)o( 13.

&he probab)l)ty that : takes a 4al,e )( the ,()o( of )(ter4als (−∞,67−a] [67+a,∞)w)ll be

*e(ote* P(X≤67−a orX≥67+a).Use )g,re 12.2 Q%,-,lat)4e Nor-al Probab)l)tyQ to +(*

the follow)(g probab)l)t)es of th)s type. Sketch the *e(s)ty c,r4e w)th rele4a(t reg)o(s

sha*e* to )ll,strate the co-p,tat)o(. #eca,se of the sy--etry of the *e(s)ty c,r4e

yo, (ee* to ,se )g,re 12.2 Q%,-,lat)4e Nor-al Probab)l)tyQ o(ly o(e t)-e for eachpart.

a P(X<57 orX>77)

b P(X<47 orX>87)

c P(X<49 orX>85)

d P(X<37 orX>97)

1; : )s a (or-ally *)str)b,te* ra(*o- 4ar)able w)th -ea( 266 a(* sta(*ar* *e4)at)o( .

&he probab)l)ty that : takes a 4al,e )( the ,()o( of )(ter4als (−∞,288−a] [288+a,∞)w)ll

be *e(ote* P(X≤288−a orX≥288+a).Use )g,re 12.2 Q%,-,lat)4e Nor-al Probab)l)tyQ to

+(* the follow)(g probab)l)t)es of th)s type. Sketch the *e(s)ty c,r4e w)th rele4a(t

reg)o(s sha*e* to )ll,strate the co-p,tat)o(. #eca,se of the sy--etry of the *e(s)ty

c,r4e yo, (ee* to ,se )g,re 12.2 Q%,-,lat)4e Nor-al Probab)l)tyQ o(ly o(e t)-e for

each part.

a P(X<278 orX>298)

b P(X<268 orX>308)

c P(X<273 orX>303)

d P(X<280 orX>296)

A22!&CAT&1NS






11 &he a-o,(t : of be4erage )( a ca( labele* 12 o,(ces )s (or-ally *)str)b,te* w)th -ea(

12.1 o,(ces a(* sta(*ar* *e4)at)o( ;.;0 o,(ce. A ca( )s selecte* at ra(*o-.

a )(* the probab)l)ty that the ca( co(ta)(s at least 12 o,(ces.

b )(* the probab)l)ty that the ca( co(ta)(s betwee( 11.9 a(* 12.1 o,(ces.

12 &he le(gth of gestat)o( for sw)(e )s (or-ally *)str)b,te* w)th -ea( 11 *ays a(*sta(*ar* *e4)at)o( ;.0 *ay. )(* the probab)l)ty that a l)tter w)ll be bor( w)th)( o(e *ay

of the -ea( of 11.

13 &he systol)c bloo* press,re : of a*,lts )( a reg)o( )s (or-ally *)str)b,te* w)th -ea( 112

-- @g a(* sta(*ar* *e4)at)o( 10 -- @g. A perso( )s co(s)*ere* prehyperte(s)4e )f

h)s systol)c bloo* press,re )s betwee( 12; a(* 13; -- @g. )(* the probab)l)ty that the

bloo* press,re of a ra(*o-ly selecte* perso( )s prehyperte(s)4e.

1 @e)ghts : of a*,lt wo-e( are (or-ally *)str)b,te* w)th -ea( 3. )(ches a(* sta(*ar*

*e4)at)o( 2.1 )(ches. Ro-eo who )s 9.20 )(ches tall w)shes to *ate o(ly wo-e( who

are shorter tha( he b,t w)th)( )(ches of h)s he)ght. )(* the probab)l)ty that the (e7t

wo-a( he -eets w)ll ha4e s,ch a he)ght.

10 @e)ghts : of a*,lt -e( are (or-ally *)str)b,te* w)th -ea( 9.1 )(ches a(* sta(*ar*

*e4)at)o( 2.92 )(ches. $,l)et who )s 3.20 )(ches tall w)shes to *ate o(ly -e( who are

taller tha( she b,t w)th)( )(ches of her he)ght. )(* the probab)l)ty that the (e7t -a(

she -eets w)ll ha4e s,ch a he)ght.

1 A reg,lat)o( hockey p,ck -,st we)gh betwee( 0.0 a(* o,(ces. &he we)ghts : of

p,cks -a*e by a part)c,lar process are (or-ally *)str)b,te* w)th -ea( 0.0 o,(ces

a(* sta(*ar* *e4)at)o( ;.11 o,(ce. )(* the probab)l)ty that a p,ck -a*e by th)s

process w)ll -eet the we)ght sta(*ar*.

1 A reg,lat)o( golf ball -ay (ot we)gh -ore tha( 1.2; o,(ces. &he we)ghts : of golf

balls -a*e by a part)c,lar process are (or-ally *)str)b,te* w)th -ea( 1.31 o,(ces

a(* sta(*ar* *e4)at)o( ;.;9 o,(ce. )(* the probab)l)ty that a golf ball -a*e by th)s

process w)ll -eet the we)ght sta(*ar*.

16 &he le(gth of t)-e that the battery )( @)ppolytas cell pho(e w)ll hol* e(o,gh charge

to operate acceptably )s (or-ally *)str)b,te* w)th -ea( 20. ho,rs a(* sta(*ar*

*e4)at)o( ;.32 ho,r. @)ppolyta forgot to charge her pho(e yester*ay so that at the

-o-e(t she +rst w)shes to ,se )t to*ay )t has bee( 2 ho,rs 16 -)(,tes s)(ce the






pho(e was last f,lly charge*. )(* the probab)l)ty that the pho(e w)ll operate

properly.

19 &he a-o,(t of (o(=-ortgage *ebt per ho,sehol* for ho,sehol*s )( a part)c,lar

)(co-e bracket )( o(e part of the co,(try )s (or-ally *)str)b,te* w)th -ea( 2630;

a(* sta(*ar* *e4)at)o( 320. )(* the probab)l)ty that a ra(*o-ly selecte* s,ch

ho,sehol* has betwee( 2;;;; a(* 3;;;; )( (o(=-ortgage *ebt.

2; #)rth we)ghts of f,ll=ter- bab)es )( a certa)( reg)o( are (or-ally *)str)b,te* w)th

-ea( .120 lb a(* sta(*ar* *e4)at)o( 1.29; lb. )(* the probab)l)ty that a ra(*o-ly

selecte* (ewbor( w)ll we)gh less tha( 0.0 lb the h)stor)c *e+()t)o( of pre-at,r)ty.

21 &he *)sta(ce fro- the seat back to the fro(t of the k(ees of seate* a*,lt -ales )s

(or-ally *)str)b,te* w)th -ea( 23.6 )(ches a(* sta(*ar* *e4)at)o( 1.22 )(ches. &he

*)sta(ce fro- the seat back to the back of the (e7t seat forwar* )( all seats o(

a)rcraft [ow( by a b,*get a)rl)(e )s 2 )(ches. )(* the proport)o( of a*,lt -e( [y)(g

w)th th)s a)rl)(e whose k(ees w)ll to,ch the back of the seat )( fro(t of the-.

22 &he *)sta(ce fro- the seat to the top of the hea* of seate* a*,lt -ales )s (or-ally

*)str)b,te* w)th -ea( 3.0 )(ches a(* sta(*ar* *e4)at)o( 1.39 )(ches. &he *)sta(ce

fro- the seat to the roof of a part)c,lar -ake a(* -o*el car )s ;.0 )(ches. )(* the

proport)o( of a*,lt -e( who whe( s)tt)(g )( th)s car w)ll ha4e at least o(e )(ch of

hea*roo- *)sta(ce fro- the top of the hea* to the roof5.

A&"NAL EKER%SES

23 &he ,sef,l l)fe of a part)c,lar -ake a(* type of a,to-ot)4e t)re )s (or-ally *)str)b,te*

w)th -ea( 00;; -)les a(* sta(*ar* *e4)at)o( 90; -)les.

a )(* the probab)l)ty that s,ch a t)re w)ll ha4e a ,sef,l l)fe of betwee( 0;;;

a(* 06;;; -)les.

b @a-let b,ys fo,r s,ch t)res. Ass,-)(g that the)r l)fet)-es are )(*epe(*e(t

+(* the probab)l)ty that all fo,r w)ll last betwee( 0;;; a(* 06;;; -)les. f

so the best t)re w)ll ha4e (o -ore tha( 1;;; -)les left o( )t whe( the +rst t)re






fa)ls.5 @)(t: &here )s a b)(o-)al ra(*o- 4ar)able here whose 4al,e of p co-es

fro- part a5.

2 A -ach)(e pro*,ces large faste(ers whose le(gth -,st be w)th)( ;.0 )(ch of 22 )(ches.

&he le(gths are (or-ally *)str)b,te* w)th -ea( 22.; )(ches a(* sta(*ar* *e4)at)o( ;.1

)(ch.

a )(* the probab)l)ty that a ra(*o-ly selecte* faste(er pro*,ce* by the

-ach)(e w)ll ha4e a( acceptable le(gth.

b &he -ach)(e pro*,ces 2; faste(ers per ho,r. &he le(gth of each o(e )s

)(specte*. Ass,-)(g le(gths of faste(ers are )(*epe(*e(t +(* the probab)l)ty

that all 2; w)ll ha4e acceptable le(gth. @)(t: &here )s a b)(o-)al ra(*o-

4ar)able here whose 4al,e of p co-es fro- part a5.

20 &he le(gths of t)-e take( by st,*e(ts o( a( algebra pro+c)e(cy e7a- )f (ot force* to

stop before co-plet)(g )t5 are (or-ally *)str)b,te* w)th -ea( 26 -)(,tes a(* sta(*ar*

*e4)at)o( 1.0 -)(,tes.

a )(* the proport)o( of st,*e(ts who w)ll +()sh the e7a- )f a 3;=-)(,te t)-e

l)-)t )s set.

b S)7 st,*e(ts are tak)(g the e7a- to*ay. )(* the probab)l)ty that all s)7 w)ll

+()sh the e7a- w)th)( the 3;=-)(,te l)-)t ass,-)(g that t)-es take( by

st,*e(ts are )(*epe(*e(t. @)(t: &here )s a b)(o-)al ra(*o- 4ar)able here

whose 4al,e of p co-es fro- part a5.

2 @e)ghts of a*,lt -e( betwee( 16 a(* 3 years of age are (or-ally *)str)b,te* w)th -ea(

9.1 )(ches a(* sta(*ar* *e4)at)o( 2.92 )(ches. "(e re<,)re-e(t for e(l)st-e(t )( the

-)l)tary )s that -e( -,st sta(* betwee( ; a(* 6; )(ches tall.

a )(* the probab)l)ty that a ra(*o-ly electe* -a( -eets the he)ght

re<,)re-e(t for -)l)tary ser4)ce.

b &we(ty=three -e( )(*epe(*e(tly co(tact a recr,)ter th)s week. )(* the

probab)l)ty that all of the- -eet the he)ght re<,)re-e(t. @)(t: &here )s a

b)(o-)al ra(*o- 4ar)able here whose 4al,e of p co-es fro- part a5.

2 A reg,lat)o( hockey p,ck -,st we)gh betwee( 0.0 a(* o,(ces. ( a( alter(at)4e

-a(,fact,r)(g process the -ea( we)ght of p,cks pro*,ce* )s 0.0 o,(ce. &he we)ghts of

p,cks ha4e a (or-al *)str)b,t)o( whose sta(*ar* *e4)at)o( ca( be *ecrease* by






)(creas)(gly str)(ge(t a(* e7pe(s)4e5 co(trols o( the -a(,fact,r)(g process. )(* the

-a7)-,- allowable sta(*ar* *e4)at)o( so that at -ost ;.;;0 of all p,cks w)ll fa)l to -eet

the we)ght sta(*ar*. @)(t: &he *)str)b,t)o( )s sy--etr)c a(* )s ce(tere* at the -)**le of

the )(ter4al of acceptable we)ghts.5

26 &he a-o,(t of gasol)(e : *el)4ere* by a -etere* p,-p whe( )t reg)sters 0 gallo(s )s a

(or-ally *)str)b,te* ra(*o- 4ar)able. &he sta(*ar* *e4)at)o( σ of : -eas,res the

prec)s)o( of the p,-pD the s-aller σ )s the s-aller the 4ar)at)o( fro- *el)4ery to *el)4ery.

A typ)cal sta(*ar* for p,-ps )s that whe( they show that 0 gallo(s of f,el has bee(

*el)4ere* the act,al a-o,(t -,st be betwee( .9 a(* 0.;3 gallo(s wh)ch correspo(*s

to be)(g o8 by at -ost abo,t half a c,p5. S,ppos)(g that the -ea( of : )s 0 +(* the

largest that σ ca( be so that 9.9 _ : _ 0.;35 )s 1.;;;; to fo,r *ec)-al places whe(

co-p,te* ,s)(g )g,re 12.2 Q%,-,lat)4e Nor-al Probab)l)tyQ wh)ch -ea(s that the

p,-p )s s,?c)e(tly acc,rate. @)(t: &he z =score of 0.;3 w)ll be the s-allest 4al,e of C sothat )g,re 12.2 Q%,-,lat)4e Nor-al Probab)l)tyQ g)4es P(Z<z)=1.0000.E











8.7 Areas o Tails o (istributions

!+A/N&N: 1';+CT&<+

1 &o lear( how to +(* for a (or-al ra(*o- 4ar)able : a(* a( area a the

4al,e x*of : so that P(X<x*)=aor that P(X>x*)=a wh)che4er )s re<,)re*.

e+()t)o(

The left tail of a density curve y=f(x)of a continuous random variable Bcut off by a

value x*of B is the region under the curve that is to the left of x*$ as shown by the shading

in !igure ."4 G-ight and 'eft Tails of a (istributionG DaE. The right tail cut off by x*is defined

similarly, as indicated by the shading in !igure ."4 G-ight and 'eft Tails of a (istributionG DbE.

!igure ."4 -ight and 'eft Tails of a (istribution






The probabilities tabulated in /igure 1!.! 07umulative 9ormal robability0 are areas of left tails in

the standard normal distribution.

Tails o the Standard Normal (istribution

%t times it is important to be able to solve the kind of problem illustrated by /igure 8.!5. (e have a

certain specific area in mind$ in this case the area 5.51!8 of the shaded region in the figure$ and we want

to find the valuez*of C that produces it. This is exactly the reverse of the kind of problems encountered so

far. ,nstead of knowing a value z*of C and finding a corresponding area$ we know the area and want to

findz*.,n the case at hand$ in the terminology of the definition #ust above$ we wish to find the

valuez*that cuts off a left tail of area 5.51!8 in the standard normal distribution.

The idea for solving such a problem is fairly simple$ although sometimes its implementation can be a bit

complicated. ,n a nutshell$ one reads the cumulative probability table for C in reverse$ looking up the

relevant area in the interior of the table and reading off the value of C from the margins.

!igure .&5 C 0alue that $roduces a Jnown Area






EKAPLE 12

)(* the 4al,e z*of C as *eter-)(e* by )g,re 0.2;: the 4al,e z*that c,ts o8 a left

ta)l of area ;.;120 )( the sta(*ar* (or-al *)str)b,t)o(. ( sy-bols +(* the

(,-ber z*s,ch that P(Z<z*)=0.0125.

Sol,t)o(:

&he (,-ber that )s k(ow( ;.;120 )s the area of a left ta)l a(* as alrea*y

-e(t)o(e* the probab)l)t)es tab,late* )( )g,re 12.2 Q%,-,lat)4e Nor-al

Probab)l)tyQ are areas of left ta)ls. &h,s to sol4e th)s proble- we (ee* o(ly search

)( the )(ter)or of )g,re 12.2 Q%,-,lat)4e Nor-al Probab)l)tyQ for the (,-ber

;.;120. t l)es )( the row w)th the hea*)(g X2.2 a(* )( the col,-( w)th the

hea*)(g ;.;. &h)s -ea(s that 9 C _ X2.25 ;.;120 he(ce z*=−2.24.

+A>2!+ %3

)(* the 4al,e z*of C as *eter-)(e* by )g,re 0.21: the 4al,e z*that c,ts o8 a

r)ght ta)l of area ;.;20; )( the sta(*ar* (or-al *)str)b,t)o(. ( sy-bols +(* the

(,-ber z*s,ch that P(Z>z*)=0.0250.

Fiigure ;.21 C <alue t!at 9roduces a @no&n 0rea






Sol,t)o(:

&he )-porta(t *)st)(ct)o( betwee( th)s e7a-ple a(* the pre4)o,s o(e )s that here

)t )s the area of a rig!t ta)l that )s k(ow(. ( or*er to be able to ,se )g,re 12.2

Q%,-,lat)4e Nor-al Probab)l)tyQ we -,st +rst +(* that area of the left ta)l c,t o8

by the ,(k(ow( (,-ber z*.S)(ce the total area ,(*er the *e(s)ty c,r4e )s 1 that

area )s 1−0.0250=0.9750. &h)s )s the (,-ber we look for )( the )(ter)or of )g,re 12.2

Q%,-,lat)4e Nor-al Probab)l)tyQ. t l)es )( the row w)th the hea*)(g 1.9 a(* )( the

col,-( w)th the hea*)(g ;.;. &herefore z*=1.96.

e+()t)o(






The value of the standard normal random variable C that cuts off a right tail of area c is denoted z c. 9y

symmetry, value of C that cuts off a left tail of area c is −zc. %ee !igure .&& GThe Iumbers G .

!igure .&&The Iumbers z c and −zc

+A>2!+ %7

)(* z.01a(* −z.01 the 4al,es of C that c,t o8 r)ght a(* left ta)ls of area ;.;1 )( the

sta(*ar* (or-al *)str)b,t)o(.

Sol,t)o(:

S)(ce −z.01c,ts o8 a left ta)l of area ;.;1 a(* )g,re 12.2 Q%,-,lat)4e Nor-al

Probab)l)tyQ )s a table of left ta)ls we look for the (,-ber ;.;1;; )( the )(ter)or of

the table. t )s (ot there b,t falls betwee( the two (,-bers ;.;1;2 a(* ;.;;99 )(

the row w)th hea*)(g X2.3. &he (,-ber ;.;;99 )s closer to ;.;1;; tha( ;.;1;2 )s

so for the h,(*re*ths place )( −z.01we ,se the hea*)(g of the col,-( that co(ta)(s

;.;;99 (a-ely ;.;3 a(* wr)te −z.01≈−2.33.

&he a(swer to the seco(* half of the proble- )s a,to-at)c: s)(ce −z.01=−2.33 we

co(cl,*e )--e*)ately that z.01=2.33.






Be co,l* >,st as well ha4e sol4e* th)s proble- by look)(g for z.01+rst a(* )t )s

)(str,ct)4e to rework the proble- th)s way. &o beg)( w)th we -,st +rst s,btract

;.;1 fro- 1 to +(* the area 1−0.0100=0.9900of the left ta)l c,t o8 by the ,(k(ow(

(,-ber z.01.See )g,re 0.23 Q%o-p,tat)o( of the N,-ber Q. &he( we search for

the area ;.99;; )( )g,re 12.2 Q%,-,lat)4e Nor-al Probab)l)tyQ. t )s (ot there b,t

falls betwee( the (,-bers ;.9696 a(* ;.99;1 )( the row w)th hea*)(g 2.3. S)(ce

;.99;1 )s closer to ;.99;; tha( ;.9696 )s we ,se the col,-( hea*)(g abo4e )t

;.;3 to obta)( the appro7)-at)o( z.01≈2.33. &he( +(ally −z.01≈−2.33.

Figure ;.2*,omputation of t!e 'umber z.01

Tails o :eneral Normal (istributions

The problem of finding the valuex*of a general normally distributed random variable B that cuts off

a tail of a specified area also arises. This problem may be solved in two steps.

+uppose B is a normally distributed random variable with mean and standard deviation 6 . To find the

valuex*of B that cuts off a left or right tail of area c in the distribution of B :






1 find the valuez*of C that cuts off a left or right tail of area c in the standard normal distribution

! z*is the z -score ofx*0 computex*using the destandardi'ation formula

x*=µ+z*σ

+A>2!+ %8

)(* x*s,ch that P(X<x*)=0.9332@ where : )s a (or-al ra(*o- 4ar)able w)th

-ea( μ 1; a(* sta(*ar* *e4)at)o( σ 2.0.

Sol,t)o(:

All the )*eas for the sol,t)o( are )ll,strate* )( )g,re 0.2 Q&a)l of a Nor-ally

)str)b,te* Ra(*o- 'ar)ableQ. S)(ce ;.9332 )s the area of a left ta)l we ca(

+(* z*s)-ply by look)(g for ;.9332 )( the )(ter)or of )g,re 12.2 Q%,-,lat)4e

Nor-al Probab)l)tyQ. t )s )( the row a(* col,-( w)th hea*)(gs 1.0 a(* ;.;;

he(ce z*=1.50. &h,s x*)s 1.0; sta(*ar* *e4)at)o(s abo4e the -ea( so

x*=µ+z*σ=10+1.50D2.5=13.75.

Figure ;.28)ail of a 'ormall/ Distributed 6andom <ariable






+A>2!+ %9

)(* x*s,ch that P(X>x*)=0.65@ where : )s a (or-al ra(*o- 4ar)able w)th -ea( μ

10 a(* sta(*ar* *e4)at)o( σ 12.

Sol,t)o(:

&he s)t,at)o( )s )ll,strate* )( )g,re 0.20 Q&a)l of a Nor-ally )str)b,te* Ra(*o-

'ar)ableQ. S)(ce ;.0 )s the area of a r)ght ta)l we +rst s,btract )t fro- 1 to

obta)( 1−0.65=0.35 the area of the co-ple-e(tary left ta)l. Be +(* z*by look)(g for

;.30;; )( the )(ter)or of )g,re 12.2 Q%,-,lat)4e Nor-al Probab)l)tyQ. t )s (ot

prese(t b,t l)es betwee( table e(tr)es ;.302; a(* ;.363. &he e(try ;.363 w)th

row a(* col,-( hea*)(gs X;.3 a(* ;.;9 )s closer to ;.30;; tha( the other e(try

)s so z*≈−0.39. &h,s x*)s ;.39 sta(*ar* *e4)at)o(s below the -ea( so

x*=µ+z*σ=175+(−0.39)D12=170.32

Figure ;.2;)ail of a 'ormall/ Distributed 6andom <ariable






+A>2!+ %4

Scores o( a sta(*ar*)Ve* college e(tra(ce e7a-)(at)o( ,((5 are (or-ally

*)str)b,te* w)th -ea( 01; a(* sta(*ar* *e4)at)o( ;. A select)4e ,()4ers)ty

*ec)*es to g)4e ser)o,s co(s)*erat)o( for a*-)ss)o( to appl)ca(ts whose ,((

scores are )( the top 0G of all ,(( scores. )(* the -)()-,- score that -eets th)s

cr)ter)o( for ser)o,s co(s)*erat)o( for a*-)ss)o(.

Sol,t)o(:

Let : *e(ote the score -a*e o( the ,(( by a ra(*o-ly selecte* )(*)4)*,al.

&he( : )s (or-ally *)str)b,te* w)th -ea( 01; a(* sta(*ar* *e4)at)o( ;. &he

probab)l)ty that : l)e )( a part)c,lar )(ter4al )s the sa-e as the proport)o( of all

e7a- scores that l)e )( that )(ter4al. &h,s the -)()-,- score that )s )( the top

0G of all ,(( )s the score x*that c,ts o8 a r)ght ta)l )( the *)str)b,t)o( of : of area

;.;0 0G e7presse* as a proport)o(5. See )g,re 0.2 Q&a)l of a Nor-ally

)str)b,te* Ra(*o- 'ar)ableQ.

Figure ;.25)ail of a 'ormall/ Distributed 6andom <ariable

S)(ce ;.;0;; )s the area of a r)ght ta)l we +rst s,btract )t fro- 1 to

obta)( 1−0.0500=0.9500 the area of the co-ple-e(tary left ta)l. Be +(* z*=z.05by

look)(g for ;.90;; )( the )(ter)or of )g,re 12.2 Q%,-,lat)4e Nor-al Probab)l)tyQ. t






)s (ot prese(t a(* l)es e7actly half=way betwee( the two (earest e(tr)es that are

;.990 a(* ;.90;0. ( the case of a t)e l)ke th)s we w)ll always a4erage the

4al,es of C correspo(*)(g to the two table e(tr)es obta)()(g here the

4al,e z*=1.645.Us)(g th)s 4al,e we co(cl,*e that x*)s 1.0 sta(*ar* *e4)at)o(s

abo4e the -ea( so

x*=µ+z*σ=510+1.645D60=608.7

+A>2!+ %5

All boys at a -)l)tary school -,st r,( a +7e* co,rse as fast as they ca( as part of

a phys)cal e7a-)(at)o(. )()sh)(g t)-es are (or-ally *)str)b,te* w)th -ea( 29

-)(,tes a(* sta(*ar* *e4)at)o( 2 -)(,tes. &he -)**le 0G of all +()sh)(g t)-es

are class)+e* as a4erage. )(* the ra(ge of t)-es that are a4erage +()sh)(gt)-es by th)s *e+()t)o(.

Sol,t)o(:

Let : *e(ote the +()sh t)-e of a ra(*o-ly selecte* boy. &he( : )s (or-ally

*)str)b,te* w)th -ea( 29 a(* sta(*ar* *e4)at)o( 2. &he probab)l)ty that : l)e )( a

part)c,lar )(ter4al )s the sa-e as the proport)o( of all +()sh t)-es that l)e )( that

)(ter4al. &h,s the s)t,at)o( )s as show( )( )g,re 0.2 Q)str)b,t)o( of &)-es to

R,( a %o,rseQ. #eca,se the area )( the -)**le correspo(*)(g to a4erage t)-es

)s ;.0 the areas of the two ta)ls a** ,p to 1 X ;.0 ;.20 )( all. #y the

sy--etry of the *e(s)ty c,r4e each ta)l -,st ha4e half of th)s total or area ;.120

each. &h,s the fastest t)-e that )s a4erage has z =score −z.125 wh)ch by )g,re

12.2 Q%,-,lat)4e Nor-al Probab)l)tyQ )s X1.10 a(* the slowest t)-e that )s

a4erage has z =score z.125=1.15. &he fastest a(* slowest t)-es that are st)ll

co(s)*ere* a4erage are

x fast=µ+(−z.125)σ=29+(−1.15)D2=26.7

a(*

x slow=µ+z.125σ=29+(1.15)D2=31.3

Figure ;.27Distribution of )imes to 6un a ,ourse






A boy has a( a4erage +()sh)(g t)-e )f he r,(s the co,rse w)th a t)-e betwee(

2. a(* 31.3 -)(,tes or e<,)4ale(tly betwee( 2 -)(,tes 2 seco(*s a(* 31

-)(,tes 16 seco(*s.

*+, TA*+AA,S

• &he proble- of +(*)(g the (,-ber z*so that the probab)l)ty P(Z<z*))s a spec)+e*

4al,e c )s sol4e* by look)(g for the (,-ber c )( the )(ter)or of )g,re 12.2

Q%,-,lat)4e Nor-al Probab)l)tyQ a(* rea*)(g z*fro- the -arg)(s.

• &he proble- of +(*)(g the (,-ber z*so that the probab)l)ty P(Z>z*))s a spec)+e*

4al,e c )s sol4e* by look)(g for the co-ple-e(tary probab)l)ty 1−c)( the )(ter)or

of )g,re 12.2 Q%,-,lat)4e Nor-al Probab)l)tyQ a(* rea*)(g z*fro- the -arg)(s.

• or a (or-al ra(*o- 4ar)able : w)th -ea( μ a(* sta(*ar* *e4)at)o( σ the

proble- of +(*)(g the (,-ber x*so that P(X<x*))s a spec)+e* 4al,e c or so

that P(X>x*))s a spec)+e* 4al,e c5 )s sol4e* )( two steps: 15 sol4e the

correspo(*)(g proble- for C w)th the sa-e 4al,e of c thereby obta)()(g the z =

score z* of x*D 25 +(* x*,s)(g x*=µ+z*Yσ.

• &he 4al,e of C that c,ts o8 a r)ght ta)l of area c )( the sta(*ar* (or-al

*)str)b,t)o( )s *e(ote* z c.











9 : )s a (or-ally *)str)b,te* ra(*o- 4ar)able : w)th -ea( 10 a(* sta(*ar* *e4)at)o(

;.20. )(* the 4al,es L a(* 6 of : that are sy--etr)cally locate* w)th respect to the

-ea( of : a(* sat)sfy 9 L _ : _ 65 ;.6;. @)(t. )rst sol4e the correspo(*)(g

proble- for C .5

1; : )s a (or-ally *)str)b,te* ra(*o- 4ar)able : w)th -ea( 26 a(* sta(*ar* *e4)at)o( 3..

)(* the 4al,es L a(* 6 of : that are sy--etr)cally locate* w)th respect to the -ea(

of : a(* sat)sfy 9 L _ : _ 65 ;.0. @)(t. )rst sol4e the correspo(*)(g proble-

for C .5

A22!&CAT&1NS

11 Scores o( a (at)o(al e7a- are (or-ally *)str)b,te* w)th -ea( 362 a(* sta(*ar*

*e4)at)o( 2.

a )(* the score that )s the 0;th perce(t)le.

b )(* the score that )s the 9;th perce(t)le.

12 @e)ghts of wo-e( are (or-ally *)str)b,te* w)th -ea( 3. )(ches a(* sta(*ar*

*e4)at)o( 2. )(ches.






a )(* the he)ght that )s the 1;th perce(t)le.

b )(* the he)ght that )s the 6;th perce(t)le.

13 &he -o(thly a-o,(t of water ,se* per ho,sehol* )( a s-all co--,()ty )s (or-ally

*)str)b,te* w)th -ea( ;9 gallo(s a(* sta(*ar* *e4)at)o( 06 gallo(s. )(* the three

<,art)les for the a-o,(t of water ,se*.1 &he <,a(t)ty of gasol)(e p,rchase* )( a s)(gle sale at a cha)( of +ll)(g stat)o(s )( a

certa)( reg)o( )s (or-ally *)str)b,te* w)th -ea( 11. gallo(s a(* sta(*ar* *e4)at)o( 2.6

gallo(s. )(* the three <,art)les for the <,a(t)ty of gasol)(e p,rchase* )( a s)(gle sale.

10 Scores o( the co--o( +(al e7a- g)4e( )( a large e(roll-e(t -,lt)ple sect)o( co,rse

were (or-ally *)str)b,te* w)th -ea( 9.30 a(* sta(*ar* *e4)at)o( 12.93. &he

*epart-e(t has the r,le that )( or*er to rece)4e a( A )( the co,rse h)s score -,st be )(

the top 1;G of all e7a- scores. )(* the -)()-,- e7a- score that -eets th)s

re<,)re-e(t.

1 &he a4erage +()sh)(g t)-e a-o(g all h)gh school boys )( a part)c,lar track e4e(t )( a

certa)( state )s 0 -)(,tes 1 seco(*s. &)-es are (or-ally *)str)b,te* w)th sta(*ar*

*e4)at)o( 12 seco(*s.

a &he <,al)fy)(g t)-e )( th)s e4e(t for part)c)pat)o( )( the state -eet )s to be set

so that o(ly the fastest 0G of all r,((ers <,al)fy. )(* the <,al)fy)(g t)-e.

@)(t: %o(4ert seco(*s to -)(,tes.5

b ( the wester( reg)o( of the state the t)-es of all boys r,(()(g )( th)s e4e(t

are (or-ally *)str)b,te* w)th sta(*ar* *e4)at)o( 12 seco(*s b,t w)th -ea( 0

-)(,tes 22 seco(*s. )(* the proport)o( of boys fro- th)s reg)o( who <,al)fy

to r,( )( th)s e4e(t )( the state -eet.

1 &ests of a (ew t)re *e4elope* by a t)re -a(,fact,rer le* to a( est)-ate* -ea( trea*

l)fe of 30; -)les a(* sta(*ar* *e4)at)o( of 112; -)les. &he -a(,fact,rer w)ll

a*4ert)se the l)fet)-e of the t)re for e7a-ple a 0;;;; -)le t)re5 ,s)(g the largest

4al,e for wh)ch )t )s e7pecte* that 96G of the t)res w)ll last at least that lo(g.

Ass,-)(g t)re l)fe )s (or-ally *)str)b,te* +(* that a*4ert)se* 4al,e.

16 &ests of a (ew l)ght le* to a( est)-ate* -ea( l)fe of 1321 ho,rs a(* sta(*ar*

*e4)at)o( of 1; ho,rs. &he -a(,fact,rer w)ll a*4ert)se the l)fet)-e of the b,lb ,s)(g

the largest 4al,e for wh)ch )t )s e7pecte* that 9;G of the b,lbs w)ll last at least that

lo(g. Ass,-)(g b,lb l)fe )s (or-ally *)str)b,te* +(* that a*4ert)se* 4al,e.

19 &he we)ghts : of eggs pro*,ce* at a part)c,lar far- are (or-ally *)str)b,te* w)th

-ea( 1.2 o,(ces a(* sta(*ar* *e4)at)o( ;.12 o,(ce. Eggs whose we)ghts l)e )( the

-)**le 0G of the *)str)b,t)o( of we)ghts of all eggs are class)+e* as -e*),-. )(*






the -a7)-,- a(* -)()-,- we)ghts of s,ch eggs. &hese we)ghts are e(*po)(ts of

a( )(ter4al that )s sy--etr)c abo,t the -ea( a(* )( wh)ch the we)ghts of 0G of the

eggs pro*,ce* at th)s far- l)e.5

2; &he le(gths : of har*woo* [oor)(g str)ps are (or-ally *)str)b,te* w)th -ea( 26.9

)(ches a(* sta(*ar* *e4)at)o( .12 )(ches. Str)ps whose le(gths l)e )( the -)**le

6;G of the *)str)b,t)o( of le(gths of all str)ps are class)+e* as a4erage=le(gth

str)ps. )(* the -a7)-,- a(* -)()-,- le(gths of s,ch str)ps. &hese le(gths are

e(*po)(ts of a( )(ter4al that )s sy--etr)c abo,t the -ea( a(* )( wh)ch the le(gths

of 6;G of the har*woo* str)ps l)e.5

21 All st,*e(ts )( a large e(roll-e(t -,lt)ple sect)o( co,rse take co--o( )(=class

e7a-s a(* a co--o( +(al a(* s,b-)t co--o( ho-ework ass)g(-e(ts. %o,rse

gra*es are ass)g(e* base* o( st,*e(ts +(al o4erall scores wh)ch are appro7)-ately

(or-ally *)str)b,te*. &he *epart-e(t ass)g(s a % to st,*e(ts whose scores co(st)t,te

the -)**le 2/3 of all scores. f scores th)s se-ester ha* -ea( 2.0 a(* sta(*ar*

*e4)at)o( .1 +(* the )(ter4al of scores that w)ll be ass)g(e* a %.

22 Researchers w)sh to )(4est)gate the o4erall health of )(*)4)*,als w)th ab(or-ally h)gh

or low le4els of gl,cose )( the bloo* strea-. S,ppose gl,cose le4els are (or-ally

*)str)b,te* w)th -ea( 9 a(* sta(*ar* *e4)at)o( 6.0 -g/* ℓ a(* that (or-al )s

*e+(e* as the -)**le 9;G of the pop,lat)o(. )(* the )(ter4al of (or-al gl,cose

le4els that )s the )(ter4al ce(tere* at 9 that co(ta)(s 9;G of all gl,cose le4els )(the pop,lat)o(.

A((&T&1NA! ++/C&S+S

23 A -ach)(e for +ll)(g 2=l)ter bottles of soft *r)(k *el)4ers a( a-o,(t to each bottle that

4ar)es fro- bottle to bottle accor*)(g to a (or-al *)str)b,t)o( w)th sta(*ar* *e4)at)o(

;.;;2 l)ter a(* -ea( whate4er a-o,(t the -ach)(e )s set to *el)4er.

a f the -ach)(e )s set to *el)4er 2 l)ters so the -ea( a-o,(t *el)4ere* )s 2

l)ters5 what proport)o( of the bottles w)ll co(ta)( at least 2 l)ters of soft *r)(kC

b )(* the -)()-,- sett)(g of the -ea( a-o,(t *el)4ere* by the -ach)(e so

that at least 99G of all bottles w)ll co(ta)( at least 2 l)ters.

2 A (,rsery has obser4e* that the -ea( (,-ber of *ays )t -,st *arke( the e(4)ro(-e(t

of a spec)es po)(sett)a pla(t *a)ly )( or*er to ha4e )t rea*y for -arket )s 1 *ays.

S,ppose the le(gths of s,ch per)o*s of *arke()(g are (or-ally *)str)b,te* w)th sta(*ar*

*e4)at)o( 2 *ays. )(* the (,-ber of *ays )( a*4a(ce of the pro>ecte* *el)4ery *ates of






the pla(ts to -arket that the (,rsery -,st beg)( the *a)ly *arke()(g process )( or*er

that at least 90G of the pla(ts w)ll be rea*y o( t)-e. Po)(sett)as are so lo(g=l)4e* that

o(ce rea*y for -arket the pla(t re-a)(s salable )(*e+()tely.5






Chapter 9

Samplin$ (istributions

% statistic$ such as the sample mean or the sample standard deviation$ is a number computed from asample. +ince a sample is random$ every statistic is a random variable: it varies from sample to

sample in a way that cannot be predicted with certainty. %s a random variable it has a mean$ a

standard deviation$ and a probability distribution. The probability distribution of a statistic is called

itssampling distribution. Typically sample statistics are not ends in themselves$ but are computed in

order to estimate the corresponding population parameters$ as illustrated in the grand picture of

statistics presented in /igure 1.1 0The rand icture of +tatistics0 in 7hapter 1 0,ntroduction0.

This chapter introduces the concepts of the mean$ the standard deviation$ and the sampling

distribution of a sample statistic$ with an emphasis on the sample mean x −.

9.% The >ean and Standard (eviation o the Sample >ean

LEARNN! "#$E%&'ES

1 &o beco-e fa-)l)ar w)th the co(cept of the probab)l)ty *)str)b,t)o( of the sa-ple

-ea(.

2 &o ,(*ersta(* the -ea()(g of the for-,las for the -ea( a(* sta(*ar* *e4)at)o(

of the sa-ple -ea(.

+uppose we wish to estimate the mean of a population. ,n actual practice we would typically take

#ust one sample. ,magine however that we take sample after sample$ all of the same si'e n$ and

compute the sample mean x −of each one. (e will likely get a different value of x −each time. The

sample mean x −is a random variable: it varies from sample to sample in a way that cannot be

predicted with certainty. (e will write X −− when the sample mean is thought of as a random

variable$ and write x −for the values that it takes. The random variable X −−has a mean$






denotedµX −−$ and a standard deviation$ denotedσX −−.Eere is an example with such a small

population and small sample si'e that we can actually write down every single sample.

EKAPLE 1

A row)(g tea- co(s)sts of fo,r rowers who we)gh 102 10 1; a(* 1 po,(*s.

)(* all poss)ble ra(*o- sa-ples w)th replace-e(t of s)Ve two a(* co-p,te the

sa-ple -ea( for each o(e. Use the- to +(* the probab)l)ty *)str)b,t)o( the

-ea( a(* the sta(*ar* *e4)at)o( of the sa-ple -ea( X −−.

Sol,t)o(

&he follow)(g table shows all poss)ble sa-ples w)th replace-e(t of s)Ve two

alo(g w)th the -ea( of each:

Sample >ean Sample >ean Sample >ean Sample >ean

102 102 102 10 102 10 1; 102 10 1 102 106

102 10 10 10 10 10 1; 10 106 1 10 1;

102 1; 10 10 1; 106 1; 1; 1; 1 1; 12

102 1 106 10 1 1; 1; 1 12 1 1 1
















*+, TA*+AA,S

• &he sa-ple -ea( )s a ra(*o- 4ar)ableD as s,ch )t )s wr)tte( X−− a(* x− sta(*s for

)(*)4)*,al 4al,es )t takes.

• As a ra(*o- 4ar)able the sa-ple -ea( has a probab)l)ty *)str)b,t)o( a -ea( µX−−

a(* a sta(*ar* *e4)at)o( σX−−.

• &here are for-,las that relate the -ea( a(* sta(*ar* *e4)at)o( of the sa-ple

-ea( to the -ea( a(* sta(*ar* *e4)at)o( of the pop,lat)o( fro- wh)ch the

sa-ple )s *raw(.

++/C&S+S

1 Ra(*o- sa-ples of s)Ve 220 are *raw( fro- a pop,lat)o( w)th -ea( 1;; a(* sta(*ar*

*e4)at)o( 2;. )(* the -ea( a(* sta(*ar* *e4)at)o( of the sa-ple -ea(.

2 Ra(*o- sa-ples of s)Ve are *raw( fro- a pop,lat)o( w)th -ea( 32 a(* sta(*ar*

*e4)at)o( 0. )(* the -ea( a(* sta(*ar* *e4)at)o( of the sa-ple -ea(.

3 A pop,lat)o( has -ea( 0 a(* sta(*ar* *e4)at)o( 12.

a Ra(*o- sa-ples of s)Ve 121 are take(. )(* the -ea( a(* sta(*ar* *e4)at)o(

of the sa-ple -ea(.






b @ow wo,l* the a(swers to part a5 cha(ge )f the s)Ve of the sa-ples were ;;

)(stea* of 121C

A pop,lat)o( has -ea( 0.0 a(* sta(*ar* *e4)at)o( 1.;2.

a Ra(*o- sa-ples of s)Ve 61 are take(. )(* the -ea( a(* sta(*ar* *e4)at)o(

of the sa-ple -ea(.b @ow wo,l* the a(swers to part a5 cha(ge )f the s)Ve of the sa-ples were 20

)(stea* of 61C

9.0 The Samplin$ (istribution o the Sample >ean

LEARNN! "#$E%&'ES

1 &o lear( what the sa-pl)(g *)str)b,t)o( of X −− )s whe( the sa-ple s)Ve )s large.

2 &o lear( what the sa-pl)(g *)str)b,t)o( of X −− )s whe( the pop,lat)o( )s (or-al.

The Central !imit Theorem

,n 9ote ;.8 0xample 10 in +ection ;.1 0The ean and +tandard 2eviation of the +ample ean0 we

constructed the probability distribution of the sample mean for samples of si'e two drawn from the

population of four rowers. The probability distribution is:











Eistograms illustrating these distributions are shown in /igure ;.! 02istributions of the +ample

ean0.






!igure /.& (istributions of the %ample 2ean

%s n increases the sampling distribution of X −−evolves in an interesting way: the probabilities on

the lower and the upper ends shrink and the probabilities in the middle become larger in relation to

them. ,f we were to continue to increase nthen the shape of the sampling distribution would become

smoother and more bell-shaped.

(hat we are seeing in these examples does not depend on the particular population distributions

involved. ,n general$ one may start with any distribution and the sampling distribution of the sample

mean will increasingly resemble the bell-shaped normal curve as the sample si'e increases. This is

the content of the 7entral imit Theorem.











The dashed vertical lines in the figures locate the population mean. Cegardless of the distribution of

the population$ as the sample si'e is increased the shape of the sampling distribution of the sample

mean becomes increasingly bell-shaped$ centered on the population mean. Typically by the time the

sample si'e is 35 the distribution of the sample mean is practically the same as a normal distribution.

The importance of the 7entral imit Theorem is that it allows us to make probability statements

about the sample mean$ specifically in relation to its value in comparison to the population mean$ as






we will see in the examples. 4ut to use the result properly we must first reali'e that there are two

separate random variables *and therefore two probability distributions) at play:

1 B $ the measurement of a single element selected at random from the population the distribution

of B is the distribution of the population$ with mean the population mean and standard deviationthe population standard deviation 6

! X−−$ the mean of the measurements in a sample of si'e n the distribution of X−−is its sampling

distribution$ with meanµX−−=µand standard deviationσX−−=σ/n√.











Normally (istributed 2opulations

The 7entral imit Theorem says that no matter what the distribution of the population is$ as long as

the sample is Glarge$H meaning of si'e 35 or more$ the sample mean is approximately normally

distributed. ,f the population is normal to begin with then the sample mean also has a normal

distribution$ regardless of the sample si'e.

/or samples of any si'e drawn from a normally distributed population$ the sample mean is normally

distributed$ with meanµX −−=µand standard deviationσX−−=σ/√n$ where n is the sample si'e.

The effect of increasing the sample si'e is shown in /igure ;.6 02istribution of +ample eans for a

9ormal opulation0.






!igure /. (istribution of %ample 2eans for a Iormal $opulation











*+, TA*+AA,S

• Bhe( the sa-ple s)Ve )s at least 3; the sa-ple -ea( )s (or-ally *)str)b,te*.

• Bhe( the pop,lat)o( )s (or-al the sa-ple -ea( )s (or-ally *)str)b,te*

regar*less of the sa-ple s)Ve.






++/C&S+S

'AS&C


a )(* the -ea( a(* sta(*ar* *e4)at)o( of X−− for sa-ples of s)Ve 3.

b )(* the probab)l)ty that the -ea( of a sa-ple of s)Ve 3 w)ll be w)th)( 1;

,()ts of the pop,lat)o( -ea( that )s betwee( 116 a(* 136.


a )(* the -ea( a(* sta(*ar* *e4)at)o( of X−− for sa-ples of s)Ve 1;;.

b )(* the probab)l)ty that the -ea( of a sa-ple of s)Ve 1;; w)ll be w)th)( 1;;

,()ts of the pop,lat)o( -ea( that )s betwee( 12 a(* 12.

3 A pop,lat)o( has -ea( 3.0 a(* sta(*ar* *e4)at)o( 2.0.

a )(* the -ea( a(* sta(*ar* *e4)at)o( of X−− for sa-ples of s)Ve 3;.

b )(* the probab)l)ty that the -ea( of a sa-ple of s)Ve 3; w)ll be less tha( 2.

A pop,lat)o( has -ea( 6. a(* sta(*ar* *e4)at)o( .3.

a )(* the -ea( a(* sta(*ar* *e4)at)o( of X−− for sa-ples of s)Ve .

b )(* the probab)l)ty that the -ea( of a sa-ple of s)Ve w)ll be less tha(

..

0 A (or-ally *)str)b,te* pop,lat)o( has -ea( 20. a(* sta(*ar* *e4)at)o( 3.3.

a )(* the probab)l)ty that a s)(gle ra(*o-ly selecte* ele-e(t : of the

pop,lat)o( e7cee*s 3;.

b )(* the -ea( a(* sta(*ar* *e4)at)o( of X−− for sa-ples of s)Ve 9.

c )(* the probab)l)ty that the -ea( of a sa-ple of s)Ve 9 *raw( fro- th)s

pop,lat)o( e7cee*s 3;.

A (or-ally *)str)b,te* pop,lat)o( has -ea( 0. a(* sta(*ar* *e4)at)o( 12.1.


pop,lat)o( )s less tha( 0.



pop,lat)o( )s less tha( 0.

A pop,lat)o( has -ea( 00 a(* sta(*ar* *e4)at)o( 30.







b )(* the probab)l)ty that the -ea( of a sa-ple of s)Ve 0; w)ll be -ore tha(

0;.

6 A pop,lat)o( has -ea( 1 a(* sta(*ar* *e4)at)o( 1..


b )(* the probab)l)ty that the -ea( of a sa-ple of s)Ve 6; w)ll be -ore tha(

1..

9 A (or-ally *)str)b,te* pop,lat)o( has -ea( 121 a(* sta(*ar* *e4)at)o( 122.


pop,lat)o( )s betwee( 11;; a(* 13;;.



pop,lat)o( )s betwee( 11;; a(* 13;;.

1; A (or-ally *)str)b,te* pop,lat)o( has -ea( 06;; a(* sta(*ar* *e4)at)o( 0;.


pop,lat)o( )s betwee( 0;;; a(* 06;;;.

b )(* the -ea( a(* sta(*ar* *e4)at)o( of X−− for sa-ples of s)Ve 1;;.

c )(* the probab)l)ty that the -ea( of a sa-ple of s)Ve 1;; *raw( fro- th)s

pop,lat)o( )s betwee( 0;;; a(* 06;;;.

11 A pop,lat)o( has -ea( 2 a(* sta(*ar* *e4)at)o( .

a )(* the -ea( a(* sta(*ar* *e4)at)o( of X−− for sa-ples of s)Ve 0.

b )(* the probab)l)ty that the -ea( of a sa-ple of s)Ve 0 w)ll *)8er fro- the

pop,lat)o( -ea( 2 by at least 2 ,()ts that )s )s e)ther less tha( ; or -ore

tha( . @)(t: "(e way to sol4e the proble- )s to +rst +(* the probab)l)ty of

the co-ple-e(tary e4e(t.5

12 A pop,lat)o( has -ea( 12 a(* sta(*ar* *e4)at)o( 1.0.


b )(* the probab)l)ty that the -ea( of a sa-ple of s)Ve 9; w)ll *)8er fro- the

pop,lat)o( -ea( 12 by at least ;.3 ,()t that )s )s e)ther less tha( 11. or






-ore tha( 12.3. @)(t: "(e way to sol4e the proble- )s to +rst +(* the

probab)l)ty of the co-ple-e(tary e4e(t.5

A22!&CAT&1NS

13 S,ppose the -ea( (,-ber of *ays to ger-)(at)o( of a 4ar)ety of see* )s 22 w)th

sta(*ar* *e4)at)o( 2.3 *ays. )(* the probab)l)ty that the -ea( ger-)(at)o( t)-e of a

sa-ple of 1; see*s w)ll be w)th)( ;.0 *ay of the pop,lat)o( -ea(.

1 S,ppose the -ea( le(gth of t)-e that a caller )s place* o( hol* whe( telepho()(g a

c,sto-er ser4)ce ce(ter )s 23.6 seco(*s w)th sta(*ar* *e4)at)o( . seco(*s. )(* the

probab)l)ty that the -ea( le(gth of t)-e o( hol* )( a sa-ple of 12;; calls w)ll be w)th)(

;.0 seco(* of the pop,lat)o( -ea(.

10 S,ppose the -ea( a-o,(t of cholesterol )( eggs labele* large )s 16 -)ll)gra-s w)th

sta(*ar* *e4)at)o( -)ll)gra-s. )(* the probab)l)ty that the -ea( a-o,(t of

cholesterol )( a sa-ple of 1 eggs w)ll be w)th)( 2 -)ll)gra-s of the pop,lat)o( -ea(.

1 S,ppose that )( o(e reg)o( of the co,(try the -ea( a-o,(t of cre*)t car* *ebt per

ho,sehol* )( ho,sehol*s ha4)(g cre*)t car* *ebt )s 1020; w)th sta(*ar* *e4)at)o(

120. )(* the probab)l)ty that the -ea( a-o,(t of cre*)t car* *ebt )( a sa-ple of

1;; s,ch ho,sehol*s w)ll be w)th)( 3;; of the pop,lat)o( -ea(.

1 S,ppose spee*s of 4eh)cles o( a part)c,lar stretch of roa*way are (or-ally *)str)b,te*

w)th -ea( 3. -ph a(* sta(*ar* *e4)at)o( 1. -ph.

a )(* the probab)l)ty that the spee* : of a ra(*o-ly selecte* 4eh)cle )s

betwee( 30 a(* ; -ph.

b )(* the probab)l)ty that the -ea( spee* X−− of 2; ra(*o-ly selecte* 4eh)cles

)s betwee( 30 a(* ; -ph.

16 a(y sharks e(ter a state of to()c )--ob)l)ty whe( )(4erte*. S,ppose that )( a

part)c,lar spec)es of sharks the t)-e a shark re-a)(s )( a state of to()c )--ob)l)ty whe(

)(4erte* )s (or-ally *)str)b,te* w)th -ea( 11.2 -)(,tes a(* sta(*ar* *e4)at)o( 1.1

-)(,tes.

a f a b)olog)st )(*,ces a state of to()c )--ob)l)ty )( s,ch a shark )( or*er to

st,*y )t +(* the probab)l)ty that the shark w)ll re-a)( )( th)s state for

betwee( 1; a(* 13 -)(,tes.

b Bhe( a b)olog)st w)shes to est)-ate the -ea( t)-e that s,ch sharks stay

)--ob)le by )(*,c)(g to()c )--ob)l)ty )( each of a sa-ple of 12 sharks +(*

the probab)l)ty that -ea( t)-e of )--ob)l)ty )( the sa-ple w)ll be betwee( 1;

a(* 13 -)(,tes.






19 S,ppose the -ea( cost across the co,(try of a 3;=*ay s,pply of a ge(er)c *r,g )s

.06 w)th sta(*ar* *e4)at)o( .6. )(* the probab)l)ty that the -ea( of a sa-ple of

1;; pr)ces of 3;=*ay s,ppl)es of th)s *r,g w)ll be betwee( 0 a(* 0;.

2; S,ppose the -ea( le(gth of t)-e betwee( s,b-)ss)o( of a state ta7 ret,r( re<,est)(g a

ref,(* a(* the )ss,a(ce of the ref,(* )s *ays w)th sta(*ar* *e4)at)o( *ays. )(*the probab)l)ty that )( a sa-ple of 0; ret,r(s re<,est)(g a ref,(* the -ea( s,ch t)-e

w)ll be -ore tha( 0; *ays.

21 Scores o( a co--o( +(al e7a- )( a large e(roll-e(t -,lt)ple=sect)o( fresh-a( co,rse

are (or-ally *)str)b,te* w)th -ea( 2. a(* sta(*ar* *e4)at)o( 13.1.

a )(* the probab)l)ty that the score : o( a ra(*o-ly selecte* e7a- paper )s

betwee( ; a(* 6;.

b )(* the probab)l)ty that the -ea( score X−− of 36 ra(*o-ly selecte* e7a-

papers )s betwee( ; a(* 6;.

22 S,ppose the -ea( we)ght of school ch)l*re(Ws bookbags )s 1. po,(*s w)th sta(*ar*

*e4)at)o( 2.2 po,(*s. )(* the probab)l)ty that the -ea( we)ght of a sa-ple of 3;

bookbags w)ll e7cee* 1 po,(*s.

23 S,ppose that )( a certa)( reg)o( of the co,(try the -ea( *,rat)o( of +rst -arr)ages that

e(* )( *)4orce )s .6 years sta(*ar* *e4)at)o( 1.2 years. )(* the probab)l)ty that )( a

sa-ple of 0 *)4orces the -ea( age of the -arr)ages )s at -ost 6 years.

2 #orach)o eats at the sa-e fast foo* resta,ra(t e4ery *ay. S,ppose the t)-e : betwee(

the -o-e(t #orach)o e(ters the resta,ra(t a(* the -o-e(t he )s ser4e* h)s foo* )s

(or-ally *)str)b,te* w)th -ea( .2 -)(,tes a(* sta(*ar* *e4)at)o( 1.3 -)(,tes.

a )(* the probab)l)ty that whe( he e(ters the resta,ra(t to*ay )t w)ll be at least

0 -)(,tes ,(t)l he )s ser4e*.

b )(* the probab)l)ty that a4erage t)-e ,(t)l he )s ser4e* )( e)ght ra(*o-ly

selecte* 4)s)ts to the resta,ra(t w)ll be at least 0 -)(,tes.

A((&T&1NA! ++/C&S+S

20 A h)gh=spee* pack)(g -ach)(e ca( be set to *el)4er betwee( 11 a(* 13 o,(ces of a

l)<,)*. or a(y *el)4ery sett)(g )( th)s ra(ge the a-o,(t *el)4ere* )s (or-ally *)str)b,te*

w)th -ea( so-e a-o,(t μ a(* w)th sta(*ar* *e4)at)o( ;.;6 o,(ce. &o cal)brate the

-ach)(e )t )s set to *el)4er a part)c,lar a-o,(t -a(y co(ta)(ers are +lle* a(* 20co(ta)(ers are ra(*o-ly selecte* a(* the a-o,(t they co(ta)( )s -eas,re*. )(* the

probab)l)ty that the sa-ple -ea( w)ll be w)th)( ;.;0 o,(ce of the act,al -ea( a-o,(t

be)(g *el)4ere* to all co(ta)(ers.

2 A t)re -a(,fact,rer states that a certa)( type of t)re has a -ea( l)fet)-e of ;;;;

-)les. S,ppose l)fet)-es are (or-ally *)str)b,te* w)th sta(*ar* *e4)at)o( σ= 3,500-)les.






a )(* the probab)l)ty that )f yo, b,y o(e s,ch t)re )t w)ll last o(ly 0;;; or

fewer -)les. f yo, ha* th)s e7per)e(ce )s )t part)c,larly stro(g e4)*e(ce that

the t)re )s (ot as goo* as cla)-e*C

b A co(s,-er gro,p b,ys +4e s,ch t)res a(* tests the-. )(* the probab)l)ty

that a4erage l)fet)-e of the +4e t)res w)ll be 0;;; -)les or less. f the -ea()s so low )s that part)c,larly stro(g e4)*e(ce that the t)re )s (ot as goo* as

cla)-e*C






.3 &he Sa-ple Proport)o(

LEARNN! "#$E%&'ES

1 &o recog()Ve that the sa-ple proport)o( Pˆ)s a ra(*o- 4ar)able.

2 &o ,(*ersta(* the -ea()(g of the for-,las for the -ea( a(* sta(*ar* *e4)at)o(

of the sa-ple proport)o(.

3 &o lear( what the sa-pl)(g *)str)b,t)o( of P )s whe( the sa-ple s)Ve )s large.

Often sampling is done in order to estimate the proportion of a population that has a specific

characteristic$ such as the proportion of all items coming off an assembly line that are defective or

the proportion of all people entering a retail store who make a purchase before leaving. The

population proportion is denoted p and the sample proportion is denoted p.Thus if in reality 63B of

people entering a store make a purchase before leaving$ p M 5.63 if in a sample of !55 people

entering the store$ @? make a purchase$ p=78/200=0.39.

The sample proportion is a random variable: it varies from sample to sample in a way that cannot be

predicted with certainty. Diewed as a random variable it will be written P .,t has a mean µP and

a standard deviation σP.Eere are formulas for their values.











/igure ;.8 02istribution of +ample roportions0 shows that when p M 5.1 a sample of si'e 18 is too

small but a sample of si'e 155 is acceptable. /igure ;.; 02istribution of +ample roportions for

0 shows that when p M 5.8 a sample of si'e 18 is acceptable.






!igure /. (istribution of %ample $roportions

!igure /./ (istribution of %ample $roportions for p H 5. and n H "











+A>2!+ 5

A( o(l)(e reta)ler cla)-s that 9;G of all or*ers are sh)ppe* w)th)( 12 ho,rs of

be)(g rece)4e*. A co(s,-er gro,p place* 121 or*ers of *)8ere(t s)Ves a(* at

*)8ere(t t)-es of *ayD 1;2 or*ers were sh)ppe* w)th)( 12 ho,rs.

a %o-p,te the sa-ple proport)o( of )te-s sh)ppe* w)th)( 12 ho,rs.

b %o(+r- that the sa-ple )s large e(o,gh to ass,-e that the sa-ple

proport)o( )s (or-ally *)str)b,te*. Use p ;.9; correspo(*)(g to the

ass,-pt)o( that the reta)lerWs cla)- )s 4al)*.

c Ass,-)(g the reta)lerWs cla)- )s tr,e +(* the probab)l)ty that a sa-ple of

s)Ve 121 wo,l* pro*,ce a sa-ple proport)o( so low as was obser4e* )(

th)s sa-ple.

* #ase* o( the a(swer to part c5 *raw a co(cl,s)o( abo,t the reta)lerWs

cla)-.


























A22!&CAT&1NS

13 S,ppose that 6G of all -ales s,8er so-e for- of color bl)(*(ess. )(* the probab)l)ty

that )( a ra(*o- sa-ple of 20; -e( at least 1;G w)ll s,8er so-e for- of color

bl)(*(ess. )rst 4er)fy that the sa-ple )s s,?c)e(tly large to ,se the (or-al *)str)b,t)o(.

1 S,ppose that 29G of all res)*e(ts of a co--,()ty fa4or a((e7at)o( by a (earby

-,()c)pal)ty. )(* the probab)l)ty that )( a ra(*o- sa-ple of 0; res)*e(ts at least 30G

w)ll fa4or a((e7at)o(. )rst 4er)fy that the sa-ple )s s,?c)e(tly large to ,se the (or-al

*)str)b,t)o(.

10 S,ppose that 2G of all cell pho(e co((ect)o(s by a certa)( pro4)*er are *roppe*. )(*

the probab)l)ty that )( a ra(*o- sa-ple of 10;; calls at -ost ; w)ll be *roppe*. )rst

4er)fy that the sa-ple )s s,?c)e(tly large to ,se the (or-al *)str)b,t)o(.

1 S,ppose that )( 2;G of all tra?c acc)*e(ts )(4ol4)(g a( )(>,ry *r)4er *)stract)o( )( so-e

for- for e7a-ple cha(g)(g a ra*)o stat)o( or te7t)(g5 )s a factor. )(* the probab)l)ty

that )( a ra(*o- sa-ple of 20 s,ch acc)*e(ts betwee( 10G a(* 20G )(4ol4e *r)4er

*)stract)o( )( so-e for-. )rst 4er)fy that the sa-ple )s s,?c)e(tly large to ,se the

(or-al *)str)b,t)o(.

1 A( a)rl)(e cla)-s that 2G of all )ts [)ghts to a certa)( reg)o( arr)4e o( t)-e. ( a ra(*o-

sa-ple of 3; rece(t arr)4als 19 were o( t)-e. Jo, -ay ass,-e that the (or-al

*)str)b,t)o( appl)es.a %o-p,te the sa-ple proport)o(.

b Ass,-)(g the a)rl)(eWs cla)- )s tr,e +(* the probab)l)ty of a sa-ple of s)Ve 3;

pro*,c)(g a sa-ple proport)o( so low as was obser4e* )( th)s sa-ple.

16 A h,-a(e soc)ety reports that 19G of all pet *ogs were a*opte* fro- a( a()-al shelter.

Ass,-)(g the tr,th of th)s assert)o( +(* the probab)l)ty that )( a ra(*o- sa-ple of 6;






pet *ogs betwee( 10G a(* 2;G were a*opte* fro- a shelter. Jo, -ay ass,-e that the

(or-al *)str)b,t)o( appl)es.

19 ( o(e st,*y )t was fo,(* that 6G of all ho-es ha4e a f,(ct)o(al s-oke *etector.

S,ppose th)s proport)o( )s 4al)* for all ho-es. )(* the probab)l)ty that )( a ra(*o-

sa-ple of ;; ho-es betwee( 6;G a(* 9;G w)ll ha4e a f,(ct)o(al s-oke *etector. Jo,-ay ass,-e that the (or-al *)str)b,t)o( appl)es.

2; A state )(s,ra(ce co--)ss)o( est)-ates that 13G of all -otor)sts )( )ts state are

,()(s,re*. S,ppose th)s proport)o( )s 4al)*. )(* the probab)l)ty that )( a ra(*o- sa-ple

of 0; -otor)sts at least 0 w)ll be ,()(s,re*. Jo, -ay ass,-e that the (or-al

*)str)b,t)o( appl)es.

21 A( o,ts)*e +(a(c)al a,*)tor has obser4e* that abo,t G of all *oc,-e(ts he e7a-)(es

co(ta)( a( error of so-e sort. Ass,-)(g th)s proport)o( to be acc,rate +(* the

probab)l)ty that a ra(*o- sa-ple of ;; *oc,-e(ts w)ll co(ta)( at least 3; w)th so-e

sort of error. Jo, -ay ass,-e that the (or-al *)str)b,t)o( appl)es.

22 S,ppose G of all ho,sehol*s ha4e (o ho-e telepho(e b,t *epe(* co-pletely o( cell

pho(es. )(* the probab)l)ty that )( a ra(*o- sa-ple of 0; ho,sehol*s betwee( 20

a(* 30 w)ll ha4e (o ho-e telepho(e. Jo, -ay ass,-e that the (or-al *)str)b,t)o(

appl)es.

A((&T&1NA! ++/C&S+S

23 So-e co,(tr)es allow )(*)4)*,al packages of prepackage* goo*s to we)gh less tha( what

)s state* o( the package s,b>ect to certa)( co(*)t)o(s s,ch as the a4erage of all

packages be)(g the state* we)ght or greater. S,ppose that o(e re<,)re-e(t )s that at

-ost G of all packages -arke* 0;; gra-s ca( we)gh less tha( 9; gra-s. Ass,-)(g

that a pro*,ct act,ally -eets th)s re<,)re-e(t +(* the probab)l)ty that )( a ra(*o-

sa-ple of 10; s,ch packages the proport)o( we)gh)(g less tha( 9; gra-s )s at least 3G.

Jo, -ay ass,-e that the (or-al *)str)b,t)o( appl)es.

2 A( eco(o-)st w)shes to )(4est)gate whether people are keep)(g cars lo(ger (ow tha( )(

the past. @e k(ows that +4e years ago 36G of all passe(ger 4eh)cles )( operat)o( were

at least te( years ol*. @e co--)ss)o(s a st,*y )( wh)ch 320 a,to-ob)les are ra(*o-ly

sa-ple*. "f the- 132 are te( years ol* or ol*er.

a )(* the sa-ple proport)o(.

b )(* the probab)l)ty that whe( a sa-ple of s)Ve 320 )s *raw( fro- a

pop,lat)o( )( wh)ch the tr,e proport)o( )s ;.36 the sa-ple proport)o( w)ll be

as large as the 4al,e yo, co-p,te* )( part a5. Jo, -ay ass,-e that the


c !)4e a( )(terpretat)o( of the res,lt )( part b5. s there stro(g e4)*e(ce that

people are keep)(g the)r cars lo(ger tha( was the case +4e years agoC






20 A state p,bl)c health *epart-e(t w)shes to )(4est)gate the e8ect)4e(ess of a ca-pa)g(

aga)(st s-ok)(g. @)stor)cally 22G of all a*,lts )( the state reg,larly s-oke* c)gars or

c)garettes. ( a s,r4ey co--)ss)o(e* by the p,bl)c health *epart-e(t 29 of 10;;

ra(*o-ly selecte* a*,lts state* that they s-oke reg,larly.

a )(* the sa-ple proport)o(.b )(* the probab)l)ty that whe( a sa-ple of s)Ve 10;; )s *raw( fro- a

pop,lat)o( )( wh)ch the tr,e proport)o( )s ;.22 the sa-ple proport)o( w)ll be

(o larger tha( the 4al,e yo, co-p,te* )( part a5. Jo, -ay ass,-e that the


c !)4e a( )(terpretat)o( of the res,lt )( part b5. @ow stro(g )s the e4)*e(ce that

the ca-pa)g( to re*,ce s-ok)(g has bee( e8ect)4eC

2 ( a( e8ort to re*,ce the pop,lat)o( of ,(wa(te* cats a(* *ogs a gro,p of 4eter)(ar)a(s

set ,p a low=cost spay/(e,ter cl)()c. At the )(cept)o( of the cl)()c a s,r4ey of pet ow(ers

)(*)cate* that 6G of all pet *ogs a(* cats )( the co--,()ty were spaye* or (e,tere*.

After the low=cost cl)()c ha* bee( )( operat)o( for three years that +g,re ha* r)se( to

6G.

a Bhat )(for-at)o( )s -)ss)(g that yo, wo,l* (ee* to co-p,te the probab)l)ty

that a sa-ple *raw( fro- a pop,lat)o( )( wh)ch the proport)o( )s 6G

correspo(*)(g to the ass,-pt)o( that the low=cost cl)()c ha* ha* (o e8ect5 )s

as h)gh as 6GC

b I(ow)(g that the s)Ve of the or)g)(al sa-ple three years ago was 10; a(* that

the s)Ve of the rece(t sa-ple was 120 co-p,te the probab)l)ty -e(t)o(e* )(

part a5. Jo, -ay ass,-e that the (or-al *)str)b,t)o( appl)es.

c !)4e a( )(terpretat)o( of the res,lt )( part b5. @ow stro(g )s the e4)*e(ce that

the prese(ce of the low=cost cl)()c has )(crease* the proport)o( of pet *ogs

a(* cats that ha4e bee( spaye* or (e,tere*C

2 A( or*)(ary *)e )s fa)r or bala(ce* )f each face has a( e<,al cha(ce of la(*)(g o( top

whe( the *)e )s rolle*. &h,s the proport)o( of t)-es a three )s obser4e* )( a large (,-ber

of tosses )s e7pecte* to be close to 1/ or 0.16−.S,ppose a *)e )s rolle* 2; t)-es a(*

shows three o( top 3 t)-es for a sa-ple proport)o( of ;.10.

a )(* the probab)l)ty that a fa)r *)e wo,l* pro*,ce a proport)o( of ;.10 or less. Jo,

-ay ass,-e that the (or-al *)str)b,t)o( appl)es.

b !)4e a( )(terpretat)o( of the res,lt )( part b5. @ow stro(g )s the e4)*e(ce that the

*)e )s (ot fa)rC

c S,ppose the sa-ple proport)o( ;.10 ca-e fro- roll)(g the *)e 2;; t)-es )(stea*

of o(ly 2; t)-es. Rework part a5 ,(*er these c)rc,-sta(ces.






* !)4e a( )(terpretat)o( of the res,lt )( part c5. @ow stro(g )s the e4)*e(ce that the

*)e )s (ot fa)rC











Chapter 4

+stimation

,f we wish to estimate the mean of a population for which a census is impractical$ say the averageheight of all 1?-year-old men in the country$ a reasonable strategy is to take a sample$ compute its

meanx−$ and estimate the unknown number by the known numberx−./or example$ if the average

height of 155 randomly selected men aged 1? is @5.; inches$ then we would say that the average

height of all 1?-year-old men is *at least approximately) @5.; inches.

stimating a population parameter by a single number like this is called point estimation in the

case at hand the statistic x −is a point estimate of the parameter . The terminology arises because

a single number corresponds to a single point on the number line.

% problem with a point estimate is that it gives no indication of how reliable the estimate is. ,n

contrast$ in this chapter we learn about interval estimation. ,n brief$ in the case of estimating a

population mean we use a formula to compute from the data a number ; $ called

the margin of error of the estimate$ and form the interval [x −−E,x−+E]. (e do this in such a way

that a certain proportion$ say F8B$ of all the intervals constructed from sample data by means of this

formula contain the unknown parameter . +uch an interval is called

a "#1 confidence interval f or .

7ontinuing with the example of the average height of 1?-year-old men$ suppose that the sample of

155 men mentioned above for which x−=70.6inches also had sample standard deviation s M 1.@

inches. ,t then turns out that ; M 5.33 and we would state that we are F8B confident that the average

height of all 1?-year-old men is in the interval formed by 70.6±0.33inches$ that is$ the average is

between @5.!@ and @5.F3 inches. ,f the sample statistics had come from a smaller sample$ say a

sample of 85 men$ the lower reliability would show up in the F8B confidence interval being longer$






hence less precise in its estimate. ,n this example the F8B confidence interval for the same sample

statistics but with n M 85 is70.6±0.47inches$ or from @5.13 to @1.5@ inches.

4.% !ar$e Sample +stimation o a 2opulation >ean

LEARNN! "#$E%&'ES

1 &o beco-e fa-)l)ar w)th the co(cept of a( )(ter4al est)-ate of the pop,lat)o(

-ea(.

2 &o ,(*ersta(* how to apply for-,las for a co(+*e(ce )(ter4al for a pop,lat)o(

-ea(.






/igure @.! 07omputer +imulation of 65 F8B 7onfidence ,ntervals for a ean0shows the intervals

generated by a computer simulation of drawing 65 samples from a normally distributed population

and constructing the F8B confidence interval for each one. (e expect that about (0.05)(40)=2of the

intervals so constructed would fail to contain the population mean $ and in this simulation two of the

intervals$ shown in red$ do.






!igure 1.& <omputer %imulation of 5 4K <onfidence >ntervals for a 2ean

,t is standard practice to identify the level of confidence in terms of the area αin the two tails of the

distribution of X^−− when the middle part specified by the level of confidence is taken out. This is

shown in /igure @.3$ drawn for the general situation$ and in /igure @.6$ drawn for F8B confidence.

Cemember from +ection 8.6.1 0Tails of the +tandard 9ormal 2istribution0 in 7hapter 8 07ontinuousCandom Dariables0 that the z -value that cuts off a right tail of area c is denoted z c. Thus the number

1.F;5 in the example isz.025$ which iszα 2forα=1−0.95=0.05.

!igure 1.*






100(1−α)α/2.

!igure 1.

α/2=0.025.











EKAPLE 2

Use )g,re 12.3 Q%r)t)cal 'al,es of Q to +(* the (,-ber zα/2(ee*e* )( co(str,ct)o(

of a co(+*e(ce )(ter4al:

a. whe( the le4el of co(+*e(ce )s 9;GD






b. whe( the le4el of co(+*e(ce )s 99G.

Sol,t)o(:

a. ( the (e7t sect)o( we w)ll lear( abo,t a co(t)(,o,s ra(*o- 4ar)able that has a

probab)l)ty *)str)b,t)o( calle* the St,*e(t t =*)str)b,t)o(. )g,re 12.3 Q%r)t)cal 'al,es of

Q g)4es the 4al,e t c that c,ts o8 a r)ght ta)l of area c for *)8ere(t 4al,es of c. &he last l)(e

of that table the o(e whose hea*)(g )s the sy-bol ∞ for )(+()ty a(* [z] g)4es the

correspo(*)(g z =4al,e z c that c,ts o8 a r)ght ta)l of the sa-e area c. ( part)c,lar z ;.;0 )s

the (,-ber )( that row a(* )( the col,-( w)th the hea*)(g t ;.;0. Be rea* o8 *)rectly

that z0.05=1.645.

b. ( )g,re 12.3 Q%r)t)cal 'al,es of Q z ;.;;0 )s the (,-ber )( the last row a(* )( the col,-(

hea*e* t ;.;;0 (a-ely 2.0.

/igure 1!.3 07ritical Dalues of 0 can be used to find z c only for those values of cfor which there is a

column with the heading t c appearing in the table otherwise we must use /igure 1!.! 07umulative

9ormal robability0 in reverse. 4ut when it can be done it is both faster and more accurate to use the

last line of /igure 1!.3 07ritical Dalues of 0 to find z c than it is to do so using /igure 1!.! 07umulative

9ormal robability0 in reverse.











++/C&S+S'AS&C

1 A ra(*o- sa-ple )s *raw( fro- a pop,lat)o( of k(ow( sta(*ar* *e4)at)o( 11.3.

%o(str,ct a 9;G co(+*e(ce )(ter4al for the pop,lat)o( -ea( base* o( the )(for-at)o(

g)4e( (ot all of the )(for-at)o( g)4e( (ee* be ,se*5.

a n 3 x−=105.2 s 11.2

b n 1;; x−=105.2 s 11.2

2 A ra(*o- sa-ple )s *raw( fro- a pop,lat)o( of k(ow( sta(*ar* *e4)at)o( 22.1.

%o(str,ct a 90G co(+*e(ce )(ter4al for the pop,lat)o( -ea( base* o( the )(for-at)o(

g)4e( (ot all of the )(for-at)o( g)4e( (ee* be ,se*5.

a n 121 x−=82.4 s 21.9

b n 61 x−=82.4 s 21.9

3 A ra(*o- sa-ple )s *raw( fro- a pop,lat)o( of ,(k(ow( sta(*ar* *e4)at)o(. %o(str,ct

a 99G co(+*e(ce )(ter4al for the pop,lat)o( -ea( base* o( the )(for-at)o( g)4e(.

a n 9 x−=17.1 s 2.1






b n 19 x−=17.1 s 2.1

A ra(*o- sa-ple )s *raw( fro- a pop,lat)o( of ,(k(ow( sta(*ar* *e4)at)o(.

%o(str,ct a 96G co(+*e(ce )(ter4al for the pop,lat)o( -ea( base* o( the

)(for-at)o( g)4e(.

a n 220 x−=92.0 s 6.

b n x−=92.0 s 6.

0 A ra(*o- sa-ple of s)Ve 1 )s *raw( fro- a pop,lat)o( whose *)str)b,t)o( -ea(

a(* sta(*ar* *e4)at)o( are all ,(k(ow(. &he s,--ary stat)st)cs are x−=58.2a(* s

2..

a %o(str,ct a( 6;G co(+*e(ce )(ter4al for the pop,lat)o( -ea( μ.

b %o(str,ct a 9;G co(+*e(ce )(ter4al for the pop,lat)o( -ea( μ.

c %o--e(t o( why o(e )(ter4al )s lo(ger tha( the other.

A ra(*o- sa-ple of s)Ve 20 )s *raw( fro- a pop,lat)o( whose *)str)b,t)o( -ea(

a(* sta(*ar* *e4)at)o( are all ,(k(ow(. &he s,--ary stat)st)cs are x−=1011a(* s

3.

a %o(str,ct a 9;G co(+*e(ce )(ter4al for the pop,lat)o( -ea( μ.

b %o(str,ct a 99G co(+*e(ce )(ter4al for the pop,lat)o( -ea( μ.

c %o--e(t o( why o(e )(ter4al )s lo(ger tha( the other.

APPL%A&"NS

A go4er(-e(t age(cy was charge* by the leg)slat,re w)th est)-at)(g the le(gth of t)-e

)t takes c)t)Ve(s to +ll o,t 4ar)o,s for-s. &wo h,(*re* ra(*o-ly selecte* a*,lts were

t)-e* as they +lle* o,t a part)c,lar for-. &he t)-es re<,)re* ha* -ea( 12.6 -)(,tes

w)th sta(*ar* *e4)at)o( 1. -)(,tes. %o(str,ct a 9;G co(+*e(ce )(ter4al for the -ea(

t)-e take( for all a*,lts to +ll o,t th)s for-.

6 o,r h,(*re* ra(*o-ly selecte* work)(g a*,lts )( a certa)( state )(cl,*)(g those who

worke* at ho-e were aske* the *)sta(ce fro- the)r ho-e to the)r workplace. &he

a4erage *)sta(ce was 6.6 -)les w)th sta(*ar* *e4)at)o( 2.; -)les. %o(str,ct a 99G






co(+*e(ce )(ter4al for the -ea( *)sta(ce fro- ho-e to work for all res)*e(ts of th)s

state.

9 "( e4ery passe(ger 4eh)cle that )t tests a( a,to-ot)4e -agaV)(e -eas,res at tr,e

spee* 00 -ph the *)8ere(ce betwee( the tr,e spee* of the 4eh)cle a(* the spee*

)(*)cate* by the spee*o-eter. or 3 4eh)cles teste* the -ea( *)8ere(ce was X1.2

-ph w)th sta(*ar* *e4)at)o( ;.2 -ph. %o(str,ct a 9;G co(+*e(ce )(ter4al for the -ea(

*)8ere(ce betwee( tr,e spee* a(* )(*)cate* spee* for all 4eh)cles.

1; A corporat)o( -o()tors t)-e spe(t by o?ce workers brows)(g the web o( the)r

co-p,ters )(stea* of work)(g. ( a sa-ple of co-p,ter recor*s of 0; workers the

a4erage a-o,(t of t)-e spe(t brows)(g )( a( e)ght=ho,r work *ay was 2.6 -)(,tes

w)th sta(*ar* *e4)at)o( 6.2 -)(,tes. %o(str,ct a 99.0G co(+*e(ce )(ter4al for the

-ea( t)-e spe(t by all o?ce workers )( brows)(g the web )( a( e)ght=ho,r *ay.

11 A sa-ple of 20; workers age* 1 a(* ol*er pro*,ce* a( a4erage le(gth of t)-e w)th

the c,rre(t e-ployer >ob te(,re5 of . years w)th sta(*ar* *e4)at)o( 3.6 years.

%o(str,ct a 99.9G co(+*e(ce )(ter4al for the -ea( >ob te(,re of all workers age* 1 or

ol*er.

12 &he a-o,(t of a part)c,lar b)oche-)cal s,bsta(ce relate* to bo(e break*ow( was

-eas,re* )( 3; healthy wo-e(. &he sa-ple -ea( a(* sta(*ar* *e4)at)o( were 3.3

(a(ogra-s per -)ll)l)ter (g/-L5 a(* 1. (g/-L. %o(str,ct a( 6;G co(+*e(ce )(ter4al

for the -ea( le4el of th)s s,bsta(ce )( all healthy wo-e(.

13 A corporat)o( that ow(s apart-e(t co-ple7es w)shes to est)-ate the a4erage le(gth of

t)-e res)*e(ts re-a)( )( the sa-e apart-e(t before -o4)(g o,t. A sa-ple of 10; re(tal

co(tracts ga4e a -ea( le(gth of occ,pa(cy of 3. years w)th sta(*ar* *e4)at)o( 1.2

years. %o(str,ct a 90G co(+*e(ce )(ter4al for the -ea( le(gth of occ,pa(cy of

apart-e(ts ow(e* by th)s corporat)o(.






1 &he *es)g(er of a garbage tr,ck that l)fts roll=o,t co(ta)(ers -,st est)-ate the -ea(

we)ght the tr,ck w)ll l)ft at each collect)o( po)(t. A ra(*o- sa-ple of 320 co(ta)(ers of

garbage o( c,rre(t collect)o( ro,tes y)el*e* x−=75.3lb s 12.6 lb. %o(str,ct a 99.6G

co(+*e(ce )(ter4al for the -ea( we)ght the tr,cks -,st l)ft each t)-e.

10 ( or*er to est)-ate the -ea( a-o,(t of *a-age s,sta)(e* by 4eh)cles whe( a *eer )s

str,ck a( )(s,ra(ce co-pa(y e7a-)(e* the recor*s of 0; s,ch occ,rre(ces a(*

obta)(e* a sa-ple -ea( of 260 w)th sa-ple sta(*ar* *e4)at)o( 221. %o(str,ct a

90G co(+*e(ce )(ter4al for the -ea( a-o,(t of *a-age )( all s,ch acc)*e(ts.

1 ( or*er to est)-ate the -ea( %" cre*)t score of )ts -e-bers a cre*)t ,()o( sa-ples

the scores of 90 -e-bers a(* obta)(s a sa-ple -ea( of 36.2 w)th sa-ple sta(*ar*

*e4)at)o( .2. %o(str,ct a 99G co(+*e(ce )(ter4al for the -ea( %" score of all of )ts

-e-bers.











LAR!E A&A S E& EKE R%SES

23 Large ata Set 1 recor*s the SA& scores of 1;;; st,*e(ts. Regar*)(g )t as a ra(*o-

sa-ple of all h)gh school st,*e(ts ,se )t to co(str,ct a 99G co(+*e(ce )(ter4al for the

-ea( SA& score of all st,*e(ts.

http://www.1.7ls






2 Large ata Set 1 recor*s the !PAs of 1;;; college st,*e(ts. Regar*)(g )t as a ra(*o-

sa-ple of all college st,*e(ts ,se )t to co(str,ct a 90G co(+*e(ce )(ter4al for the

-ea( !PA of all st,*e(ts.

http://www.1.7ls

20 Large ata Set 1 l)sts the SA& scores of 1;;; st,*e(ts.

http://www.1.7ls

a Regar* the *ata as ar)s)(g fro- a ce(s,s of all st,*e(ts at a h)gh school )(

wh)ch the SA& score of e4ery st,*e(t was -eas,re*. %o-p,te the pop,lat)o(

-ea( μ.

b Regar* the +rst 3 st,*e(ts as a ra(*o- sa-ple a(* ,se )t to co(str,ct a

99G co(+*e(ce for the -ea( μ of all 1;;; SA& scores. oes )t act,ally

capt,re the -ea( μC


http://www.1.7ls

a Regar* the *ata as ar)s)(g fro- a ce(s,s of all fresh-a( at a s-all college at the

e(* of the)r +rst aca*e-)c year of college st,*y )( wh)ch the !PA of e4ery s,ch

perso( was -eas,re*. %o-p,te the pop,lat)o( -ea( μ.

b Regar* the +rst 3 st,*e(ts as a ra(*o- sa-ple a(* ,se )t to co(str,ct a 90G

co(+*e(ce for the -ea( μ of all 1;;; !PAs. oes )t act,ally capt,re the -ea( μC






4.0 Small Sample +stimation o a 2opulation>ean

!+A/N&N: 1';+CT&<+S

1 &o beco-e fa-)l)ar w)th St,*e(tWs t =*)str)b,t)o(.

2 &o ,(*ersta(* how to apply a**)t)o(al for-,las for a co(+*e(ce )(ter4al for a

pop,lat)o( -ea(.

The confidence interval formulas in the previous section are based on the 7entral imit Theorem$ the

statement that for large samples X −−is normally distributed with mean and standard

deviationσ/√n. (hen the population mean is estimated with a small sample *n V 35)$ the 7entral

imit Theorem does not apply. ,n order to proceed we assume that the numerical population from

which the sample is taken has a normal distribution to begin with. ,f this condition is satisfied then






when the population standard deviation 6 is known the old formula x−±zα 2 σ √n)can still be

used to construct a100(1−α)1 confidence interval for .

,f the population standard deviation is unknown and the sample si'e n is small then when we

substitute the sample standard deviation s for 6 the normal approximation is no longer valid. The

solution is to use a different distribution$ called Students t(

distribution *ith n−1degrees of freedom. +tudent&s t -distribution is very much like the

standard normal distribution in that it is centered at 5 and has the same "ualitative bell shape$ but it

has heavier tails than the standard normal distribution does$ as indicated by /igure @.8 0+tudent&s 0$

in which the curve *in brown) that meets the dashed vertical line at the lowest point is the t -

distribution with two degrees of freedom$ the next curve *in blue) is the t -distribution with five

degrees of freedom$ and the thin curve *in red) is the standard normal distribution. %s also indicated

by the figure$ as the sample si'e n increases$ +tudent&s t -distribution ever more closely resembles thestandard normal distribution. %lthough there is a different t -distribution for every value of n$ once the

sample si'e is 35 or more it is typically acceptable to use the standard normal distribution instead$ as

we will always do in this text.

!igure 1. %tudent=s t :(istribution

Rust as the symbol z c stands for the value that cuts off a right tail of area c in the standard normal

distribution$ so the symbol t c stands for the value that cuts off a right tail of area c in the standard

normal distribution. This gives us the following confidence interval formulas.
















7ompare 9ote @.F 0xample 60 in +ection @.1 0arge +ample stimation of a opulation

ean0 and 9ote @.1; 0xample ;0. The summary statistics in the two samples are the same$ but the

F5B confidence interval for the average % of all students at the university in 9ote @.F 0xample

60 in +ection @.1 0arge +ample stimation of a opulation ean0$(2.63,2.79) is shorter than the

F5B confidence interval(2.45,2.97) in 9ote @.1; 0xample ;0. This is partly because in 9ote @.F






0xample 60 the sample si'e is larger there is more information pertaining to the true value of in

the large data set than in the small one.

*+, TA*+AA,S

• ( select)(g the correct for-,la for co(str,ct)o( of a co(+*e(ce )(ter4al for a

pop,lat)o( -ea( ask two <,est)o(s: )s the pop,lat)o( sta(*ar* *e4)at)o( σ k(ow(

or ,(k(ow( a(* )s the sa-ple large or s-allC

• Be ca( co(str,ct co(+*e(ce )(ter4als w)th s-all sa-ples o(ly )f the pop,lat)o( )s

(or-al.














































4.3 !ar$e Sample +stimation o a 2opulation2roportion

!+A/N&N: 1';+CT&<+

1 &o ,(*ersta(* how to apply the for-,la for a co(+*e(ce )(ter4al for a pop,lat)o(

proport)o(.

+ince from +ection ;.3 0The +ample roportion0 in 7hapter ; 0+ampling 2istributions0 we know the

mean$ standard deviation$ and sampling distribution of the sample proportion p$ the ideas of the

previous two sections can be applied to produce a confidence interval for a population proportion.

Eere is the formula.






*+, TA*+AA,S

• Be ha4e a s)(gle for-,la for a co(+*e(ce )(ter4al for a pop,lat)o( proport)o(

wh)ch )s 4al)* whe( the sa-ple )s large.

• &he co(*)t)o( that a sa-ple be large )s (ot that )ts s)Ve n be at least 3; b,t that

the *e(s)ty f,(ct)o( +t )(s)*e the )(ter4al [0,1].











a !)4e a po)(t est)-ate of the proport)o( p of all people who co,l* rea* wor*s

*)sg,)se* )( th)s way.

b Show that the sa-ple )s (ot s,?c)e(tly large to co(str,ct a co(+*e(ce

)(ter4al for the proport)o( of all people who co,l* rea* wor*s *)sg,)se* )( th)sway.

6 ( a ra(*o- sa-ple of 9;; a*,lts 2 *e+(e* the-sel4es as 4egetar)a(s.

a !)4e a po)(t est)-ate of the proport)o( of all a*,lts who wo,l* *e+(e

the-sel4es as 4egetar)a(s.






b 'er)fy that the sa-ple )s s,?c)e(tly large to ,se )t to co(str,ct a co(+*e(ce

)(ter4al for that proport)o(.

c %o(str,ct a( 6;G co(+*e(ce )(ter4al for the proport)o( of all a*,lts who

wo,l* *e+(e the-sel4es as 4egetar)a(s.

9 ( a ra(*o- sa-ple of 20; e-ploye* people 1 sa)* that they br)(g work ho-e w)ththe- at least occas)o(ally.

a !)4e a po)(t est)-ate of the proport)o( of all e-ploye* people who br)(g work

ho-e w)th the- at least occas)o(ally.

b %o(str,ct a 99G co(+*e(ce )(ter4al for that proport)o(.

1; ( a ra(*o- sa-ple of 120; ho,sehol* -o4es 622 were -o4es to a locat)o( w)th)( the

sa-e co,(ty as the or)g)(al res)*e(ce.

a !)4e a po)(t est)-ate of the proport)o( of all ho,sehol* -o4es that are to a

locat)o( w)th)( the sa-e co,(ty as the or)g)(al res)*e(ce.

b %o(str,ct a 96G co(+*e(ce )(ter4al for that proport)o(.

11 ( a ra(*o- sa-ple of 12 h)p replace-e(t or re4)s)o( s,rgery proce*,res

(at)o(w)*e 12 pat)e(ts *e4elope* a s,rg)cal s)te )(fect)o(.

a !)4e a po)(t est)-ate of the proport)o( of all pat)e(ts ,(*ergo)(g a h)p

s,rgery proce*,re who *e4elop a s,rg)cal s)te )(fect)o(.



c %o(str,ct a 90G co(+*e(ce )(ter4al for the proport)o( of all pat)e(ts

,(*ergo)(g a h)p s,rgery proce*,re who *e4elop a s,rg)cal s)te )(fect)o(.

12 ( a certa)( reg)o( prepackage* pro*,cts labele* 0;; g -,st co(ta)( o( a4erage at least

0;; gra-s of the pro*,ct a(* at least 9;G of all packages -,st we)gh at least 9;

gra-s. ( a ra(*o- sa-ple of 3;; packages 266 we)ghe* at least 9; gra-s.

a !)4e a po)(t est)-ate of the proport)o( of all packages that we)gh at least 9;

gra-s.



c %o(str,ct a 99.6G co(+*e(ce )(ter4al for the proport)o( of all packages that

we)gh at least 9; gra-s.






10 ( or*er to est)-ate the proport)o( of e(ter)(g st,*e(ts who gra*,ate w)th)( s)7 years

the a*-)()strat)o( at a state ,()4ers)ty e7a-)(e* the recor*s of ;; ra(*o-ly selecte*

st,*e(ts who e(tere* the ,()4ers)ty s)7 years ago a(* fo,(* that 312 ha* gra*,ate*.

a !)4e a po)(t est)-ate of the s)7=year gra*,at)o( rate the proport)o( of e(ter)(g

st,*e(ts who gra*,ate w)th)( s)7 years.

b Ass,-)(g that the sa-ple )s s,?c)e(tly large co(str,ct a 96G co(+*e(ce

)(ter4al for the s)7=year gra*,at)o( rate.

1 ( a ra(*o- sa-ple of 23;; -ortgages take( o,t )( a certa)( reg)o( last year 16

were a*>,stable=rate -ortgages.






a !)4e a po)(t est)-ate of the proport)o( of all -ortgages take( o,t )( th)s reg)o(

last year that were a*>,stable=rate -ortgages.

b Ass,-)(g that the sa-ple )s s,?c)e(tly large co(str,ct a 99.9G co(+*e(ce

)(ter4al for the proport)o( of all -ortgages take( o,t )( th)s reg)o( last year that

were a*>,stable=rate -ortgages.1 ( a research st,*y )( cattle bree*)(g 109 of 23 cows )( se4eral her*s that were )(

estr,s were *etecte* by -ea(s of a( )(te(s)4e o(ce a *ay o(e=ho,r obser4at)o( of the

her*s )( early -or()(g.

a !)4e a po)(t est)-ate of the proport)o( of all cattle )( estr,s who are *etecte* by

th)s -etho*.

b Ass,-)(g that the sa-ple )s s,?c)e(tly large co(str,ct a 9;G co(+*e(ce

)(ter4al for the proport)o( of all cattle )( estr,s who are *etecte* by th)s -etho*.

16 A s,r4ey of 2120; ho,sehol*s co(cer()(g telepho(e ser4)ce ga4e the res,lts show( )(

the table.

2andline No 2andline

e"" $hone 12474 5844

No !e"" $hone 2529 403

a !)4e a po)(t est)-ate for the proport)o( of all ho,sehol*s )( wh)ch there )s a cell

pho(e b,t (o la(*l)(e.

b Ass,-)(g the sa-ple )s s,?c)e(tly large co(str,ct a 99.9G co(+*e(ce )(ter4al

for the proport)o( of all ho,sehol*s )( wh)ch there )s a cell pho(e b,t (o la(*l)(e.

c !)4e a po)(t est)-ate for the proport)o( of all ho,sehol*s )( wh)ch there )s (o

telepho(e ser4)ce of e)ther k)(*.

* Ass,-)(g the sa-ple )s s,?c)e(tly large co(str,ct a 99.9G co(+*e(ce )(ter4al

for the proport)o( of all all ho,sehol*s )( wh)ch there )s (o telepho(e ser4)ce of

e)ther k)(*.

A((&T&1NA! ++/C&S+S

19 ( a ra(*o- sa-ple of 9;; a*,lts 2 *e+(e* the-sel4es as 4egetar)a(s. "f these 2

29 were wo-e(.

a !)4e a po)(t est)-ate of the proport)o( of all self=*escr)be* 4egetar)a(s who

are wo-e(.








c %o(str,ct a 9;G co(+*e(ce )(ter4al for the proport)o( of all all self=*escr)be*

4egetar)a(s who are wo-e(.

2; A ra(*o- sa-ple of 160 college soccer players who ha* s,8ere* )(>,r)es that res,lte*

)( loss of play)(g t)-e was -a*e w)th the res,lts show( )( the table. (>,r)es are

class)+e* accor*)(g to se4er)ty of the )(>,ry a(* the co(*)t)o( ,(*er wh)ch )t wass,sta)(e*.

#inor #oderate Serious

*ra!'i!e 48 20 6

(a%e 62 32 17

a !)4e a po)(t est)-ate for the proport)o( p of all )(>,r)es to college soccer

players that are s,sta)(e* )( pract)ce.

b %o(str,ct a 90G co(+*e(ce )(ter4al for the proport)o( p of all )(>,r)es to

college soccer players that are s,sta)(e* )( pract)ce.

c !)4e a po)(t est)-ate for the proport)o( p of all )(>,r)es to college soccer

players that are e)ther -o*erate or ser)o,s.

21 &he bo*y -ass )(*e7 #5 was -eas,re* )( 12;; ra(*o-ly selecte* a*,lts w)th

the res,lts show( )( the table.

6#/

Under 78+ 78+*!+ Over !+

en 36 165 315

o%en 75 274 335

a !)4e a po)(t est)-ate for the proport)o( of all -e( whose # )s o4er 20.

b Ass,-)(g the sa-ple )s s,?c)e(tly large co(str,ct a 99G co(+*e(ce )(ter4al for the

proport)o( of all -e( whose # )s o4er 20.

c !)4e a po)(t est)-ate for the proport)o( of all a*,lts regar*less of ge(*er whose # )s

o4er 20.

* Ass,-)(g the sa-ple )s s,?c)e(tly large co(str,ct a 99G co(+*e(ce )(ter4al for the

proport)o( of all a*,lts regar*less of ge(*er whose # )s o4er 20.





















4.7 Sample Sie Considerations

LEARNN! "#$E%&'E






1 &o lear( how to apply for-,las for est)-at)(g the s)Ve sa-ple that w)ll be (ee*e*

)( or*er to co(str,ct a co(+*e(ce )(ter4al for a pop,lat)o( -ea( or proport)o(

that -eets g)4e( cr)ter)a.

+ampling is typically done with a set of clear ob#ectives in mind. /or example$ an economist might

wish to estimate the mean yearly income of workers in a particular industry at F5B confidence and

to within >855. +ince sampling costs time$ effort$ and money$ it would be useful to be able to

estimate the smallest si'e sample that is likely to meet these criteria.





















There is a dilemma here: the formula for estimating how large a sample to take contains the

number p$ which we know only after we have taken the sample. There are two ways out of this

dilemma. Typically the researcher will have some idea as to the value of the population proportion p$

hence of what the sample proportion p is likely to be. /or example$ if last month 3@B of all voters

thought that state taxes are too high$ then it is likely that the proportion with that opinion this month

will not be dramatically different$ and we would use the value 5.3@ for p in the formula.

The second approach to resolving the dilemma is simply to replace p in the formula by 5.8. This is

because if p is large then1− p is small$ and vice versa$ which limits their product to a maximum

value of 5.!8$ which occurs when p=0.5.This is called the most conservative estimate$ since it

gives the largest possible estimate of n.











*+, TA*+AA,S• f the pop,lat)o( sta(*ar* *e4)at)o( σ )s k(ow( or ca( be est)-ate* the( the

-)()-,- sa-ple s)Ve (ee*e* to obta)( a co(+*e(ce )(ter4al for the pop,lat)o(

-ea( w)th a g)4e( -a7)-,- error of the est)-ate a(* a g)4e( le4el of co(+*e(ce

ca( be est)-ate*.

• &he -)()-,- sa-ple s)Ve (ee*e* to obta)( a co(+*e(ce )(ter4al for a pop,lat)o(

proport)o( w)th a g)4e( -a7)-,- error of the est)-ate a(* a g)4e( le4el of

co(+*e(ce ca( always be est)-ate*. f there )s pr)or k(owle*ge of the pop,lat)o(

proport)o( p the( the est)-ate ca( be sharpe(e*.

++/C&S+S

'AS&C

1 Est)-ate the -)()-,- sa-ple s)Ve (ee*e* to for- a co(+*e(ce )(ter4al for the -ea(

of a pop,lat)o( ha4)(g the sta(*ar* *e4)at)o( show( -eet)(g the cr)ter)a g)4e(.

a σ 3; 90G co(+*e(ce ( 1;

b σ 3; 99G co(+*e(ce ( 1;






c σ 3; 90G co(+*e(ce ( 0

2 Est)-ate the -)()-,- sa-ple s)Ve (ee*e* to for- a co(+*e(ce )(ter4al for the -ea(

of a pop,lat)o( ha4)(g the sta(*ar* *e4)at)o( show( -eet)(g the cr)ter)a g)4e(.

a σ 90G co(+*e(ce ( 1

b σ 99G co(+*e(ce ( 1c σ 90G co(+*e(ce ( ;.0

3 Est)-ate the -)()-,- sa-ple s)Ve (ee*e* to for- a co(+*e(ce )(ter4al for the

proport)o( of a pop,lat)o( that has a part)c,lar character)st)c -eet)(g the cr)ter)a

g)4e(.

a p F ;.3 6;G co(+*e(ce ( ;.;0

b p F ;.3 9;G co(+*e(ce ( ;.;0

c p F ;.3 6;G co(+*e(ce ( ;.;1

Est)-ate the -)()-,- sa-ple s)Ve (ee*e* to for- a co(+*e(ce )(ter4al for the


g)4e(.

a p F ;.61 90G co(+*e(ce ( ;.;2

b p F ;.61 99G co(+*e(ce ( ;.;2

c p F ;.61 90G co(+*e(ce ( ;.;1

0 Est)-ate the -)()-,- sa-ple s)Ve (ee*e* to for- a co(+*e(ce )(ter4al for the


g)4e(.

a 6;G co(+*e(ce ( ;.;0

b 9;G co(+*e(ce ( ;.;0

c 6;G co(+*e(ce ( ;.;1

Est)-ate the -)()-,- sa-ple s)Ve (ee*e* to for- a co(+*e(ce )(ter4al for the


g)4e(.

a 90G co(+*e(ce ( ;.;2

b 99G co(+*e(ce ( ;.;2

c 90G co(+*e(ce ( ;.;1

A22!&CAT&1NS






A software e(g)(eer w)shes to est)-ate to w)th)( 0 seco(*s the -ea( t)-e that a (ew

appl)cat)o( takes to start ,p w)th 90G co(+*e(ce. Est)-ate the -)()-,- s)Ve sa-ple

re<,)re* )f the sta(*ar* *e4)at)o( of start ,p t)-es for s)-)lar software )s 12 seco(*s.

6 A real estate age(t w)shes to est)-ate to w)th)( 2.0; the -ea( reta)l cost per s<,are

foot of (ewly b,)lt ho-es w)th 6;G co(+*e(ce. @e est)-ates the sta(*ar* *e4)at)o( of

s,ch costs at 0.;;. Est)-ate the -)()-,- s)Ve sa-ple re<,)re*.

9 A( eco(o-)st w)shes to est)-ate to w)th)( 2 -)(,tes the -ea( t)-e that e-ploye*

perso(s spe(* co--,t)(g each *ay w)th 90G co(+*e(ce. "( the ass,-pt)o( that the

sta(*ar* *e4)at)o( of co--,t)(g t)-es )s 6 -)(,tes est)-ate the -)()-,- s)Ve sa-ple

re<,)re*.

1; A -otor cl,b w)shes to est)-ate to w)th)( 1 ce(t the -ea( pr)ce of 1 gallo( of reg,lar

gasol)(e )( a certa)( reg)o( w)th 96G co(+*e(ce. @)stor)cally the 4ar)ab)l)ty of pr)ces )s

-eas,re* by σ=$0.03.Est)-ate the -)()-,- s)Ve sa-ple re<,)re*.

11 A ba(k w)shes to est)-ate to w)th)( 20 the -ea( a4erage -o(thly bala(ce )( )ts

check)(g acco,(ts w)th 99.6G co(+*e(ce. Ass,-)(g σ=$250 est)-ate the -)()-,- s)Ve

sa-ple re<,)re*.

12 A reta)ler w)shes to est)-ate to w)th)( 10 seco(*s the -ea( *,rat)o( of telepho(e

or*ers take( at )ts call ce(ter w)th 99.0G co(+*e(ce. ( the past the sta(*ar* *e4)at)o(

of call le(gth has bee( abo,t 1.20 -)(,tes. Est)-ate the -)()-,- s)Ve sa-ple

re<,)re*. #e caref,l to e7press all the )(for-at)o( )( the sa-e ,()ts.5

13 &he a*-)()strat)o( at a college w)shes to est)-ate to w)th)( two perce(tage po)(ts the

proport)o( of all )ts e(ter)(g fresh-e( who gra*,ate w)th)( fo,r years w)th 9;G

co(+*e(ce. Est)-ate the -)()-,- s)Ve sa-ple re<,)re*.1 A cha)( of a,to-ot)4e repa)r stores w)shes to est)-ate to w)th)( +4e perce(tage po)(ts

the proport)o( of all passe(ger 4eh)cles )( operat)o( that are at least +4e years ol* w)th

96G co(+*e(ce. Est)-ate the -)()-,- s)Ve sa-ple re<,)re*.

10 A( )(ter(et ser4)ce pro4)*er w)shes to est)-ate to w)th)( o(e perce(tage po)(t the

c,rre(t proport)o( of all e-a)l that )s spa- w)th 99.9G co(+*e(ce. Last year the

proport)o( that was spa- was 1G. Est)-ate the -)()-,- s)Ve sa-ple re<,)re*.






1 A( agro(o-)st w)shes to est)-ate to w)th)( o(e perce(tage po)(t the proport)o( of a

(ew 4ar)ety of see* that w)ll ger-)(ate whe( pla(te* w)th 90G co(+*e(ce. A typ)cal

ger-)(at)o( rate )s 9G. Est)-ate the -)()-,- s)Ve sa-ple re<,)re*.

1 A char)table orga()Vat)o( w)shes to est)-ate to w)th)( half a perce(tage po)(t the

proport)o( of all telepho(e sol)c)tat)o(s to )ts *o(ors that res,lt )( a g)ft w)th 9;G

co(+*e(ce. Est)-ate the -)()-,- sa-ple s)Ve re<,)re* ,s)(g the )(for-at)o( that )(

the past the respo(se rate has bee( abo,t 3;G.

16 A go4er(-e(t age(cy w)shes to est)-ate the proport)o( of *r)4ers age* 12 who

ha4e bee( )(4ol4e* )( a tra?c acc)*e(t )( the last year. t w)shes to -ake the est)-ate

to w)th)( o(e perce(tage po)(t a(* at 9;G co(+*e(ce. )(* the -)()-,- sa-ple s)Ve

re<,)re* ,s)(g the )(for-at)o( that se4eral years ago the proport)o( was ;.12.A((&T&1NA! ++/C&S+S

19 A( eco(o-)st w)shes to est)-ate to w)th)( s)7 -o(ths the -ea( t)-e betwee( sales of

e7)st)(g ho-es w)th 90G co(+*e(ce. Est)-ate the -)()-,- s)Ve sa-ple re<,)re*. (

h)s e7per)e(ce 4)rt,ally all ho,ses are re=sol* w)th)( ; -o(ths so ,s)(g the E-p)r)cal

R,le he w)ll est)-ate σ by o(e=s)7th the ra(ge or 40/6=6.7.

2; A w)l*l)fe -a(ager w)shes to est)-ate the -ea( le(gth of +sh )( a large lake to w)th)(

o(e )(ch w)th 6;G co(+*e(ce. Est)-ate the -)()-,- s)Ve sa-ple re<,)re*. ( h)s

e7per)e(ce 4)rt,ally (o +sh ca,ght )( the lake )s o4er 23 )(ches lo(g so ,s)(g the

E-p)r)cal R,le he w)ll est)-ate σ by o(e=s)7th the ra(ge or 23/6=3.8.

21 Jo, w)sh to est)-ate the c,rre(t -ea( b)rth we)ght of all (ewbor(s )( a certa)( reg)o(

to w)th)( 1 o,(ce 1/1 po,(*5 a(* w)th 90G co(+*e(ce. A sa-ple w)ll cost ;; pl,s

1.0; for e4ery (ewbor( we)ghe*. Jo, bel)e4e the sta(*ar* *e4)at)o(s of we)ght to be

(o -ore tha( 1.20 po,(*s. Jo, ha4e 20;; to spe(* o( the st,*y.

a %a( yo, a8or* the sa-ple re<,)re*C

b f (ot what are yo,r opt)o(sC

22 Jo, w)sh to est)-ate a pop,lat)o( proport)o( to w)th)( three perce(tage po)(ts at 90G

co(+*e(ce. A sa-ple w)ll cost 0;; pl,s 0; ce(ts for e4ery sa-ple ele-e(t -eas,re*. Jo, ha4e 1;;; to spe(* o( the st,*y.

a %a( yo, a8or* the sa-ple re<,)re*C

b f (ot what are yo,r opt)o(sC











Chapter 5

Testin$ =ypotheses

% manufacturer of emergency e"uipment asserts that a respirator that it makes delivers pure air for

@8 minutes on average. % government regulatory agency is charged with testing such claims$ in this

case to verify that the average time is not less than @8 minutes. To do so it would select a random

sample of respirators$ compute the mean time that they deliver pure air$ and compare that mean to

the asserted time @8 minutes.

,n the sampling that we have studied so far the goal has been to estimate a population parameter.

4ut the sampling done by the government agency has a somewhat different ob#ective$ not so much

to estimate the population mean as totest an assertionWor a hypothesisWabout it$ namely$ whether

it is as large as @8 or not. The agency is not necessarily interested in the actual value of $ #ust

whether it is as claimed. Their sampling is done to perform a test of hypotheses$ the sub#ect of this

chapter.






5.% The +lements o =ypothesis Testin$

LEARNN! "#$E%&'ES

1 &o ,(*ersta(* the log)cal fra-ework of tests of hypotheses.

2 &o lear( bas)c ter-)(ology co((ecte* w)th hypothes)s test)(g.

3 &o lear( f,(*a-e(tal facts abo,t hypothes)s test)(g.

Types o =ypotheses

% hypothesis about the value of a population parameter is an assertion about its value. %s in the

introductory example we will be concerned with testing the truth of two competing hypotheses$ only one

of which can be true.






(e)nitionThe null hypothesis, denoted + 5, is the statement about the population parameter that is assumed to

be true unless there is convincing evidence to the contrary.

The alternative hypothesis, denoted + a, is a statement about the population parameter that is

contradictory to the null hypothesis, and is accepted as true only if there is convincing evidence in favor

of it.

(e)nition

3ypothesis testing is a statistical procedure in which a choice is made between a null hypothesis and

an alternative hypothesis based on information in a sample.

The end result of a hypotheses testing procedure is a choice of one of the following two possible

conclusions:

1 Ce#ect + 5 *and therefore accept + a)$ or

! /ail to re#ect + 5 *and therefore fail to accept + a).

The null hypothesis typically represents the status "uo$ or what has historically been true. ,n the

example of the respirators$ we would believe the claim of the manufacturer unless there is reason not

to do so$ so the null hypotheses is H 0:µ=75.The alternative hypothesis in the example is the

contradictory statementH a:µ<75.The null hypothesis will always be an assertion containing an

e"uals sign$ but depending on the situation the alternative hypothesis can have any one of three

forms: with the symbol GV$H as in the example #ust discussed$ with the symbol GX$H or with the symbol

GYH The following two examples illustrate the latter two cases.

+A>2!+ %

A p,bl)sher of college te7tbooks cla)-s that the a4erage pr)ce of all har*bo,(*

college te7tbooks )s 12.0;. A st,*e(t gro,p bel)e4es that the act,al -ea( )s

h)gher a(* w)shes to test the)r bel)ef. State the rele4a(t (,ll a(* alter(at)4e

hypotheses.

Sol,t)o(:

&he *efa,lt opt)o( )s to accept the p,bl)sherWs cla)- ,(less there )s co-pell)(g

e4)*e(ce to the co(trary. &h,s the (,ll hypothes)s )s H 0:µ=127.50.S)(ce the st,*e(t






gro,p th)(ks that the a4erage te7tbook pr)ce )s greater tha( the p,bl)sherWs +g,re

the alter(at)4e hypothes)s )( th)s s)t,at)o( )s H a:µ>127.50.

+A>2!+ 0

&he rec)pe for a bakery )te- )s *es)g(e* to res,lt )( a pro*,ct that co(ta)(s 6

gra-s of fat per ser4)(g. &he <,al)ty co(trol *epart-e(t sa-ples the pro*,ct

per)o*)cally to )(s,re that the pro*,ct)o( process )s work)(g as *es)g(e*. State

the rele4a(t (,ll a(* alter(at)4e hypotheses.

Sol,t)o(:

&he *efa,lt opt)o( )s to ass,-e that the pro*,ct co(ta)(s the a-o,(t of fat )t was

for-,late* to co(ta)( ,(less there )s co-pell)(g e4)*e(ce to the co(trary. &h,s

the (,ll hypothes)s )s H 0:µ=8.0.S)(ce to co(ta)( e)ther -ore fat tha( *es)re* or to

co(ta)( less fat tha( *es)re* are both a( )(*)cat)o( of a fa,lty pro*,ct)o( process

the alter(at)4e hypothes)s )( th)s s)t,at)o( )s that the -ea( )s dierent fro- 6.;

so H a:µ≠8.0.

,n 9ote ?.? 0xample 10$ the textbook example$ it might seem more natural that the publisher&s

claim be that the average price is at most >[email protected]$ not exactly >[email protected]. ,f the claim were made this

way$ then the null hypothesis would beH 0:µ≤127.50$ and the value >[email protected] given in the example

would be the one that is least favorable to the publisher&s claim$ the null hypothesis. ,t is always true

that if the null hypothesis is retained for its least favorable value$ then it is retained for every other

value.

Thus in order to make the null and alternative hypotheses easy for the student to distinguish$ in

every example and problem in this text we will always present one of the two competing claims about

the value of a parameter with an e"uality. The claim expressed with an equality is the null

hypothesis. This is the same as always stating the null hypothesis in the least favorable light. +o in the

introductory example about the respirators$ we stated the manufacturer&s claim as Gthe average is @8minutesH instead of the perhaps more natural Gthe average is at least @8 minutes$H essentially

reducing the presentation of the null hypothesis to its worst case.

The first step in hypothesis testing is to identify the null and alternative hypotheses.






The !o$ic o =ypothesis Testin$

%lthough we will study hypothesis testing in situations other than for a single population mean *for

example$ for a population proportion instead of a mean or in comparing the means of two different

populations)$ in this section the discussion will always be given in terms of a single population

mean .

The null hypothesis always has the form H 0:µ=µ0for a specific number µ0 *in the respirator

exampleµ0=75$ in the textbook exampleµ0=127.50$ and in the baked goods example µ0=8.0). +ince

the null hypothesis is accepted unless there is strong evidence to the contrary$ the test procedure is

based on the initial assumption that + 5 is true. This point is so important that we will repeat it in a

display:

The test procedure is based on the initial assumption that + 5 is true.

!igure 3." The (ensity <urve for X−−if + 5 >s True






Think of the respirator example$ for which the null hypothesis isH0:µ=75$ the claim that the average

time air is delivered for all respirators is @8 minutes. ,f the sample mean is @8 or greater then we

certainly would not re#ect + 5 *since there is no issue with an emergency respirator delivering air even

longer than claimed).

,f the sample mean is slightly less than @8 then we would logically attribute the difference to

sampling error and also not re#ect + 5 either.

Dalues of the sample mean that are smaller and smaller are less and less likely to come from a

population for which the population mean is @8. Thus if the sample mean is far less than @8$ say

around ;5 minutes or less$ then we would certainly re#ect + 5$ because we know that it is highly

unlikely that the average of a sample would be so low if the population mean were @8. This is the rare

event criterionfor re#ection: what we actually observed X^−−<605 would be so rare an event if M @8

were true that we regard it as much more likely that the alternative hypothesis V @8 holds.

,n summary$ to decide between + 5 and + a in this example we would select a Gre6ection regionH of

values sufficiently far to the left of @8$ based on the rare event criterion$ and re#ect + 5 if the sample

mean X−−lies in the re#ection region$ but not re#ect + 5 if it does not.






The /eGection /e$ion

ach different form of the alternative hypothesis + a has its own kind of re#ection region:

1 if *as in the respirator example) + a has the formHa:µ<µ0$ we re#ect + 5 ifx−is far to the left ofµ0$ that is$

to the left of some number < $ so the re#ection region has the form of an interval *Z[$< \

! if *as in the textbook example) + a has the formHa:µ>µ0$ we re#ect + 5 ifx−is far to the right ofµ0$ that

is$ to the right of some number < $ so the re#ection region has the form of an interval ]< $[)

3 if *as in the baked good example) + a has the formHa:µ≠µ0$ we re#ect + 5 ifx−is far away fromµ0in

either direction$ that is$ either to the left of some number < or to the right of some other number < $so the re#ection region has the form of the union of two intervals *Z[$< \Z]< $[).

The key issue in our line of reasoning is the "uestion of how to determine the number < or

numbers < and < $ called the critical value or critical values of the statistic$ that determine the

re#ection region.

The key issue in our line of reasoning is the "uestion of how to determine the number < or

numbers < and < $ called the critical value or critical values of the statistic$ that determine the

re#ection region.

e+()t)o(

The critical value or critical values of a test of hypotheses are the number or numbers that determine

the rejection region.

+uppose the re#ection region is a single interval$ so we need to select a single number < . Eere is the

procedure for doing so. (e select a small probability$ denotedα$ say 1B$ which we take as our

definition of Grare event:H an event is GrareH if its probability of occurrence is less than α.*,n all the

examples and problems in this text the value ofα will be given already.) The probability that X^−






− takes a value in an interval is the area under its density curve and above that interval$ so as shown

in /igure ?.! *drawn under the assumption that + 5 is true$ so that the curve centers at µ0) the critical

value < is the value of X^−− that cuts off a tail areaαin the probability density curve of X^−−. (hen

the re#ection region is in two pieces$ that is$ composed of two intervals$ the total area above both of

them must beα$ so the area above each one is α/2$ as also shown in /igure ?.!.

!igure 3.&






Figure ".*6eEection 6egion for t!e ,!oice α=0.10






&he *ec)s)o( proce*,re )s: take a sa-ple of s)Ve 0 a(* co-p,te the sa-ple

-ea( x−.f x− )s e)ther .69 gra-s or less or 6.11 gra-s or -ore the( re>ect the

hypothes)s that the a4erage a-o,(t of fat )( all ser4)(gs of the pro*,ct )s 6.;

gra-s )( fa4or of the alter(at)4e that )t )s *)8ere(t fro- 6.; gra-s. "therw)se *o

(ot re>ect the hypothes)s that the a4erage a-o,(t )s 6.; gra-s.

&he reaso()(g )s that )f the tr,e a4erage a-o,(t of fat per ser4)(g were 6.;

gra-s the( there wo,l* be less tha( a 1;G cha(ce that a sa-ple of s)Ve 0 wo,l*

pro*,ce a -ea( of e)ther .69 gra-s or less or 6.11 gra-s or -ore. @e(ce )f that

happe(e* )t wo,l* be -ore l)kely that the 4al,e 6.; )s )(correct always ass,-)(g

that the pop,lat)o( sta(*ar* *e4)at)o( )s ;.10 gra-5.

4ecause the re#ection regions are computed based on areas in tails of distributions$ as shown

in /igure ?.!$ hypothesis tests are classified according to the form of the alternative hypothesis in the

following way.

e+()t)o(






>f + a has the form µ≠µ0the test is called a t*o(tailed test.

>f + a has the form µ<µ0the test is called a left(tailed test.

>f + a has the form µ>µ0the test is called a right(tailed test.

;ach of the last two forms is also called a one(tailed test.

Two Types o +rrors

The format of the testing procedure in general terms is to take a sample and use the information it

contains to come to a decision about the two hypotheses. %s stated before our decision will always be

either

1 re#ect the null hypothesis + 5 in favor of the alternative + a presented$ or

! do not re#ect the null hypothesis + 5 in favor of the alternative + a presented.

There are four possible outcomes of hypothesis testing procedure$ as shown in the following table:

,r&e 'a'e o Na'&re

H 0 i+ 'r&e H 0 i+ a"+e

O&r #e!i+ion

#o no' ree!' H 0 orre!' de!i+ion ,$e error

ee!' H 0 ,$e error orre!' de!i+ion

%s the table shows$ there are two ways to be right and two ways to be wrong. Typically to

re#ect + 5 when it is actually true is a more serious error than to fail to re#ect it when it is false$ so the

former error is labeled GType ,H and the latter error GType ,,.H

(e)nition

>n a test of hypotheses, a Type , error is the decision to reject + 5 when it is in fact true. A Type ,, error is

the decision not to reject + 5 when it is in fact not true.






<nless we perform a census we do not have certain knowledge$ so we do not know whether our

decision matches the true state of nature or if we have made an error. (e re#ect + 5 if what we observe

would be a GrareH event if + 5 were true. 4ut rare events are not impossible: they occur with

probabilityα.Thus when + 5 is true$ a rare event will be observed in the proportionαof repeated

similar tests$ and + 5 will be erroneously re#ected in those tests. Thus αis the probability that in

following the testing procedure to decide between + 5 and + a we will make a Type , error.

e+()t)o(

The number αthat is used to determine the rejection region is called the level of significance of the test. >t

is the probability that the test procedure will result in a Type > error.

The probability of making a Type ,, error is too complicated to discuss in a beginning text$ so we will say

no more about it than this: for a fixed sample si'e$ choosingαsmaller in order to reduce the chance of

making a Type , error has the effect of increasing the chance of making a Type ,, error. The only way to

simultaneously reduce the chances of making either kind of error is to increase the sample si'e.

Standardiin$ the Test Statistic

Eypotheses testing will be considered in a number of contexts$ and great unification as well as

simplification results when the relevant sample statistic is standardized by subtracting its mean from it

and then dividing by its standard deviation. The resulting statistic is called a standardized test statistic. ,nevery situation treated in this and the following two chapters the standardi'ed test statistic will have

either the standard normal distribution or +tudent&s t -distribution.

e+()t)o(






A standardized test statistic for a hypothesis test is the statistic that is formed by subtracting from

the statistic of interest its mean and dividing by its standard deviation.











very instance of hypothesis testing discussed in this and the following two chapters will have a

re#ection region like one of the six forms tabulated in the tables above.

9o matter what the context a test of hypotheses can always be performed by applying the following

systematic procedure$ which will be illustrated in the examples in the succeeding sections.

Syste-at)c @ypothes)s &est)(g Proce*,re: %r)t)cal 'al,e

Approach

1 ,dentify the null and alternative hypotheses.

! ,dentify the relevant test statistic and its distribution.

3 7ompute from the data the value of the test statistic.

6 7onstruct the re#ection region.

8 7ompare the value computed in +tep 3 to the re#ection region constructed in +tep 6 and make a

decision. /ormulate the decision in the context of the problem$ if applicable.

The procedure that we have outlined in this section is called the G7ritical Dalue %pproachH tohypothesis testing to distinguish it from an alternative but e"uivalent approach that will be

introduced at the end of +ection ?.3 0The Observed +ignificance of a Test0.

*+, TA*+AA,S

• A test of hypotheses )s a stat)st)cal process for *ec)*)(g betwee( two co-pet)(g

assert)o(s abo,t a pop,lat)o( para-eter.

• &he test)(g proce*,re )s for-al)Ve* )( a +4e=step proce*,re.

++/C&S+S

1 State the (,ll a(* alter(at)4e hypotheses for each of the follow)(g s)t,at)o(s. &hat )s

)*e(t)fy the correct (,-ber µ0 a(* wr)te H0:µ=µ0 a(* the appropr)ate a(alogo,s e7press)o(

for a.5






a &he a4erage $,ly te-perat,re )( a reg)o( h)stor)cally has bee( .0. Perhaps

)t )s h)gher (ow.

b &he a4erage we)ght of a fe-ale a)rl)(e passe(ger w)th l,ggage was 10

po,(*s te( years ago. &he AA bel)e4es )t to be h)gher (ow.

c &he a4erage st)pe(* for *octoral st,*e(ts )( a part)c,lar *)sc)pl)(e at a state,()4ers)ty )s 10. &he *epart-e(t cha)r-a( bel)e4es that the (at)o(al

a4erage )s h)gher.

* &he a4erage roo- rate )( hotels )( a certa)( reg)o( )s 62.03. A tra4el age(t

bel)e4es that the a4erage )( a part)c,lar resort area )s *)8ere(t.

e &he a4erage far- s)Ve )( a pre*o-)(ately r,ral state was 9. acres. &he

secretary of agr)c,lt,re of that state asserts that )t )s less to*ay.

2 State the (,ll a(* alter(at)4e hypotheses for each of the follow)(g s)t,at)o(s. &hat )s

)*e(t)fy the correct (,-ber µ0 a(* wr)te H0:µ=µ0 a(* the appropr)ate a(alogo,s e7press)o(

for a.5

a &he a4erage t)-e workers spe(t co--,t)(g to work )( 'ero(a +4e years ago

was 36.2 -)(,tes. &he 'ero(a %ha-ber of %o--erce asserts that the

a4erage )s less (ow.

b &he -ea( salary for all -e( )( a certa)( profess)o( )s 06291. A spec)al

)(terest gro,p th)(ks that the -ea( salary for wo-e( )( the sa-e profess)o(

)s *)8ere(t.

c &he accepte* +g,re for the ca8e)(e co(te(t of a( 6=o,(ce c,p of co8ee )s 133

-g. A *)et)t)a( bel)e4es that the a4erage for co8ee ser4e* )( a local

resta,ra(ts )s h)gher.

* &he a4erage y)el* per acre for all types of cor( )( a rece(t year was 11.9

b,shels. A( eco(o-)st bel)e4es that the a4erage y)el* per acre )s *)8ere(t th)s

year.

e A( )(*,stry assoc)at)o( asserts that the a4erage age of all self=*escr)be* [y

+sher-e( )s 2.6 years. A soc)olog)st s,spects that )t )s h)gher.

3 escr)be the two types of errors that ca( be -a*e )( a test of hypotheses.

U(*er what c)rc,-sta(ce )s a test of hypotheses certa)( to y)el* a correct *ec)s)o(C






5.0 !ar$e Sample Tests or a 2opulation >ean

!+A/N&N: 1';+CT&<+S

1 &o lear( how to apply the +4e=step test proce*,re for a test of hypotheses

co(cer()(g a pop,lat)o( -ea( whe( the sa-ple s)Ve )s large.

2 &o lear( how to )(terpret the res,lt of a test of hypotheses )( the co(te7t of the

or)g)(al (arrate* s)t,at)o(.











EKAPLE

t )s hope* that a (ewly *e4elope* pa)( rel)e4er w)ll -ore <,)ckly pro*,ce

percept)ble re*,ct)o( )( pa)( to pat)e(ts after -)(or s,rger)es tha( a sta(*ar*

pa)( rel)e4er. &he sta(*ar* pa)( rel)e4er )s k(ow( to br)(g rel)ef )( a( a4erage of

3.0 -)(,tes w)th sta(*ar* *e4)at)o( 2.1 -)(,tes. &o test whether the (ew pa)(






rel)e4er works -ore <,)ckly tha( the sta(*ar* o(e 0; pat)e(ts w)th -)(or

s,rger)es were g)4e( the (ew pa)( rel)e4er a(* the)r t)-es to rel)ef were recor*e*.

&he e7per)-e(t y)el*e* sa-ple -ea( x −=3.1-)(,tes a(* sa-ple sta(*ar*

*e4)at)o( s 1.0 -)(,tes. s there s,?c)e(t e4)*e(ce )( the sa-ple to )(*)cate

at the 0G le4el of s)g()+ca(ce that the (ewly *e4elope* pa)( rel)e4er *oes

*el)4er percept)ble rel)ef -ore <,)cklyC

Sol,t)o(:

Be perfor- the test of hypotheses ,s)(g the +4e=step proce*,re g)4e( at the e(*

of Sect)o( 6.1 Q&he Ele-e(ts of @ypothes)s &est)(gQ.

• Step 1. &he (at,ral ass,-pt)o( )s that the (ew *r,g )s (o better tha( the ol*

o(e b,t -,st be pro4e* to be better. &h,s )f μ *e(otes the a4erage t)-e ,(t)l

all pat)e(ts who are g)4e( the (ew *r,g e7per)e(ce pa)( rel)ef the hypothes)s

test )s






percept)ble rel)ef fro- pa)( ,s)(g the (ew pa)( rel)e4er )s s-aller tha( the

a4erage t)-e for the sta(*ar* pa)( rel)e4er.

Figure ".;6eEection 6egion and )est %tatistic for 'ote ".27 >(ample 8>






+A>2!+ 8

A cos-et)cs co-pa(y +lls )ts best=sell)(g 6=o,(ce >ars of fac)al crea- by a(

a,to-at)c *)spe(s)(g -ach)(e. &he -ach)(e )s set to *)spe(se a -ea( of 6.1

o,(ces per >ar. U(co(trollable factors )( the process ca( sh)ft the -ea( away

fro- 6.1 a(* ca,se e)ther ,(*er+ll or o4er+ll both of wh)ch are ,(*es)rable. (

s,ch a case the *)spe(s)(g -ach)(e )s stoppe* a(* recal)brate*. Regar*less of

the -ea( a-o,(t *)spe(se* the sta(*ar* *e4)at)o( of the a-o,(t *)spe(se*

always has 4al,e ;.22 o,(ce. A <,al)ty co(trol e(g)(eer ro,t)(ely selects 3; >ars

fro- the asse-bly l)(e to check the a-o,(ts +lle*. "( o(e occas)o( the sa-ple

-ea( )s x−=8.2o,(ces a(* the sa-ple sta(*ar* *e4)at)o( )s s ;.20 o,(ce.

eter-)(e )f there )s s,?c)e(t e4)*e(ce )( the sa-ple to )(*)cate at the 1G le4el

of s)g()+ca(ce that the -ach)(e sho,l* be recal)brate*.

Sol,t)o(:

• Step 1. &he (at,ral ass,-pt)o( )s that the -ach)(e )s work)(g properly. &h,s

)f μ *e(otes the -ea( a-o,(t of fac)al crea- be)(g *)spe(se* the

hypothes)s test )s

H 0:µ = 8.1

vs.H a:µ=≠8.1@ α=0.01






Figure ".56eEection 6egion and )est %tatistic for 'ote ".2" >(ample ;>






*+, TA*+AA,S

• &here are two for-,las for the test stat)st)c )( test)(g hypotheses abo,t a

pop,lat)o( -ea( w)th large sa-ples. #oth test stat)st)cs follow the sta(*ar*

(or-al *)str)b,t)o(.

•

&he pop,lat)o( sta(*ar* *e4)at)o( )s ,se* )f )t )s k(ow( otherw)se the sa-plesta(*ar* *e4)at)o( )s ,se*.

• &he sa-e +4e=step proce*,re )s ,se* w)th e)ther test stat)st)c.

++/C&S+S

'AS&C

1 )(* the re>ect)o( reg)o( for the sta(*ar*)Ve* test stat)st)c5 for each hypothes)s test.

a H0:µ=274s. Ha:µ<27 α=0.05.

b H0:µ=524s. Ha:µ≠52 α=0.05.

c H0:µ=−1054s. Ha:µ>−105 α=0.10.

* H0:µ=78.84s. Ha:µ≠78.8 α=0.10.


a H0:µ=174s. Ha:µ<17 α=0.01.

b H0:µ=8804s. Ha:µ≠880 α=0.01.

c H0:µ=−124s. Ha:µ>−12 α=0.05.

* H0:µ=21.14s. Ha:µ≠21.1 α=0.05.


*e(t)fy the test as left=ta)le* r)ght=ta)le* or two=ta)le*.

a H0:µ=1414s. Ha:µ<141 α=0.20.






b H0:µ=−544s. Ha:µ<−54 α=0.05.

c H0:µ=98.64s. Ha:µ≠98.6 α=0.05.

* H0:µ=3.84s. Ha:µ>3.8 α=0.001.

)(* the re>ect)o( reg)o( for the sta(*ar*)Ve* test stat)st)c5 for each hypothes)s test.

*e(t)fy the test as left=ta)le* r)ght=ta)le* or two=ta)le*.

a H0:µ=−624s. Ha:µ≠−62 α=0.005.

b H0:µ=734s. Ha:µ>73 α=0.001.

c H0:µ=11244s. Ha:µ<1124 α=0.001.

* H0:µ=0.124s. Ha:µ≠0.12 α=0.001.

0 %o-p,te the 4al,e of the test stat)st)c for the )(*)cate* test base* o( the

)(for-at)o( g)4e(.

a &est)(g H0:µ=72.24s. Ha:µ>72.2 σ ,(k(ow( n 00 x−=75.1 s 9.20

b &est)(g H0:µ=584s. Ha:µ>58 σ 1.22 n ; x−=58.5 s 1.29

c &est)(g H0:µ=−19.54s. Ha:µ<−19.5 σ ,(k(ow( n 3; x−=−23.2 s 9.00

* &est)(g H0:µ=8054s. Ha:µ≠805 σ 3.0 n 0 x−=818 s 3.2

%o-p,te the 4al,e of the test stat)st)c for the )(*)cate* test base* o( the

)(for-at)o( g)4e(.

a &est)(g H0:µ=3424s. Ha:µ<342 σ 11.2 n ; x−=339 s 1;.3

b &est)(g H0:µ=1054s. Ha:µ>105 σ 0.3 n 6; x−=107 s 0.1

c &est)(g H0:µ=−13.54s. Ha:µ≠−13.5 σ ,(k(ow( n 32 x−=−13.8 s 1.0

* &est)(g H0:µ=284s. Ha:µ≠28 σ ,(k(ow( n 6 x−=27.8 s 1.3

Perfor- the )(*)cate* test of hypotheses base* o( the )(for-at)o( g)4e(.

a &est H0:µ=2124s. Ha:µ<212 α=0.10 σ ,(k(ow( n 3 x−=211.2 s 2.2

b &est H0:µ=−184s. Ha:µ>−18 α=0.05 σ 3.3 n x−=−17.2 s 3.1

c &est H0:µ=244s. Ha:µ≠24 α=0.02 σ ,(k(ow( n 0; x−=22.8 s 1.9

6 Perfor- the )(*)cate* test of hypotheses base* o( the )(for-at)o( g)4e(.

a &est H0:µ=1054s. Ha:µ>105 α=0.05 σ ,(k(ow( n 3; x−=108 s .2

b &est H0:µ=21.64s. Ha:µ<21.6 α=0.01 σ ,(k(ow( n 6 x−=20.5 s 3.9

c &est H0:µ=−3754s. Ha:µ≠−375 α=0.01 σ 16.0 n 31 x−=−388 s 16.;

Saylor URL: http://www.saylor.org/books Saylor.org;;





A22!&CAT&1NS

9 ( the past the a4erage le(gth of a( o,tgo)(g telepho(e call fro- a b,s)(ess o?ce has

bee( 13 seco(*s. A -a(ager w)shes to check whether that a4erage has *ecrease*

after the )(tro*,ct)o( of pol)cy cha(ges. A sa-ple of 1;; telepho(e calls pro*,ce* a

-ea( of 133 seco(*s w)th a sta(*ar* *e4)at)o( of 30 seco(*s. Perfor- the rele4a(ttest at the 1G le4el of s)g()+ca(ce.

1; &he go4er(-e(t of a( )-po4er)she* co,(try reports the -ea( age at *eath a-o(g

those who ha4e s,r4)4e* to a*,lthoo* as .2 years. A rel)ef age(cy e7a-)(es 3;

ra(*o-ly selecte* *eaths a(* obta)(s a -ea( of 2.3 years w)th sta(*ar* *e4)at)o( 6.1

years. &est whether the age(cyWs *ata s,pport the alter(at)4e hypothes)s at the 1G

le4el of s)g()+ca(ce that the pop,lat)o( -ea( )s less tha( .2.

11 &he a4erage ho,sehol* s)Ve )( a certa)( reg)o( se4eral years ago was 3.1 perso(s. A

soc)olog)st w)shes to test at the 0G le4el of s)g()+ca(ce whether )t )s *)8ere(t (ow.

Perfor- the test ,s)(g the )(for-at)o( collecte* by the soc)olog)st: )( a ra(*o- sa-ple

of 0 ho,sehol*s the a4erage s)Ve was 2.96 perso(s w)th sa-ple sta(*ar* *e4)at)o(

;.62 perso(.

12 &he reco--e(*e* *a)ly calor)e )(take for tee(age g)rls )s 22;; calor)es/*ay. A

(,tr)t)o()st at a state ,()4ers)ty bel)e4es the a4erage *a)ly calor)c )(take of g)rls )( that

state to be lower. &est that hypothes)s at the 0G le4el of s)g()+ca(ce aga)(st the (,ll

hypothes)s that the pop,lat)o( a4erage )s 22;; calor)es/*ay ,s)(g the follow)(g sa-ple

*ata: n 3 x−= 2,150 s 2;3.

13 A( a,to-ob)le -a(,fact,rer reco--e(*s o)l cha(ge )(ter4als of 3;;; -)les. &o

co-pare act,al )(ter4als to the reco--e(*at)o( the co-pa(y ra(*o-ly sa-ples

recor*s of 0; o)l cha(ges at ser4)ce fac)l)t)es a(* obta)(s sa-ple -ea( 302 -)les w)th

sa-ple sta(*ar* *e4)at)o( 36 -)les. eter-)(e whether the *ata pro4)*e s,?c)e(t

e4)*e(ce at the 0G le4el of s)g()+ca(ce that the pop,lat)o( -ea( )(ter4al betwee( o)l

cha(ges e7cee*s 3;;; -)les.

1 A -e*)cal laboratory cla)-s that the -ea( t,r(=aro,(* t)-e for perfor-a(ce of a

battery of tests o( bloo* sa-ples )s 1.66 b,s)(ess *ays. &he -a(ager of a large -e*)cal

pract)ce bel)e4es that the act,al -ea( )s larger. A ra(*o- sa-ple of 0 bloo* sa-ples

y)el*e* -ea( 2.;9 a(* sa-ple sta(*ar* *e4)at)o( ;.13 *ay. Perfor- the rele4a(t test atthe 1;G le4el of s)g()+ca(ce ,s)(g these *ata.

10 A grocery store cha)( has as o(e sta(*ar* of ser4)ce that the -ea( t)-e c,sto-ers wa)t

)( l)(e to beg)( check)(g o,t (ot e7cee* 2 -)(,tes. &o 4er)fy the perfor-a(ce of a store

the co-pa(y -eas,res the wa)t)(g t)-e )( 3; )(sta(ces obta)()(g -ea( t)-e 2.1

-)(,tes w)th sta(*ar* *e4)at)o( ;. -)(,te. Use these *ata to test the (,ll hypothes)s

Saylor URL: http://www.saylor.org/books Saylor.org;1





that the -ea( wa)t)(g t)-e )s 2 -)(,tes 4ers,s the alter(at)4e that )t e7cee*s 2

-)(,tes at the 1;G le4el of s)g()+ca(ce.

1 A -agaV)(e p,bl)sher tells pote(t)al a*4ert)sers that the -ea( ho,sehol* )(co-e of )ts

reg,lar rea*ersh)p )s 10;;. A( a*4ert)s)(g age(cy w)shes to test th)s cla)- aga)(st

the alter(at)4e that the -ea( )s s-aller. A sa-ple of ; ra(*o-ly selecte* reg,larrea*ers y)el*s -ea( )(co-e 096;; w)th sta(*ar* *e4)at)o( 060;. Perfor- the

rele4a(t test at the 1G le4el of s)g()+ca(ce.

1 A,thors of a co-p,ter algebra syste- w)sh to co-pare the spee* of a (ew

co-p,tat)o(al algor)th- to the c,rre(tly )-ple-e(te* algor)th-. &hey apply the (ew

algor)th- to 0; sta(*ar* proble-sD )t a4erages 6.1 seco(*s w)th sta(*ar* *e4)at)o(

;.1 seco(*. &he c,rre(t algor)th- a4erages 6.21 seco(*s o( s,ch proble-s. &est at

the 1G le4el of s)g()+ca(ce the alter(at)4e hypothes)s that the (ew algor)th- has a

lower a4erage t)-e tha( the c,rre(t algor)th-.

16 A ra(*o- sa-ple of the start)(g salar)es of 30 ra(*o-ly selecte* gra*,ates w)th

bachelorWs *egrees last year ga4e sa-ple -ea( a(* sta(*ar* *e4)at)o( 12;2 a(*

21 respect)4ely. &est whether the *ata pro4)*e s,?c)e(t e4)*e(ce at the 0G le4el

of s)g()+ca(ce to co(cl,*e that the -ea( start)(g salary of all gra*,ates last year )s

less tha( the -ea( of all gra*,ates two years before 3069.

A((&T&1NA! ++/C&S+S

19 &he -ea( ho,sehol* )(co-e )( a reg)o( ser4e* by a cha)( of cloth)(g stores )s 60;.

( a sa-ple of ; c,sto-ers take( at 4ar)o,s stores the -ea( )(co-e of the c,sto-ers

was 010;0 w)th sta(*ar* *e4)at)o( 602.a &est at the 1;G le4el of s)g()+ca(ce the (,ll hypothes)s that the -ea(

ho,sehol* )(co-e of c,sto-ers of the cha)( )s 60; aga)(st that

alter(at)4e that )t )s *)8ere(t fro- 60;.

b &he sa-ple -ea( )s greater tha( 60; s,ggest)(g that the act,al -ea( of

people who patro()Ve th)s store )s greater tha( 60;. Perfor- th)s test also

at the 1;G le4el of s)g()+ca(ce. &he co-p,tat)o( of the test stat)st)c *o(e )(

part a5 st)ll appl)es here.5

2; &he labor charge for repa)rs at a( a,to-ob)le ser4)ce ce(ter are base* o( a sta(*ar*

t)-e spec)+e* for each type of repa)r. &he t)-e spec)+e* for replace-e(t of ,()4ersal

>o)(t )( a *r)4e shaft )s o(e ho,r. &he -a(ager re4)ews a sa-ple of 3; s,ch repa)rs. &he

a4erage of the act,al repa)r t)-es )s ;.6 ho,r w)th sta(*ar* *e4)at)o( ;.32 ho,r.

a &est at the 1G le4el of s)g()+ca(ce the (,ll hypothes)s that the act,al -ea(

t)-e for th)s repa)r *)8ers fro- o(e ho,r.






b &he sa-ple -ea( )s less tha( o(e ho,r s,ggest)(g that the -ea( act,al t)-e

for th)s repa)r )s less tha( o(e ho,r. Perfor- th)s test also at the 1G le4el of

s)g()+ca(ce. &he co-p,tat)o( of the test stat)st)c *o(e )( part a5 st)ll appl)es

here.5

!A/:+ (ATA S+T ++/C &S+S

21 Large ata Set 1 recor*s the SA& scores of 1;;; st,*e(ts. Regar*)(g )t as a ra(*o-

sa-ple of all h)gh school st,*e(ts ,se )t to test the hypothes)s that the pop,lat)o(

-ea( e7cee*s 101; at the 1G le4el of s)g()+ca(ce. &he (,ll hypothes)s )s that μ

101;.5

http://www.1.7ls

22 Large ata Set 1 recor*s the !PAs of 1;;; college st,*e(ts. Regar*)(g )t as a ra(*o-

sa-ple of all college st,*e(ts ,se )t to test the hypothes)s that the pop,lat)o( -ea( )s

less tha( 2.0; at the 1;G le4el of s)g()+ca(ce. &he (,ll hypothes)s )s that μ 2.0;.5

http://www.1.7ls

23 Large ata Set 1 l)sts the SA& scores of 1;;; st,*e(ts.

http://www.1.7ls

a Regar* the *ata as ar)s)(g fro- a ce(s,s of all st,*e(ts at a h)gh school )(

wh)ch the SA& score of e4ery st,*e(t was -eas,re*. %o-p,te the pop,lat)o(

-ea( μ.

b Regar* the +rst 0; st,*e(ts )( the *ata set as a ra(*o- sa-ple *raw( fro-

the pop,lat)o( of part a5 a(* ,se )t to test the hypothes)s that the pop,lat)o(

-ea( e7cee*s 101; at the 1;G le4el of s)g()+ca(ce. &he (,ll hypothes)s )sthat μ 101;.5

c s yo,r co(cl,s)o( )( part b5 )( agree-e(t w)th the tr,e state of (at,re wh)ch

by part a5 yo, k(ow5 or )s yo,r *ec)s)o( )( errorC f yo,r *ec)s)o( )s )( error )s

)t a &ype error or a &ype errorC


http://www.1.7ls

a Regar* the *ata as ar)s)(g fro- a ce(s,s of all fresh-a( at a s-all college at

the e(* of the)r +rst aca*e-)c year of college st,*y )( wh)ch the !PA of e4ery

s,ch perso( was -eas,re*. %o-p,te the pop,lat)o( -ea( μ.

b Regar* the +rst 0; st,*e(ts )( the *ata set as a ra(*o- sa-ple *raw( fro-

the pop,lat)o( of part a5 a(* ,se )t to test the hypothes)s that the pop,lat)o(

-ea( )s less tha( 2.0; at the 1;G le4el of s)g()+ca(ce. &he (,ll hypothes)s )s

that μ 2.0;.5






c s yo,r co(cl,s)o( )( part b5 )( agree-e(t w)th the tr,e state of (at,re wh)ch

by part a5 yo, k(ow5 or )s yo,r *ec)s)o( )( errorC f yo,r *ec)s)o( )s )( error )s

)t a &ype error or a &ype errorC






5.3 The 1bserved Si$ni)cance o a Test

LEARNN! "#$E%&'ES

1 &o lear( what the obser4e* s)g()+ca(ce of a test )s.

2 &o lear( how to co-p,te the obser4e* s)g()+ca(ce of a test.

3 &o lear( how to apply the p=4al,e approach to hypothes)s test)(g.

The 1bserved Si$ni)cance

The conceptual basis of our testing procedure is that we re#ect + 5 only if the data that we obtained would

constitute a rare event if + 5 were actually true. The level of significanceαspecifies what is meant by

Grare.H The observed significance of the test is a measure of how rare the value of the test statistic that we

have #ust observed would be if the null hypothesis were true. That is$ the observed significance of the test

#ust performed is the probability that$ if the test were repeated with a new sample$ the result of the new

test would be at least as contrary to + 5 and in support of + a as what was observed in the original test.






e+()t)o(

The observed significance or p(value of a specific test of hypotheses is the probability, on the

supposition that + 5 is true, of obtaining a result at least as contrary to + 5 and in favor of + a as the result

actually observed in the sample data.

Think back to 9ote ?.!@ 0xample 60 in +ection ?.! 0arge +ample Tests for a opulation

ean0 concerning the effectiveness of a new pain reliever. This was a left-tailed test in which the value of

the test statistic was Z1.??;. To be as contrary to + 5 and in support of + a as the resultZ=−1.886actually

observed means to obtain a value of the test statistic in the interval(−∞,−1.886].Counding Z1.??; to

Z1.?F$ we can read directly from /igure 1!.! 07umulative 9ormal

robability0 that P(Z≤−1.89)=0.0294.Thus the p-value or observed significance of the test in 9ote ?.!@

0xample 60 is 5.5!F6 or about 3B. <nder repeated sampling from this population$ if + 5 were true then

only about 3B of all samples of si'e 85 would give a result as contrary to + 5 and in favor of + a as the

sample we observed. 9ote that the probability 5.5!F6 is the area of the left tail cut off by the test statistic

in this left-tailed test.

%nalogous reasoning applies to a right-tailed or a two-tailed test$ except that in the case of a two-tailed

test being as far from 5 as the observed value of the test statistic but on the opposite side of 5 is #ust as

contrary to + 5 as being the same distance away and on the same side of 5$ hence the corresponding tail

area is doubled.

Computational (e)nition o the 1bserved Si$ni)canceo a Test o =ypothesesThe observed significance of a test of hypotheses is the area of the tail of the distribution cut off by the

test statistic *times two in the case of a two-tailed test).

+A>2!+ 9

%o-p,te the obser4e* s)g()+ca(ce of the test perfor-e* )( Note 6.26 QE7a-ple

0Q)( Sect)o( 6.2 QLarge Sa-ple &ests for a Pop,lat)o( ea(Q.






Sol,t)o(:

&he 4al,e of the test stat)st)c was z 2.9; wh)ch by )g,re 12.2 Q%,-,lat)4e

Nor-al Probab)l)tyQ c,ts o8 a ta)l of area ;.;; as show( )( )g,re 6. QArea of the

&a)l for Q. S)(ce the test was two=ta)le* the obser4e* s)g()+ca(ce )s 2×0.0064=0.0128.

Figure ".7 0rea of t!e )ail for 'ote ".*8 >(ample 5>

The p-value Approach to =ypothesis Testin$

,n 9ote ?.!@ 0xample 60 in +ection ?.! 0arge +ample Tests for a opulation ean0 the test was

performed at the 8B level of significance: the definition of GrareH event was probability α=0.05or less.

(e saw above that the observed significance of the test was p M 5.5!F6 or about 3B.






+ince p=0.0294<0.05=α*or 3B is less than 8B)$ the decision turned out to be to re#ect: what was

observed was sufficiently unlikely to "ualify as an event so rare as to be regarded as *practically)

incompatible with + 5.

,n 9ote ?.!? 0xample 80 in +ection ?.! 0arge +ample Tests for a opulation ean0 the test was

performed at the 1B level of significance: the definition of GrareH event was probability α=0.01or less.

The observed significance of the test was computed in 9ote ?.36 0xample ;0 as p M 5.51!? or about

1.3B. +ince p=0.0128>0.01=α*or 1.3B is greater than 1B)$ the decision turned out to be not to re#ect.

The event observed was unlikely$ but not sufficiently unlikely to lead to re#ection of the null

hypothesis.

The reasoning #ust presented is the basis for a slightly different but e"uivalent formulation of the

hypothesis testing process. The first three steps are the same as before$ but instead of using αto

compute critical values and construct a re#ection region$ one computes the p-value p of the test and

compares it toα$ re#ecting + 5 if p≤αand not re#ecting if p>α.

Systematic =ypothesis Testin$ 2rocedureH p-<alueApproach1 ,dentify the null and alternative hypotheses.

! ,dentify the relevant test statistic and its distribution.

3 7ompute from the data the value of the test statistic.

6 7ompute the p-value of the test.

8 7ompare the value computed in +tep 6 to significance levelαand make a decision:

re#ect + 5 if p≤αand do not re#ect + 5 if p>α./ormulate the decision in the context of the problem$ ifapplicable.











+A>2!+ 5

r. Prospero has bee( teach)(g Algebra fro- a part)c,lar te7tbook at Re-ote

sle @)gh School for -a(y years. "4er the years st,*e(ts )( h)s Algebra classesha4e co(s)ste(tly score* a( a4erage of o( the e(* of co,rse e7a- E"%5. &h)s

year r. Prospero ,se* a (ew te7tbook )( the hope that the a4erage score o( the

E"% test wo,l* be h)gher. &he a4erage E"% test score of the st,*e(ts who

took Algebra fro- r. Prospero th)s year ha* -ea( 9. a(* sa-ple sta(*ar*

*e4)at)o( .1. eter-)(e whether these *ata pro4)*e s,?c)e(t e4)*e(ce at the






1G le4el of s)g()+ca(ce to co(cl,*e that the a4erage E"% test score )s h)gher

w)th the (ew te7tbook.

Sol,t)o(:

• Step 1. Let μ be the tr,e a4erage score o( the E"% e7a- of all r. ProsperoWs

st,*e(ts who take the Algebra co,rse w)th the (ew te7tbook. &he (at,ral

state-e(t that wo,l* be ass,-e* tr,e ,(less there were stro(g e4)*e(ce to

the co(trary )s that the (ew book )s abo,t the sa-e as the ol* o(e. &he

alter(at)4e wh)ch )t takes e4)*e(ce to establ)sh )s that the (ew book )s

better wh)ch correspo(*s to a h)gher 4al,e of μ. &h,s the rele4a(t test )s

H0:µ = 67

vs.Ha:µ >67@ α=0.01






Figure ".)est %tatistic for 'ote ".*7 >(ample ">











Figure ".1A)est %tatistic for 'ote ".*" >(ample >

*+, TA*+AA,S

• &he obser4e* s)g()+ca(ce or p=4al,e of a test )s a -eas,re of how )(co(s)ste(t

the sa-ple res,lt )s w)th ; a(* )( fa4or of a.

• &he p=4al,e approach to hypothes)s test)(g -ea(s that o(e -erely co-pares

the p=4al,e to α)(stea* of co(str,ct)(g a re>ect)o( reg)o(.






• &here )s a syste-at)c +4e=step proce*,re for the p=4al,e approach to hypothes)s

test)(g.






a Perfor- the rele4a(t test of hypotheses at the 2;G le4el of s)g()+ca(ce ,s)(g

the cr)t)cal 4al,e approach.

b %o-p,te the obser4e* s)g()+ca(ce of the test.

c Perfor- the test at the 2;G le4el of s)g()+ca(ce ,s)(g the p=4al,e approach.

Jo, (ee* (ot repeat the +rst three steps alrea*y *o(e )( part a5.

9 &he -ea( score o( a 20=po)(t place-e(t e7a- )( -athe-at)cs ,se* for the past two

years at a large state ,()4ers)ty )s 1.3. &he place-e(t coor*)(ator w)shes to test

whether the -ea( score o( a re4)se* 4ers)o( of the e7a- *)8ers fro- 1.3. She g)4es






the re4)se* e7a- to 3; e(ter)(g fresh-e( early )( the s,--erD the -ea( score )s 1.

w)th sta(*ar* *e4)at)o( 2..

a Perfor- the test at the 1;G le4el of s)g()+ca(ce ,s)(g the cr)t)cal 4al,e

approach.

b %o-p,te the obser4e* s)g()+ca(ce of the test.c Perfor- the test at the 1;G le4el of s)g()+ca(ce ,s)(g the p=4al,e approach. Jo,

(ee* (ot repeat the +rst three steps alrea*y *o(e )( part a5.

1; &he -ea( )(crease )( wor* fa-)ly 4ocab,lary a-o(g st,*e(ts )( a o(e=year fore)g(

la(g,age co,rse )s 0 wor* fa-)l)es. ( or*er to est)-ate the e8ect of a (ew type of

class sche*,l)(g a( )(str,ctor -o()tors the progress of ; st,*e(tsD the sa-ple -ea(

)(crease )( wor* fa-)ly 4ocab,lary of these st,*e(ts )s 02 wor* fa-)l)es w)th sa-ple

sta(*ar* *e4)at)o( 16 wor* fa-)l)es.

a &est at the 0G le4el of s)g()+ca(ce whether the -ea( )(crease w)th the (ew

class sche*,l)(g )s *)8ere(t fro- 0 wor* fa-)l)es ,s)(g the cr)t)cal 4al,e

approach.


c Perfor- the test at the 0G le4el of s)g()+ca(ce ,s)(g the p=4al,e approach. Jo,


11 &he -ea( y)el* for har* re* w)(ter wheat )( a certa)( state )s .6 b,/acre. ( a p)lot

progra- a -o*)+e* grow)(g sche-e was )(tro*,ce* o( 30 )(*epe(*e(t plots. &he

res,lt was a sa-ple -ea( y)el* of 0. b,/acre w)th sa-ple sta(*ar* *e4)at)o( 1.

b,/acre a( appare(t )(crease )( y)el*.

a &est at the 0G le4el of s)g()+ca(ce whether the -ea( y)el* ,(*er the (ew

sche-e )s greater tha( .6 b,/acre ,s)(g the cr)t)cal 4al,e approach.




12 &he a4erage a-o,(t of t)-e that 4)s)tors spe(t look)(g at a reta)l co-pa(yWs ol* ho-e

page o( the worl* w)*e web was 23. seco(*s. &he co-pa(y co--)ss)o(s a (ew ho-e

page. "( )ts +rst *ay )( place the -ea( t)-e spe(t at the (ew page by 26 4)s)tors

was 23.0 seco(*s w)th sta(*ar* *e4)at)o( 0.1 seco(*s.

a &est at the 0G le4el of s)g()+ca(ce whether the -ea( 4)s)t t)-e for the (ew

page )s less tha( the for-er -ea( of 23. seco(*s ,s)(g the cr)t)cal 4al,e

approach.














5.7 Small Sample Tests or a 2opulation >ean

LEARNN! "#$E%&'E

1 &o lear( how to apply the +4e=step test proce*,re for test of hypotheses

co(cer()(g a pop,lat)o( -ea( whe( the sa-ple s)Ve )s s-all.

,n the previous section hypotheses testing for population means was described in the case of large

samples. The statistical validity of the tests was insured by the 7entral imit Theorem$ with

essentially no assumptions on the distribution of the population. (hen sample si'es are small$ as is

often the case in practice$ the 7entral imit Theorem does not apply. One must then impose stricter

assumptions on the population to give statistical validity to the test procedure. One commonassumption is that the population from which the sample is taken has a normal probability

distribution to begin with. <nder such circumstances$ if the population standard deviation is known$

then the test statistic x−−µ0) σ √n)still has the standard normal distribution$ as in the previous

two sections. ,f 6 is unknown and is approximated by the sample standard deviation s$ then the

resulting test statistic x−−µ0) s √n)follows +tudent&s t -distribution withn−1degrees of freedom.






!igure 3."" (istribution of the %tandardized Test %tatistic and the -ejection -egion






The p-value of a test of hypotheses for which the test statistic has +tudent&s t -distribution can be

computed using statistical software$ but it is impractical to do so using tables$ since that would

re"uire 35 tables analogous to /igure 1!.! 07umulative 9ormal robability0$ one for each degree of

freedom from 1 to 35./igure 1!.3 07ritical Dalues of 0 can be used to approximate the p-value of such

a test$ and this is typically ade"uate for making a decision using the p-value approach to hypothesis

testing$ although not always. /or this reason the tests in the two examples in this section will be

made following the critical value approach to hypothesis testing summari'ed at the end of +ection ?.1

0The lements of Eypothesis Testing0$ but after each one we will show how the p-value approach

could have been used.

EKAPLE 1;

&he pr)ce of a pop,lar te(()s racket at a (at)o(al cha)( store )s 19. Port)a bo,ght

+4e of the sa-e racket at a( o(l)(e a,ct)o( s)te for the follow)(g pr)ces:

155 179 175 175 161






Ass,-)(g that the a,ct)o( pr)ces of rackets are (or-ally *)str)b,te* *eter-)(e

whether there )s s,?c)e(t e4)*e(ce )( the sa-ple at the 0G le4el of s)g()+ca(ce to

co(cl,*e that the a4erage pr)ce of the racket )s less tha( 19 )f p,rchase* at a(

o(l)(e a,ct)o(.

Sol,t)o(:

• Step 1. &he assert)o( for wh)ch e4)*e(ce -,st be pro4)*e* )s that the a4erage

o(l)(e pr)ce μ )s less tha( the a4erage pr)ce )( reta)l stores so the hypothes)s

test )s






(−∞,−2.132].

• Step 0. As show( )( )g,re 6.12 QRe>ect)o( Reg)o( a(* &est Stat)st)c for Q the

test stat)st)c falls )( the re>ect)o( reg)o(. &he *ec)s)o( )s to re>ect ;. ( the

co(te7t of the proble- o,r co(cl,s)o( )s:

&he *ata pro4)*e s,?c)e(t e4)*e(ce at the 0G le4el of s)g()+ca(ce to

co(cl,*e that the a4erage pr)ce of s,ch rackets p,rchase* at o(l)(e a,ct)o(s )s

less tha( 19.






Figure ".126eEection 6egion and )est %tatistic for 'ote ".82 >(ample 1A>

To perform the test in 9ote ?.6! 0xample 150 using the p-value approach$ look in the row in /igure 1!.3

07ritical Dalues of 0 with the headingdf=4and search for the two t -values that bracket the unsigned value

!.18! of the test statistic. They are !.13! and !.@@;$ in the columns with headings t 5.585 and t 5.5!8. They cut

off right tails of area 5.585 and 5.5!8$ so because !.18! is between them it must cut off a tail of area

between 5.585 and 5.5!8. 4y symmetry Z!.18! cuts off a left tail of area between 5.585 and 5.5!8$ hence

the p-value corresponding tot=−2.152is between 5.5!8 and 5.58. %lthough its precise value is unknown$

it must be less than α=0.05$ so the decision is to re#ect + 5.

+A>2!+ %%

A s-all co-po(e(t )( a( electro()c *e4)ce has two s-all holes where a(other t)(y

part )s +tte*. ( the -a(,fact,r)(g process the a4erage *)sta(ce betwee( the two

holes -,st be t)ghtly co(trolle* at ;.;2 -- else -a(y ,()ts wo,l* be *efect)4e

a(* waste*. a(y t)-es thro,gho,t the *ay <,al)ty co(trol e(g)(eers take a

s-all sa-ple of the co-po(e(ts fro- the pro*,ct)o( l)(e -eas,re the *)sta(ce

betwee( the two holes a(* -ake a*>,st-e(ts )f (ee*e*. S,ppose at o(e t)-e

fo,r ,()ts are take( a(* the *)sta(ces are -eas,re* as

0.021 0.019 0.023 0.020






eter-)(e at the 1G le4el of s)g()+ca(ce )f there )s s,?c)e(t e4)*e(ce )( the

sa-ple to co(cl,*e that a( a*>,st-e(t )s (ee*e*. Ass,-e the *)sta(ces of

)(terest are (or-ally *)str)b,te*.

Sol,t)o(:

• Step 1. &he ass,-pt)o( )s that the process )s ,(*er co(trol ,(less there )s

stro(g e4)*e(ce to the co(trary. S)(ce a *e4)at)o( of the a4erage *)sta(ce to

e)ther s)*e )s ,(*es)rable the rele4a(t test )s






co(cl,s)o( )s:

&he *ata *o (ot pro4)*e s,?c)e(t e4)*e(ce at the 1G le4el of s)g()+ca(ce to

co(cl,*e that the -ea( *)sta(ce betwee( the holes )( the co-po(e(t *)8ers

fro- ;.;2 --.

Figure ".1*6eEection 6egion and )est %tatistic for 'ote ".8* >(ample 11>

To perform the test in 9ote ?.63 0xample 110 using the p-value approach$ look in the row

in /igure 1!.3 07ritical Dalues of 0 with the headingdf=3and search for the two t -values that

bracket the value 5.?@@ of the test statistic. %ctually 5.?@@ is smaller than the smallest number in

the row$ which is 5.F@?$ in the column with heading t 5.!55. The value 5.F@? cuts off a right tail of

area 5.!55$ so because 5.?@@ is to its left it must cut off a tail of area greater than 5.!55. Thus

the p-value$ which is the double of the area cut off *since the test is two-tailed)$ is greater than

5.655. %lthough its precise value is unknown$ it must be greater than α=0.01$ so the decision is not

to re#ect + 5.

*+, TA*+AA,S






• &here are two for-,las for the test stat)st)c )( test)(g hypotheses abo,t a

pop,lat)o( -ea( w)th s-all sa-ples. "(e test stat)st)c follows the sta(*ar*

(or-al *)str)b,t)o( the other St,*e(tWs t =*)str)b,t)o(.

• &he pop,lat)o( sta(*ar* *e4)at)o( )s ,se* )f )t )s k(ow( otherw)se the sa-ple

sta(*ar* *e4)at)o( )s ,se*.• E)ther +4e=step proce*,re cr)t)cal 4al,e or p=4al,e approach )s ,se* w)th e)ther

test stat)st)c.






a &est H0:µ=2504s. Ha:µ>250 α=0.05.

b Est)-ate the obser4e* s)g()+ca(ce of the test )( part a5 a(* state a *ec)s)o( base* o(

the p=4al,e approach to hypothes)s test)(g.

6 A ra(*o- sa-ple of s)Ve 12 *raw( fro- a (or-al pop,lat)o( y)el*e* the follow)(g

res,lts: x−=86.2 s ;.3.

a &est H0:µ=85.54s. Ha:µ≠85.5 α=0.01.

b Est)-ate the obser4e* s)g()+ca(ce of the test )( part a5 a(* state a *ec)s)o(

base* o( the p=4al,e approach to hypothes)s test)(g.






A22!&CAT&1NS

9 Researchers w)sh to test the e?cacy of a progra- )(te(*e* to re*,ce the le(gth of

labor )( ch)l*b)rth. &he accepte* -ea( labor t)-e )( the b)rth of a +rst ch)l* )s 10.3

ho,rs. &he -ea( le(gth of the labors of 13 +rst=t)-e -others )( a p)lot progra- was 6.6ho,rs w)th sta(*ar* *e4)at)o( 3.1 ho,rs. Ass,-)(g a (or-al *)str)b,t)o( of t)-es of

labor test at the 1;G le4el of s)g()+ca(ce test whether the -ea( labor t)-e for all

wo-e( follow)(g th)s progra- )s less tha( 10.3 ho,rs.

1; A *a)ry far- ,ses the so-at)c cell co,(t S%%5 report o( the -)lk )t pro4)*es to a

processor as o(e way to -o()tor the health of )ts her*. &he -ea( S%% fro- +4e sa-ples

of raw -)lk was 20;;;; cells per -)ll)l)ter w)th sta(*ar* *e4)at)o( 30;; cell/-l. &est

whether these *ata pro4)*e s,?c)e(t e4)*e(ce at the 1;G le4el of s)g()+ca(ce to

co(cl,*e that the -ea( S%% of all -)lk pro*,ce* at the *a)ry e7cee*s that )( the

pre4)o,s report 21;20; cell/-l. Ass,-e a (or-al *)str)b,t)o( of S%%.

11 S)7 co)(s of the sa-e type are *)sco4ere* at a( archaeolog)cal s)te. f the)r we)ghts o(

a4erage are s)g()+ca(tly *)8ere(t fro- 0.20 gra-s the( )t ca( be ass,-e* that the)r

pro4e(a(ce )s (ot the s)te )tself. &he co)(s are we)ghe* a(* ha4e -ea( .3 g w)th

sa-ple sta(*ar* *e4)at)o( ;.16 g. Perfor- the rele4a(t test at the ;.1G 1/1;th of 1G5

le4el of s)g()+ca(ce ass,-)(g a (or-al *)str)b,t)o( of we)ghts of all s,ch co)(s.

12 A( eco(o-)st w)shes to *eter-)(e whether people are *r)4)(g less tha( )( the past. (

o(e reg)o( of the co,(try the (,-ber of -)les *r)4e( per ho,sehol* per year )( the past

was 16.09 tho,sa(* -)les. A sa-ple of 10 ho,sehol*s pro*,ce* a sa-ple -ea( of

1.23 tho,sa(* -)les for the last year w)th sa-ple sta(*ar* *e4)at)o( .; tho,sa(*

-)les. Ass,-)(g a (or-al *)str)b,t)o( of ho,sehol* *r)4)(g *)sta(ces per year perfor-

the rele4a(t test at the 0G le4el of s)g()+ca(ce.

13 &he reco--e(*e* *a)ly allowa(ce of )ro( for fe-ales age* 190; )s 16 -g/*ay. A

caref,l -eas,re-e(t of the *a)ly )ro( )(take of 10 wo-e( y)el*e* a -ea( *a)ly )(take of

1.2 -g w)th sa-ple sta(*ar* *e4)at)o( . -g.

a Ass,-)(g that *a)ly )ro( )(take )( wo-e( )s (or-ally *)str)b,te* perfor- the

test that the act,al -ea( *a)ly )(take for all wo-e( )s *)8ere(t fro- 16

-g/*ay at the 1;G le4el of s)g()+ca(ce.






b &he sa-ple -ea( )s less tha( 16 s,ggest)(g that the act,al pop,lat)o( -ea(

)s less tha( 16 -g/*ay. Perfor- th)s test also at the 1;G le4el of s)g()+ca(ce.

&he co-p,tat)o( of the test stat)st)c *o(e )( part a5 st)ll appl)es here.5

1 &he target te-perat,re for a hot be4erage the -o-e(t )t )s *)spe(se* fro- a 4e(*)(g

-ach)(e )s 1;. A sa-ple of te( ra(*o-ly selecte* ser4)(gs fro- a (ew -ach)(e,(*ergo)(g a pre=sh)p-e(t )(spect)o( ga4e -ea( te-perat,re 13 w)th sa-ple

sta(*ar* *e4)at)o( .3.

a Ass,-)(g that te-perat,re )s (or-ally *)str)b,te* perfor- the test that the

-ea( te-perat,re of *)spe(se* be4erages )s *)8ere(t fro- 1; at the 1;G

le4el of s)g()+ca(ce.

b &he sa-ple -ea( )s greater tha( 1; s,ggest)(g that the act,al pop,lat)o(

-ea( )s greater tha( 1;. Perfor- th)s test also at the 1;G le4el of

s)g()+ca(ce. &he co-p,tat)o( of the test stat)st)c *o(e )( part a5 st)ll appl)es

here.5

10 &he a4erage (,-ber of *ays to co-plete reco4ery fro- a part)c,lar type of k(ee

operat)o( )s 123. *ays. ro- h)s e7per)e(ce a phys)c)a( s,spects that ,se of a top)cal

pa)( -e*)cat)o( -)ght be le(gthe()(g the reco4ery t)-e. @e ra(*o-ly selects the

recor*s of se4e( k(ee s,rgery pat)e(ts who ,se* the top)cal -e*)cat)o(. &he t)-es to

total reco4ery were:






2;;;; at the 1;G le4el of s)g()+ca(ce. Ass,-e that the SP% follows a (or-al

*)str)b,t)o(.

16 "(e water <,al)ty sta(*ar* for water that )s *)scharge* )(to a part)c,lar type of strea-

or po(* )s that the a4erage *a)ly water te-perat,re be at -ost 16%. S)7 sa-ples take(

thro,gho,t the *ay ga4e the *ata:

16.8 21.5 19.1 12.8 18.0 20.7

&he sa-ple -ea( x−=18.15e7cee*s 16 b,t perhaps th)s )s o(ly sa-pl)(g error.

eter-)(e whether the *ata pro4)*e s,?c)e(t e4)*e(ce at the 1;G le4el of

s)g()+ca(ce to co(cl,*e that the -ea( te-perat,re for the e(t)re *ay e7cee*s 16%.

A((&T&1NA! ++/C&S+S

19 A calc,lator has a b,)lt=)( algor)th- for ge(erat)(g a ra(*o- (,-ber accor*)(g to the

sta(*ar* (or-al *)str)b,t)o(. &we(ty=+4e (,-bers th,s ge(erate* ha4e -ea( ;.10 a(*






sa-ple sta(*ar* *e4)at)o( ;.9. &est the (,ll hypothes)s that the -ea( of all (,-bers

so ge(erate* )s ; 4ers,s the alter(at)4e that )t )s *)8ere(t fro- ; at the 2;G le4el of

s)g()+ca(ce. Ass,-e that the (,-bers *o follow a (or-al *)str)b,t)o(.

2; At e4ery sett)(g a h)gh=spee* pack)(g -ach)(e *el)4ers a pro*,ct )( a-o,(ts that 4ary

fro- co(ta)(er to co(ta)(er w)th a (or-al *)str)b,t)o( of sta(*ar* *e4)at)o( ;.12 o,(ce. &o co-pare the a-o,(t *el)4ere* at the c,rre(t sett)(g to the *es)re* a-o,(t .1

o,(ce a <,al)ty )(spector ra(*o-ly selects +4e co(ta)(ers a(* -eas,res the co(te(ts

of each obta)()(g sa-ple -ea( 3.9 o,(ces a(* sa-ple sta(*ar* *e4)at)o( ;.1;

o,(ce. &est whether the *ata pro4)*e s,?c)e(t e4)*e(ce at the 0G le4el of s)g()+ca(ce

to co(cl,*e that the -ea( of all co(ta)(ers at the c,rre(t sett)(g )s less tha( .1

o,(ces.

21 A -a(,fact,r)(g co-pa(y rece)4es a sh)p-e(t of 1;;; bolts of (o-)(al shear stre(gth

30; lb. A <,al)ty co(trol )(spector selects +4e bolts at ra(*o- a(* -eas,res the

shear stre(gth of each. &he *ata are:

4,320 4,290 4,360 4,350 4,320

a Ass,-)(g a (or-al *)str)b,t)o( of shear stre(gths test the (,ll hypothes)s

that the -ea( shear stre(gth of all bolts )( the sh)p-e(t )s 30; lb 4ers,s the

alter(at)4e that )t )s less tha( 30; lb at the 1;G le4el of s)g()+ca(ce.

b Est)-ate the p=4al,e obser4e* s)g()+ca(ce5 of the test of part a5.

c %o-pare the p=4al,e fo,(* )( part b5 to α=0.10a(* -ake a *ec)s)o( base* o(

the p=4al,e approach. E7pla)( f,lly.

22 A l)terary h)stor)a( e7a-)(es a (ewly *)sco4ere* *oc,-e(t poss)bly wr)tte( by "bero(

&hese,s. &he -ea( a4erage se(te(ce le(gth of the s,r4)4)(g ,(*)sp,te* works of

"bero( &hese,s )s 6.2 wor*s. &he h)stor)a( co,(ts wor*s )( se(te(ces betwee( +4e

s,ccess)4e 1;1 per)o*s )( the *oc,-e(t )( <,est)o( to obta)( a -ea( a4erage se(te(ce

le(gth of 39. wor*s w)th sta(*ar* *e4)at)o( .0 wor*s. &h,s the sa-ple s)Ve )s +4e.5

a eter-)(e )f these *ata pro4)*e s,?c)e(t e4)*e(ce at the 1G le4el of

s)g()+ca(ce to co(cl,*e that the -ea( a4erage se(te(ce le(gth )( the

*oc,-e(t )s less tha( 6.2.

b Est)-ate the p=4al,e of the test.

c #ase* o( the a(swers to parts a5 a(* b5 state whether or (ot )t )s l)kely that

the *oc,-e(t was wr)tte( by "bero( &hese,s.











5.8 !ar$e Sample Tests or a 2opulation2roportion

!+A/N&N: 1';+CT&<+S

1 &o lear( how to apply the +4e=step cr)t)cal 4al,e test proce*,re for test of

hypotheses co(cer()(g a pop,lat)o( proport)o(.

2 &o lear( how to apply the +4e=step p=4al,e test proce*,re for test of hypotheses

co(cer()(g a pop,lat)o( proport)o(.

4oth the critical value approach and the p-value approach can be applied to test hypotheses about a

population proportion p. The null hypothesis will have the form H0: p= p0for some specific

number p5 between 5 and 1. The alternative hypothesis will be one of the three

ine"ualities p< p0 p> p0 or p≠ p0 for the same number p5 that appears in the null hypothesis.











!igure 3." (istribution of the %tandardized Test %tatistic and the -ejection -egion










co(cl,*e that a -a>or)ty of a*,lts prefer the co-pa(yWs be4erage to that of

the)r co-pet)torWs.






Figure ".1;6eEection 6egion and )est %tatistic for 'ote ".87 >(ample 12>












test stat)st)c *oes (ot fall )( the re>ect)o( reg)o(. &he *ec)s)o( )s (ot to

re>ect ;. ( the co(te7t of the proble- o,r co(cl,s)o( )s:

&he *ata *o (ot pro4)*e s,?c)e(t e4)*e(ce at the 1;G le4el of s)g()+ca(ce to

co(cl,*e that the proport)o( of (ewbor(s who are -ale *)8ers fro- the h)stor)c

proport)o( )( t)-es of eco(o-)c recess)o(.






Figure ".156eEection 6egion and )est %tatistic for 'ote ".8" >(ample 1*>

EKAPLE 1

Perfor- the test of Note 6. QE7a-ple 12Q ,s)(g the p=4al,e approach.

Sol,t)o(:

Be alrea*y k(ow that the sa-ple s)Ve )s s,?c)e(tly large to 4al)*ly perfor- the

test.

• Steps 13 of the +4e=step proce*,re *escr)be* )( Sect)o( 6.3.2

Q&he Q ha4e alrea*y bee( *o(e )( Note 6. QE7a-ple 12Q so we

w)ll (ot repeat the- here b,t o(ly say that we k(ow that the test)s r)ght=ta)le* a(* that 4al,e of the test stat)st)c )s C 1.69.

• Step . S)(ce the test )s r)ght=ta)le* the p=4al,e )s the area ,(*er

the sta(*ar* (or-al c,r4e c,t o8 by the obser4e* test stat)st)c z

1.69 as )ll,strate* )()g,re 6.1. #y )g,re 12.2 Q%,-,lat)4e






Nor-al Probab)l)tyQ that area a(* therefore the p=4al,e

)s 1−0.9633=0.0367.

• Step 0. S)(ce the p=4al,e )s less tha( α=0.05the *ec)s)o( )s to

re>ect ;.

Figure ".179+<alue for 'ote ".8 >(ample 18>

+A>2!+ %8

Perfor- the test of Note 6.6 QE7a-ple 13Q ,s)(g the p=4al,e approach.

Sol,t)o(:






Be alrea*y k(ow that the sa-ple s)Ve )s s,?c)e(tly large to 4al)*ly perfor- the

test.

• Steps 13 of the +4e=step proce*,re *escr)be* )( Sect)o( 6.3.2 Q&he Q ha4e

alrea*y bee( *o(e )( Note 6.6 QE7a-ple 13Q. &hey tell ,s that the test )s two=

ta)le* a(* that 4al,e of the test stat)st)c )s C 1.02.

• Step . S)(ce the test )s two=ta)le* the p=4al,e )s the *o,ble of the area ,(*er the

sta(*ar* (or-al c,r4e c,t o8 by the obser4e* test stat)st)c z 1.02. #y)g,re

12.2 Q%,-,lat)4e Nor-al Probab)l)tyQ that area )s 1−0.9382=0.0618 as )ll,strate*

)( )g,re 6.16 he(ce the p=4al,e )s 2×0.0618=0.1236.

• Step 0. S)(ce the p=4al,e )s greater tha( α=0.10the *ec)s)o( )s (ot to re>ect;.

Figure ".1"9+<alue for 'ote ".;A >(ample 1;>

*+, TA*+AA,S

• &here )s o(e for-,la for the test stat)st)c )( test)(g hypotheses abo,t a

pop,lat)o( proport)o(. &he test stat)st)c follows the sta(*ar* (or-al *)str)b,t)o(.

• E)ther +4e=step proce*,re cr)t)cal 4al,e or p=4al,e approach ca( be ,se*.











APPL%A&"NS

11 )4e years ago 3.9G of ch)l*re( )( a certa)( reg)o( l)4e* w)th so-eo(e other tha( a

pare(t. A soc)olog)st w)shes to test whether the c,rre(t proport)o( )s *)8ere(t. Perfor-

the rele4a(t test at the 0G le4el of s)g()+ca(ce ,s)(g the follow)(g *ata: )( a ra(*o-

sa-ple of 209 ch)l*re( 119 l)4e* w)th so-eo(e other tha( a pare(t.

12 &he go4er(-e(t of a part)c,lar co,(try reports )ts l)teracy rate as 02G. A

(o(go4er(-e(tal orga()Vat)o( bel)e4es )t to be less. &he orga()Vat)o( takes a ra(*o-

sa-ple of ;; )(hab)ta(ts a(* obta)(s a l)teracy rate of 2G. Perfor- the rele4a(t test

at the ;.0G o(e=half of 1G5 le4el of s)g()+ca(ce.

13 &wo years ago 2G of ho,sehol* )( a certa)( co,(ty reg,larly part)c)pate* )( recycl)(g

ho,sehol* waste. &he co,(ty go4er(-e(t w)shes to )(4est)gate whether that proport)o(

has )(crease* after a( )(te(s)4e ca-pa)g( pro-ot)(g recycl)(g. ( a s,r4ey of 9;;






ho,sehol*s reg,larly part)c)pate )( recycl)(g. Perfor- the rele4a(t test at the 1;G

le4el of s)g()+ca(ce.

1 Pr)or to a spec)al a*4ert)s)(g ca-pa)g( 23G of all a*,lts recog()Ve* a part)c,lar

co-pa(yWs logo. At the close of the ca-pa)g( the -arket)(g *epart-e(t co--)ss)o(e*

a s,r4ey )( wh)ch 311 of 12;; ra(*o-ly selecte* a*,lts recog()Ve* the logo.

eter-)(e at the 1G le4el of s)g()+ca(ce whether the *ata pro4)*e s,?c)e(t e4)*e(ce

to co(cl,*e that -ore tha( 23G of all a*,lts (ow recog()Ve the co-pa(yWs logo.

10 A report +4e years ago state* that 30.0G of all state=ow(e* br)*ges )( a part)c,lar state

were *e+c)e(t. A( a*4ocacy gro,p took a ra(*o- sa-ple of 1;; state=ow(e* br)*ges

)( the state a(* fo,(* 33 to be c,rre(tly rate* as be)(g *e+c)e(t. &est whether the

c,rre(t proport)o( of br)*ges )( s,ch co(*)t)o( )s 30.0G 4ers,s the alter(at)4e that )t )s

*)8ere(t fro- 30.0G at the 1;G le4el of s)g()+ca(ce.

1 ( the pre4)o,s year the proport)o( of *epos)ts )( check)(g acco,(ts at a certa)( ba(k

that were -a*e electro()cally was 0G. &he ba(k w)shes to *eter-)(e )f the proport)o(

)s h)gher th)s year. t e7a-)(e* 2;;;; *epos)t recor*s a(* fo,(* that 921 were

electro()c. eter-)(e at the 1G le4el of s)g()+ca(ce whether the *ata pro4)*e

s,?c)e(t e4)*e(ce to co(cl,*e that -ore tha( 0G of all *epos)ts to check)(g acco,(ts

are (ow be)(g -a*e electro()cally.

1 Accor*)(g to the e*eral Po4erty eas,re 12G of the U.S. pop,lat)o( l)4es )( po4erty.

&he go4er(or of a certa)( state bel)e4es that the proport)o( there )s lower. ( a sa-ple

of s)Ve 100; 13 were )-po4er)she* accor*)(g to the fe*eral -eas,re.

a &est whether the tr,e proport)o( of the stateWs pop,lat)o( that )s

)-po4er)she* )s less tha( 12G at the 0G le4el of s)g()+ca(ce.

1 %o-p,te the obser4e* s)g()+ca(ce of the test.

16 A( )(s,ra(ce co-pa(y states that )t settles 60G of all l)fe )(s,ra(ce cla)-s w)th)( 3;

*ays. A co(s,-er gro,p asks the state )(s,ra(ce co--)ss)o( to )(4est)gate. ( a

sa-ple of 20; l)fe )(s,ra(ce cla)-s 2;3 were settle* w)th)( 3; *ays.

a &est whether the tr,e proport)o( of all l)fe )(s,ra(ce cla)-s -a*e to th)s

co-pa(y that are settle* w)th)( 3; *ays )s less tha( 60G at the 0G le4el of

s)g()+ca(ce.







19 A spec)al )(terest gro,p asserts that 9;G of all s-okers bega( s-ok)(g before age 16.

( a sa-ple of 60; s-okers 6 bega( s-ok)(g before age 16.

a &est whether the tr,e proport)o( of all s-okers who bega( s-ok)(g before

age 16 )s less tha( 9;G at the 1G le4el of s)g()+ca(ce.


2; ( the past 6G of a garageWs b,s)(ess was w)th for-er patro(s. &he ow(er of the

garage sa-ples 2;; repa)r )(4o)ces a(* +(*s that for o(ly 11 of the- the patro( was a

repeat c,sto-er.

a &est whether the tr,e proport)o( of all c,rre(t b,s)(ess that )s w)th repeat

c,sto-ers )s less tha( 6G at the 1G le4el of s)g()+ca(ce.


A&"NAL EKER%SES

21 A r,le of th,-b )s that for work)(g )(*)4)*,als o(e=<,arter of ho,sehol* )(co-e sho,l*

be spe(t o( ho,s)(g. A +(a(c)al a*4)sor bel)e4es that the a4erage proport)o( of )(co-e

spe(t o( ho,s)(g )s -ore tha( ;.20. ( a sa-ple of 3; ho,sehol*s the -ea( proport)o(

of ho,sehol* )(co-e spe(t o( ho,s)(g was ;.260 w)th a sta(*ar* *e4)at)o( of ;.;3.

Perfor- the rele4a(t test of hypotheses at the 1G le4el of s)g()+ca(ce. @)(t: &h)s

e7erc)se co,l* ha4e bee( prese(te* )( a( earl)er sect)o(.

22 ce crea- )s legally re<,)re* to co(ta)( at least 1;G -)lk fat by we)ght. &he

-a(,fact,rer of a( eco(o-y )ce crea- w)shes to be close to the legal l)-)t he(ce

pro*,ces )ts )ce crea- w)th a target proport)o( of ;.1; -)lk fat. A sa-ple of +4e

co(ta)(ers y)el*e* a -ea( proport)o( of ;.;9 -)lk fat w)th sta(*ar* *e4)at)o( ;.;;2.

&est the (,ll hypothes)s that the -ea( proport)o( of -)lk fat )( all co(ta)(ers )s ;.1;

aga)(st the alter(at)4e that )t )s less tha( ;.1; at the 1;G le4el of s)g()+ca(ce.

Ass,-e that the proport)o( of -)lk fat )( co(ta)(ers )s (or-ally *)str)b,te*. @)(t: &h)s

e7erc)se co,l* ha4e bee( prese(te* )( a( earl)er sect)o(.







23 Large ata Sets a(* A l)st the res,lts of 0;; tosses of a *)e. Let p *e(ote the

proport)o( of all tosses of th)s *)e that wo,l* res,lt )( a +4e. Use the sa-ple *ata to test

the hypothes)s that p )s *)8ere(t fro- 1/ at the 2;G le4el of s)g()+ca(ce.

http://www..7ls

http://www.A.7ls

2 Large ata Set recor*s res,lts of a ra(*o- s,r4ey of 2;; 4oters )( each of two

reg)o(s )( wh)ch they were aske* to e7press whether they prefer %a(*)*ate 0 for a U.S.

Se(ate seat or prefer so-e other ca(*)*ate. Use the f,ll *ata set ;; obser4at)o(s5 to

test the hypothes)s that the proport)o( p of all 4oters who prefer %a(*)*ate 0 e7cee*s

;.30. &est at the 1;G le4el of s)g()+ca(ce.

http://www..7ls

20 L)(es 2 thro,gh 03 )( Large ata Set 11 )s a sa-ple of 030 real estate sales )( a

certa)( reg)o( )( 2;;6. &hose that were foreclos,re sales are )*e(t)+e* w)th a 1 )( the

seco(* col,-(. Use these *ata to test at the 1;G le4el of s)g()+ca(ce the hypothes)s

that the proport)o( p of all real estate sales )( th)s reg)o( )( 2;;6 that were foreclos,re

sales was less tha( 20G. &he (,ll hypothes)s )s H0: p=0.25.5

http://www.11.7ls

2 L)(es 03 thro,gh 11; )( Large ata Set 11 )s a sa-ple of 0; real estate sales )( a

certa)( reg)o( )( 2;1;. &hose that were foreclos,re sales are )*e(t)+e* w)th a 1 )( the

seco(* col,-(. Use these *ata to test at the 0G le4el of s)g()+ca(ce the hypothes)s






that the proport)o( p of all real estate sales )( th)s reg)o( )( 2;1; that were foreclos,re

sales was greater tha( 23G. &he (,ll hypothes)s )s H0: p=0.23.5

http://www.11.7ls











Chapter 6

Two-Sample 2roblems

The previous two chapters treated the "uestions of estimating and making inferences about a

parameter of a single population. ,n this chapter we consider a comparison of parameters that

belong to two different populations. /or example$ we might wish to compare the average income of

all adults in one region of the country with the average income of those in another region$ or we

might wish to compare the proportion of all men who are vegetarians with the proportion of all

women who are vegetarians.

(e will study construction of confidence intervals and tests of hypotheses in four situations$

depending on the parameter of interest$ the si'es of the samples drawn from each of the populations$

and the method of sampling. (e also examine sample si'e considerations.






6.% Comparison o Two 2opulation >eansH!ar$e@ &ndependent Samples

!+A/N&N: 1';+CT&<+S

1 &o ,(*ersta(* the log)cal fra-ework for est)-at)(g the *)8ere(ce betwee( the

-ea(s of two *)st)(ct pop,lat)o(s a(* perfor-)(g tests of hypotheses co(cer()(g

those -ea(s.

2 &o lear( how to co(str,ct a co(+*e(ce )(ter4al for the *)8ere(ce )( the -ea(s of

two *)st)(ct pop,lat)o(s ,s)(g large )(*epe(*e(t sa-ples.

3 &o lear( how to perfor- a test of hypotheses co(cer()(g the *)8ere(ce betwee(

the -ea(s of two *)st)(ct pop,lat)o(s ,s)(g large )(*epe(*e(t sa-ples.

+uppose we wish to compare the means of two distinct populations. /igure F.1 0,ndependent

+ampling from Two opulations0 illustrates the conceptual framework of our investigation in this






and the next section. ach population has a mean and a standard deviation. (e arbitrarily label one

population as opulation 1 and the other as opulation !$ and subscript the parameters with the

numbers 1 and ! to tell them apart. (e draw a random sample from opulation 1 and label the

sample statistics it yields with the subscript 1. (ithout reference to the first sample we draw a

sample from opulation ! and label its sample statistics with the subscript !.

!igure 4." >ndependent %ampling from Two $opulations

(e)nition

%amples from two distinct populations are independent if each one is drawn without reference to the

other, and has no connection with the other.






EKAPLE 1

&o co-pare c,sto-er sat)sfact)o( le4els of two co-pet)(g cable tele4)s)o(

co-pa()es 1 c,sto-ers of %o-pa(y 1 a(* 300 c,sto-ers of %o-pa(y 2 were

ra(*o-ly selecte* a(* were aske* to rate the)r cable co-pa()es o( a +4e=po)(t

scale w)th 1 be)(g least sat)s+e* a(* 0 -ost sat)s+e*. &he s,r4ey res,lts are

s,--ar)Ve* )( the follow)(g table:






Company % Company 0

n1=174 n2=355

x−1=3.51 x−2=3.24

s1=0.51 s2=0.52

%o(str,ct a po)(t est)-ate a(* a 99G co(+*e(ce )(ter4al for µ1−µ2 the *)8ere(ce

)( a4erage sat)sfact)o( le4els of c,sto-ers of the two co-pa()es as -eas,re* o(

th)s +4e=po)(t scale.

Sol,t)o(:

&he po)(t est)-ate of µ1−µ2 )s

x−1−x−2=3.51−3.24=0.27.






=ypothesis Testin$

Eypotheses concerning the relative si'es of the means of two populations are tested using the same

critical value and p-value procedures that were used in the case of a single population. %ll that is

needed is to know how to express the null and alternative hypotheses and to know the formula for

the standardi'ed test statistic and the distribution that it follows.

The null and alternative hypotheses will always be expressed in terms of the difference of the two

population means. Thus the null hypothesis will always be written

H 0:µ1−µ2=D0






where (5 is a number that is deduced from the statement of the situation. %s was the case with a

single population the alternative hypothesis can take one of the three forms$ with the same

terminology:

Form o Ha Terminolo$y

Ha:µ1−µ2<D0 Left=ta)le*

Ha:µ1−µ2>D0 R)ght=ta)le*

Ha:µ1−µ2≠D0 &wo=ta)le*

%s long as the samples are independent and both are large the following formula for the standardi'ed

test statistic is valid$ and it has the standard normal distribution. *,n the relatively rare case that both

population standard deviations σ1andσ2are known they would be used instead of the sample

standard deviations.)











re>ect ;. ( the co(te7t of the proble- o,r co(cl,s)o( )s:


co(cl,*e that the -ea( c,sto-er sat)sfact)o( for %o-pa(y 1 )s h)gher tha(

that for %o-pa(y 2.






EKAPLE 3

Perfor- the test of Note 9. QE7a-ple 2Q ,s)(g the p=4al,e approach.

Sol,t)o(:

&he +rst three steps are )*e(t)cal to those )( Note 9. QE7a-ple 2Q.

• Step . &he obser4e* s)g()+ca(ce or p=4al,e of the test )s the area of the r)ght ta)l

of the sta(*ar* (or-al *)str)b,t)o( that )s c,t o8 by the test stat)st)c C 0.6.

&he (,-ber 0.6 )s too large to appear )( )g,re 12.2 Q%,-,lat)4e Nor-al

Probab)l)tyQ wh)ch -ea(s that the area of the left ta)l that )t c,ts o8 )s 1.;;;; to

fo,r *ec)-al places. &he area that we seek the area of the rig!t ta)l )s

therefore 1−1.0000=0.0000to fo,r *ec)-al places. See )g,re 9.3. &hat )s p -

value=0.0000to fo,r *ec)-al places. &he act,al 4al,e )s appro7)-ately 0.000 000 007.E

Figure .*9+<alue for 'ote .7 >(ample *>

• Step 0. S)(ce ;.;;;; _ ;.;1 p -value<αso the *ec)s)o( )s to re>ect the (,ll

hypothes)s:







co(cl,*e that the -ea( c,sto-er sat)sfact)o( for %o-pa(y 1 )s h)gher tha(

that for %o-pa(y 2.

*+, TA*+AA,S

• A po)(t est)-ate for the *)8ere(ce )( two pop,lat)o( -ea(s )s s)-ply the

*)8ere(ce )( the correspo(*)(g sa-ple -ea(s.

• ( the co(te7t of est)-at)(g or test)(g hypotheses co(cer()(g two pop,lat)o(

-ea(s large sa-ples -ea(s that bot! sa-ples are large.

• A co(+*e(ce )(ter4al for the *)8ere(ce )( two pop,lat)o( -ea(s )s co-p,te*

,s)(g a for-,la )( the sa-e fash)o( as was *o(e for a s)(gle pop,lat)o( -ea(.

• &he sa-e +4e=step proce*,re ,se* to test hypotheses co(cer()(g a s)(gle

pop,lat)o( -ea( )s ,se* to test hypotheses co(cer()(g the *)8ere(ce betwee(

two pop,lat)o( -ea(s. &he o(ly *)8ere(ce )s )( the for-,la for the sta(*ar*)Ve*

test stat)st)c.































A22!&CAT&1NS

13 ( or*er to )(4est)gate the relat)o(sh)p betwee( -ea( >ob te(,re )( years a-o(g

workers who ha4e a bachelorWs *egree or h)gher a(* those who *o (ot ra(*o- sa-ples

of each type of worker were take( w)th the follow)(g res,lts.






n x− s

:a!he"or;+ degree or higher 155 5.2 1.3

No degree 210 5.0 1.5

a %o(str,ct the 99G co(+*e(ce )(ter4al for the *)8ere(ce )( the pop,lat)o(

-ea(s base* o( these *ata.

b &est at the 1G le4el of s)g()+ca(ce the cla)- that -ea( >ob te(,re a-o(g

those w)th h)gher e*,cat)o( )s greater tha( a-o(g those w)tho,t aga)(st the

*efa,lt that there )s (o *)8ere(ce )( the -ea(s.

c %o-p,te the obser4e* s)g()+ca(ce of the test.

1 Recor*s of ; ,se* passe(ger cars a(* ; ,se* p)ck,p tr,cks (o(e ,se* co--erc)ally5

were ra(*o-ly selecte* to )(4est)gate whether there was a(y *)8ere(ce )( the -ea(

t)-e )( years that they were kept by the or)g)(al ow(er before be)(g sol*. or cars the

-ea( was 0.3 years w)th sta(*ar* *e4)at)o( 2.2 years. or p)ck,p tr,cks the -ea( was

.1 years w)th sta(*ar* *e4)at)o( 3.; years.

a %o(str,ct the 90G co(+*e(ce )(ter4al for the *)8ere(ce )( the -ea(s base*

o( these *ata.

b &est the hypothes)s that there )s a *)8ere(ce )( the -ea(s aga)(st the (,ll

hypothes)s that there )s (o *)8ere(ce. Use the 1G le4el of s)g()+ca(ce.

c %o-p,te the obser4e* s)g()+ca(ce of the test )( part b5.

10 ( pre4)o,s years the a4erage (,-ber of pat)e(ts per ho,r at a hosp)tal e-erge(cy

roo- o( weeke(*s e7cee*e* the a4erage o( week*ays by .3 4)s)ts per ho,r. A hosp)tal

a*-)()strator bel)e4es that the c,rre(t weeke(* -ea( e7cee*s the week*ay -ea( by

fewer tha( .3 ho,rs.

a %o(str,ct the 99G co(+*e(ce )(ter4al for the *)8ere(ce )( the pop,lat)o(

-ea(s base* o( the follow)(g *ata *er)4e* fro- a st,*y )( wh)ch 3; weeke(*

a(* 3; week*ay o(e=ho,r per)o*s were ra(*o-ly selecte* a(* the (,-ber of

(ew pat)e(ts )( each recor*e*.

n x− s

eeend+ 30 13.8 3.1

eeda+ 30 8.6 2.7

b &est at the 0G le4el of s)g()+ca(ce whether the c,rre(t weeke(* -ea(

e7cee*s the week*ay -ea( by fewer tha( .3 pat)e(ts per ho,r.







1 A soc)olog)st s,r4eys 0; ra(*o-ly selecte* c)t)Ve(s )( each of two co,(tr)es to

co-pare the -ea( (,-ber of ho,rs of 4ol,(teer work *o(e by a*,lts )( each. A-o(g

the 0; )(hab)ta(ts of L)ll)p,t the -ea( ho,rs of 4ol,(teer work per year was 02 w)th

sta(*ar* *e4)at)o( 11.6. A-o(g the 0; )(hab)ta(ts of #lef,sc, the -ea( (,-ber of

ho,rs of 4ol,(teer work per year was 3 w)th sta(*ar* *e4)at)o( .2.a %o(str,ct the 99G co(+*e(ce )(ter4al for the *)8ere(ce )( -ea( (,-ber of

ho,rs 4ol,(teere* by all res)*e(ts of L)ll)p,t a(* the -ea( (,-ber of ho,rs

4ol,(teere* by all res)*e(ts of #lef,sc,.

b &est at the 1G le4el of s)g()+ca(ce the cla)- that the -ea( (,-ber of ho,rs

4ol,(teere* by all res)*e(ts of L)ll)p,t )s -ore tha( te( ho,rs greater tha( the

-ea( (,-ber of ho,rs 4ol,(teere* by all res)*e(ts of #lef,sc,.

c %o-p,te the obser4e* s)g()+ca(ce of the test )( part b5.

1 A ,()4ers)ty a*-)()strator asserte* that ,pperclass-e( spe(* -ore t)-e st,*y)(g

tha( ,(*erclass-e(.

a &est th)s cla)- aga)(st the *efa,lt that the a4erage (,-ber of ho,rs of st,*y

per week by the two gro,ps )s the sa-e ,s)(g the follow)(g )(for-at)o(

base* o( ra(*o- sa-ples fro- each gro,p of st,*e(ts. &est at the 1G le4el of

s)g()+ca(ce.

n x− s

U$$er!"a++%en 35 15.6 2.9

Under!"a++%en 35 12.3 4.1


16 A( k)(es)olog)st cla)-s that the rest)(g heart rate of -e( age* 16 to 20 who e7erc)se

reg,larly )s -ore tha( +4e beats per -)(,te less tha( that of -e( who *o (ot

e7erc)se reg,larly. e( )( each category were selecte* at ra(*o- a(* the)r rest)(g

heart rates were -eas,re* w)th the res,lts show(.

n x− s

eg&"ar e<er!i+e 40 63 1.0

No reg&"ar e<er!i+e 30 71 1.2a Perfor- the rele4a(t test of hypotheses at the 1G le4el of s)g()+ca(ce.


19 %h)l*re( )( two ele-e(tary school classroo-s were g)4e( two 4ers)o(s of the sa-e

test b,t w)th the or*er of <,est)o(s arra(ge* fro- eas)er to -ore *)?c,lt )(






'ers)o( 0 a(* )( re4erse or*er )( 'ers)o( -. Ra(*o-ly selecte* st,*e(ts fro- each

class were g)4e( 'ers)o( 0 a(* the rest 'ers)o( -. &he res,lts are show( )( the table.

n x− s

=er+ion A 31 83 4.6=er+ion B 32 78 4.3

a %o(str,ct the 9;G co(+*e(ce )(ter4al for the *)8ere(ce )( the -ea(s of the

pop,lat)o(s of all ch)l*re( tak)(g 'ers)o( 0 of s,ch a test a(* of all ch)l*re(

tak)(g 'ers)o( - of s,ch a test.

b &est at the 1G le4el of s)g()+ca(ce the hypothes)s that the 0 4ers)o( of the

test )s eas)er tha( the - 4ers)o( e4e( tho,gh the <,est)o(s are the sa-e5.


2; &he ,()c)pal &ra(s)t A,thor)ty wa(ts to k(ow )f o( week*ays -ore passe(gers r)*e

the (orthbo,(* bl,e l)(e tra)( towar*s the c)ty ce(ter that *eparts at 6:10 a.-. or the

o(e that *eparts at 6:3; a.-. &he follow)(g sa-ple stat)st)cs are asse-ble* by the

&ra(s)t A,thor)ty.

n x− s

8>15 a.%. 'rain 30 323 41

8>30 a.%. 'rain 45 356 45

a %o(str,ct the 9;G co(+*e(ce )(ter4al for the *)8ere(ce )( the -ea( (,-ber

of *a)ly tra4ellers o( the 6:10 tra)( a(* the -ea( (,-ber of *a)ly tra4ellers o(the 6:3; tra)(.

b &est at the 0G le4el of s)g()+ca(ce whether the *ata pro4)*e s,?c)e(t

e4)*e(ce to co(cl,*e that -ore passe(gers r)*e the 6:3; tra)(.


21 ( co-par)(g the aca*e-)c perfor-a(ce of college st,*e(ts who are a?l)ate* w)th

frater()t)es a(* those -ale st,*e(ts who are ,(a?l)ate* a ra(*o- sa-ple of

st,*e(ts was *raw( fro- each of the two pop,lat)o(s o( a ,()4ers)ty ca-p,s.

S,--ary stat)st)cs o( the st,*e(t !PAs are g)4e( below.

n x− s

ra'erni' 645 2.90 0.47

Unai"ia'ed 450 2.88 0.42

22 &est at the 0G le4el of s)g()+ca(ce whether the *ata pro4)*e s,?c)e(t e4)*e(ce to

co(cl,*e that there )s a *)8ere(ce )( a4erage !PA betwee( the pop,lat)o( of






frater()ty st,*e(ts a(* the pop,lat)o( of ,(a?l)ate* -ale st,*e(ts o( th)s ,()4ers)ty

ca-p,s.

23 ( co-par)(g the aca*e-)c perfor-a(ce of college st,*e(ts who are a?l)ate* w)th

soror)t)es a(* those fe-ale st,*e(ts who are ,(a?l)ate* a ra(*o- sa-ple of

st,*e(ts was *raw( fro- each of the two pop,lat)o(s o( a ,()4ers)ty ca-p,s.S,--ary stat)st)cs o( the st,*e(t !PAs are g)4e( below.

n x− s

orori' 330 3.18 0.37

Unai"ia'ed 550 3.12 0.41

2 &est at the 0G le4el of s)g()+ca(ce whether the *ata pro4)*e s,?c)e(t e4)*e(ce to

co(cl,*e that there )s a *)8ere(ce )( a4erage !PA betwee( the pop,lat)o( of soror)ty

st,*e(ts a(* the pop,lat)o( of ,(a?l)ate* fe-ale st,*e(ts o( th)s ,()4ers)ty

ca-p,s.

20 &he ow(er of a profess)o(al football tea- bel)e4es that the leag,e has beco-e -ore

o8e(se or)e(te* s)(ce +4e years ago. &o check h)s bel)ef 32 ra(*o-ly selecte*

ga-es fro- o(e yearWs sche*,le were co-pare* to 32 ra(*o-ly selecte* ga-es

fro- the sche*,le +4e years later. S)(ce -ore o8e(se pro*,ces -ore po)(ts per

ga-e the ow(er a(alyVe* the follow)(g )(for-at)o( o( po)(ts per ga-e ppg5.

n x− s

$$g $revio&+" 32 20.62 4.17 $$g re!en'" 32 22.05 4.01

2 &est at the 1;G le4el of s)g()+ca(ce whether the *ata o( po)(ts per ga-e pro4)*e

s,?c)e(t e4)*e(ce to co(cl,*e that the ga-e has beco-e -ore o8e(se or)e(te*.

2 &he ow(er of a profess)o(al football tea- bel)e4es that the leag,e has beco-e -ore

o8e(se or)e(te* s)(ce +4e years ago. &o check h)s bel)ef 32 ra(*o-ly selecte*

ga-es fro- o(e yearWs sche*,le were co-pare* to 32 ra(*o-ly selecte* ga-es

fro- the sche*,le +4e years later. S)(ce -ore o8e(se pro*,ces -ore o8e(s)4e yar*s

per ga-e the ow(er a(alyVe* the follow)(g )(for-at)o( o( o8e(s)4e yar*s per ga-e

oypg5.

n x− s

o$g $revio&+" 32 316 40

o$g re!en'" 32 336 35






26 &est at the 1;G le4el of s)g()+ca(ce whether the *ata o( o8e(s)4e yar*s per ga-e

pro4)*e s,?c)e(t e4)*e(ce to co(cl,*e that the ga-e has beco-e -ore o8e(se

or)e(te*.

!A/:+ (ATA S+T ++/C &S+S20 Large ata Sets 1A a(* 1# l)st the SA& scores for 1;;; ra(*o-ly selecte* st,*e(ts.

e(ote the pop,lat)o( of all -ale st,*e(ts as Pop,lat)o( 1 a(* the pop,lat)o( of all

fe-ale st,*e(ts as Pop,lat)o( 2.

http://www.1A.7ls

http://www.1#.7ls

a Restr)ct)(g atte(t)o( to >,st the -ales +(* n1 x−1 a(* s1. Restr)ct)(g atte(t)o(

to >,st the fe-ales +(* n2 x−2 a(* s2.

b Let µ1*e(ote the -ea( SA& score for all -ales a(* µ2the -ea( SA& score for

all fe-ales. Use the res,lts of part a5 to co(str,ct a 9;G co(+*e(ce )(ter4al

for the *)8ere(ce µ1−µ2.

c &est at the 0G le4el of s)g()+ca(ce the hypothes)s that the -ea( SA& scores

a-o(g -ales e7cee*s that of fe-ales.

2 Large ata Sets 1A a(* 1# l)st the !PAs for 1;;; ra(*o-ly selecte* st,*e(ts. e(ote

the pop,lat)o( of all -ale st,*e(ts as Pop,lat)o( 1 a(* the pop,lat)o( of all fe-ale

st,*e(ts as Pop,lat)o( 2.

http://www.1A.7ls

http://www.1#.7ls

a Restr)ct)(g atte(t)o( to >,st the -ales +(* n1 x−1 a(* s1. Restr)ct)(g atte(t)o(

to >,st the fe-ales +(* n2 x−2 a(* s2.

b Let µ1*e(ote the -ea( !PA for all -ales a(* µ2the -ea( !PA for all fe-ales.

Use the res,lts of part a5 to co(str,ct a 90G co(+*e(ce )(ter4al for the

*)8ere(ce µ1−µ2.

c &est at the 1;G le4el of s)g()+ca(ce the hypothes)s that the -ea( !PAs

a-o(g -ales a(* fe-ales *)8er.

2 Large ata Sets A a(* # l)st the s,r4)4al t)-es for 0 -ale a(* 0 fe-ale laboratory

-)ce w)th thy-)c le,ke-)a. e(ote the pop,lat)o( of all s,ch -ale -)ce as Pop,lat)o( 1

a(* the pop,lat)o( of all s,ch fe-ale -)ce as Pop,lat)o( 2.






http://www.A.7ls

http://www.#.7ls

a Restr)ct)(g atte(t)o( to >,st the -ales +(* n1x−1

a(* s1. Restr)ct)(g atte(t)o(to >,st the fe-ales +(* n2 x−2 a(* s2.

b Let µ1*e(ote the -ea( s,r4)4al for all -ales a(* µ2the -ea( s,r4)4al t)-e for

all fe-ales. Use the res,lts of part a5 to co(str,ct a 99G co(+*e(ce )(ter4al

for the *)8ere(ce µ1−µ2.

c &est at the 1G le4el of s)g()+ca(ce the hypothes)s that the -ea( s,r4)4al

t)-e for -ales e7cee*s that for fe-ales by -ore tha( 162 *ays half a year5.

* %o-p,te the obser4e* s)g()+ca(ce of the test )( part c5.
















6.0 Comparison o Two 2opulation >eansHSmall@ &ndependent Samples

!+A/N&N: 1';+CT&<+S


two *)st)(ct pop,lat)o(s ,s)(g s-all )(*epe(*e(t sa-ples.

2 &o lear( how to perfor- a test of hypotheses co(cer()(g the *)8ere(ce betwee(

the -ea(s of two *)st)(ct pop,lat)o(s ,s)(g s-all )(*epe(*e(t sa-ples.

(hen one or the other of the sample si'es is small$ as is often the case in practice$ the 7entral imit

Theorem does not apply. (e must then impose conditions on the population to give statistical

validity to the test procedure. (e will assume that both populations from which the samples are

taken have a normal probability distribution and that their standard deviations are e"ual.

Con)dence &ntervals

(hen the two populations are normally distributed and have e"ual standard deviations$ the following

formula for a confidence interval for µ1−µ2is valid.






EKAPLE

A software co-pa(y -arkets a (ew co-p,ter ga-e w)th two e7per)-e(tal

packag)(g *es)g(s. es)g( 1 )s se(t to 11 storesD the)r a4erage sales the +rst

-o(th )s 02 ,()ts w)th sa-ple sta(*ar* *e4)at)o( 12 ,()ts. es)g( 2 )s se(t to

storesD the)r a4erage sales the +rst -o(th )s ,()ts w)th sa-ple sta(*ar*

*e4)at)o( 1; ,()ts. %o(str,ct a po)(t est)-ate a(* a 90G co(+*e(ce )(ter4al for

the *)8ere(ce )( a4erage -o(thly sales betwee( the two package *es)g(s.

Sol,t)o(:






















co(cl,*e that the -ea( sales per -o(th of the two *es)g(s are *)8ere(t.

EKAPLE


Sol,t)o(:






&he +rst three steps are )*e(t)cal to those )( Note 9.13 QE7a-ple 0Q.

• Step . #eca,se the test )s two=ta)le* the obser4e* s)g()+ca(ce or p=4al,e of

the test )s the *o,ble of the area of the r)ght ta)l of St,*e(tWst =*)str)b,t)o(

w)th 10 *egrees of free*o- that )s c,t o8 by the test stat)st)c ) 1.;;. Be

ca( o(ly appro7)-ate th)s (,-ber. Look)(g )( the row of )g,re 12.3 Q%r)t)cal

'al,es of Q hea*e* df=15 the (,-ber 1.;; )s betwee( the (,-bers ;.6 a(*

1.31 correspo(*)(g tot ;.2;; a(* t ;.1;;.

&he area c,t o8 by t ;.6 )s ;.2;; a(* the area c,t o8 by t 1.31 )s

;.1;;. S)(ce 1.;; )s betwee( ;.6 a(* 1.31 the area )t c,ts o8 )s betwee(

;.2;; a(* ;.1;;. &h,s the p=4al,e s)(ce the area -,st be *o,ble*5 )s betwee(

;.;; a(* ;.2;;.

• Step 0. S)(ce p>0.200>0.01 p>α so the *ec)s)o( )s (ot to re>ect the (,ll

hypothes)s:


co(cl,*e that the -ea( sales per -o(th of the two *es)g(s are *)8ere(t.

IEJ &AIE ABAJS

• ( the co(te7t of est)-at)(g or test)(g hypotheses co(cer()(g two pop,lat)o(

-ea(s s-all sa-ples -ea(s that at least one sa-ple )s s-all. ( part)c,lar

e4e( )f o(e sa-ple )s of s)Ve 3; or -ore )f the other )s of s)Ve less tha( 3; the

for-,las of th)s sect)o( -,st be ,se*.

• A co(+*e(ce )(ter4al for the *)8ere(ce )( two pop,lat)o( -ea(s )s co-p,te*

























































6.3 Comparison o Two 2opulation >eansH2aired Samples

!+A/N&N: 1';+CT&<+S

1 &o lear( the *)st)(ct)o( betwee( )(*epe(*e(t sa-ples a(* pa)re* sa-ples.


two *)st)(ct pop,lat)o(s ,s)(g pa)re* sa-ples.

3 &o lear( how to perfor- a test of hypotheses co(cer()(g the *)8ere(ce )( the

-ea(s of two *)st)(ct pop,lat)o(s ,s)(g pa)re* sa-ples.

+uppose chemical engineers wish to compare the fuel economy obtained by two different

formulations of gasoline. +ince fuel economy varies widely from car to car$ if the mean fuel economy

of two independent samples of vehicles run on the two types of fuel were compared$ even if one

formulation were better than the other the large variability from vehicle to vehicle might make any

difference arising from difference in fuel difficult to detect. Rust imagine one random sample having

many more large vehicles than the other. ,nstead of independent random samples$ it would make

more sense to select pairs of cars of the same make and model and driven under similar

circumstances$ and compare the fuel economy of the two cars in each pair. Thus the data would look

something like Table F.1 0/uel conomy of airs of Dehicles0$ where the first car in each pair is






operated on one formulation of the fuel *call it Type 1 gasoline) and the second car is operated on the

second *call it Type ! gasoline).

Table F.1 /uel conomy of airs of Dehicles

#ake and #odel Car Car !

:&i! ?aro++e 17.0 17.0

#odge =i$er 13.2 12.9

Honda -@ 35.3 35.4

H&%%er H 3 13.6 13.2

?e<&+ A 32.7 32.5

aBda A-9 18.4 18.1

aab 9-3 22.5 22.5

,oo'a oro""a 26.8 26.7

=o"vo A 90 15.1 15.0

The first column of numbers form a sample from opulation 1$ the population of all cars operated on

Type 1 gasoline the second column of numbers form a sample from opulation !$ the population of

all cars operated on Type ! gasoline. ,t would be incorrect to analy'e the data using the formulas

from the previous section$ however$ since the samples were not drawn independently.

(hat is correct is to compute the difference in the numbers in each pair *subtracting in the same

order each time) to obtain the third column of numbers as shown in Table F.! 0/uel conomy of

airs of Dehicles0 and treat the differences as the data. %t this point$ the new sample ofdifferencesd1=0.0,…,d9=0.1in the third column of Table F.! 0/uel conomy of airs of Dehicles0 may

be considered as a random sample of si'e n M F selected from a population with mean µd=µ1−µ2.This

approach essentially transforms the paired two-sample problem into a one-sample problem as

discussed in the previous two chapters.

Table F.! /uel conomy of airs of Dehicles

#ake and #odel Car Car ! %ifference

:&i! ?aro++e 17.0 17.0 0.0

#odge =i$er 13.2 12.9 0.3

Honda -@ 35.3 35.4 C0.1

H&%%er H 3 13.6 13.2 0.4

?e<&+ A 32.7 32.5 0.2

aBda A-9 18.4 18.1 0.3






#ake and #odel Car Car ! %ifference

aab 9-3 22.5 22.5 0.0

,oo'a oro""a 26.8 26.7 0.1

=o"vo A 90 15.1 15.0 0.1

9ote carefully that although it does not matter what order the subtraction is done$ it must be done in

the same order for all pairs. This is why there are both positive and negative "uantities in the third

column of numbers in Table F.! 0/uel conomy of airs of Dehicles0.

Con)dence &ntervals

(hen the population of differences is normally distributed the following formula for a confidence interval

forµd=µ1−µ2is valid.

EKAPLE

Us)(g the *ata )( &able 9.1 Q,el Eco(o-y of Pa)rs of 'eh)clesQ co(str,ct a po)(t

est)-ate a(* a 90G co(+*e(ce )(ter4al for the *)8ere(ce )( a4erage f,el

eco(o-y betwee( cars operate* o( &ype 1 gasol)(e a(* cars operate* o( &ype 2

gasol)(e.






Sol,t)o(:

Be ha4e referre* to the *ata )( &able 9.1 Q,el Eco(o-y of Pa)rs of

'eh)clesQbeca,se that )s the way that the *ata are typ)cally prese(te* b,t we

e-phas)Ve that w)th pa)re* sa-pl)(g o(e )--e*)ately co-p,tes the *)8ere(ces

as g)4e( )( &able 9.2 Q,el Eco(o-y of Pa)rs of 'eh)clesQ a(* ,ses the *)8ere(ces

as the *ata.

&he -ea( a(* sta(*ar* *e4)at)o( of the *)8ere(ces are

=ypothesis Testin$






Testing hypotheses concerning the difference of two population means using paired difference

samples is done precisely as it is done for independent samples$ although now the null and

alternative hypotheses are expressed in terms ofµdinstead ofµ1−µ2.Thus the null hypothesis will

always be written

H 0:µd=D0

The three forms of the alternative hypothesis$ with the terminology for each case$ are:

Form ofHa Terminolog1

Ha:µd<D0 ?e'-'ai"ed

Ha:µd>D0 igh'-'ai"ed

Ha:µd≠D0 ,/o-'ai"ed

The same conditions on the population of differences that was re"uired for constructing a confidence

interval for the difference of the means must also be met when hypotheses are tested. Eere is the

standardi'ed test statistic that is used in the test.

EKAPLE 6






Us)(g the *ata of &able 9.2 Q,el Eco(o-y of Pa)rs of 'eh)clesQ test the hypothes)s

that -ea( f,el eco(o-y for &ype 1 gasol)(e )s greater tha( that for &ype 2

gasol)(e aga)(st the (,ll hypothes)s that the two for-,lat)o(s of gasol)(e y)el*

the sa-e -ea( f,el eco(o-y. &est at the 0G le4el of s)g()+ca(ce ,s)(g the

cr)t)cal 4al,e approach.

Sol,t)o(:

&he o(ly part of the table that we ,se )s the th)r* col,-( the *)8ere(ces.

• Step 1. S)(ce the *)8ere(ces were co-p,te* )( the or*er Type 1 mpg− Type 2 mpg

better f,el eco(o-y w)th &ype 1 f,el correspo(*s to µd=µ1−µ2>0. &h,s the test )s

H 0:µd = 0

vs.H a:µd>0@ α=0.05

f the *)8ere(ces ha* bee( co-p,te* )( the oppos)te or*er the( the

alter(at)4e hypotheses wo,l* ha4e bee( H a:µd<0.E






Figure .;6eEection 6egion and )est %tatistic for 'ote .2A >(ample ">






&he *ata pro4)*e s,?c)e(t e4)*e(ce at the 0G le4el of s)g()+ca(ce to co(cl,*e that

the -ea( f,el eco(o-y pro4)*e* by &ype 1 gasol)(e )s greater tha( that for &ype 2

gasol)(e.

EKAPLE 9Perfor- the test of Note 9.2; QE7a-ple 6Q ,s)(g the p=4al,e approach.

Sol,t)o(:

&he +rst three steps are )*e(t)cal to those )( Note 9.2; QE7a-ple 6Q.

•Step . #eca,se the test )s o(e=ta)le* the obser4e* s)g()+ca(ce or p=4al,e ofthe test )s >,st the area of the r)ght ta)l of St,*e(tWs t =*)str)b,t)o( w)th 6

*egrees of free*o- that )s c,t o8 by the test stat)st)c ) 2.;;. Be ca( o(ly

appro7)-ate th)s (,-ber. Look)(g )( the row of )g,re 12.3 Q%r)t)cal 'al,es of

Q hea*e* df=8 the (,-ber 2.;; )s betwee( the (,-bers 2.3; a(* 2.69

correspo(*)(g tot ;.;20 a(* t ;.;1;.

&he area c,t o8 by t 2.3; )s ;.;20 a(* the area c,t o8 by t 2.69 )s;.;1;. S)(ce 2.;; )s betwee( 2.3; a(* 2.69 the area )t c,ts o8 )s betwee(

;.;20 a(* ;.;1;. &h,s the p=4al,e )s betwee( ;.;20 a(* ;.;1;. ( part)c,lar )t

)s less tha( ;.;20. See )g,re 9..

Figure .59+<alue for 'ote .21 >(ample >






• Step 0. S)(ce ;.;20 _ ;.;0 p<α so the *ec)s)o( )s to re>ect the (,ll hypothes)s:

&he *ata pro4)*e s,?c)e(t e4)*e(ce at the 0G le4el of s)g()+ca(ce to co(cl,*e that

the -ea( f,el eco(o-y pro4)*e* by &ype 1 gasol)(e )s greater tha( that for &ype 2

gasol)(e.

The paired two-sample experiment is a very powerful study design. ,t bypasses many unwanted

sources of Gstatistical noiseH that might otherwise influence the outcome of the experiment$ and

focuses on the possible difference that might arise from the one factor of interest.

,f the sample is large *meaning that n _ 35) then in the formula for the confidence interval we may

replacetα/2 byzα/2./or hypothesis testing when the number of pairs is at least 35$ we may use the same

statistic as for small samples for hypothesis testing$ except now it follows a standard normal

distribution$ so we use the last line of /igure 1!.3 07ritical Dalues of 0 to compute critical values$

and p-values can be computed exactly with /igure 1!.! 07umulative 9ormal robability0$ not merely

estimated using /igure 1!.3 07ritical Dalues of 0.

*+, TA*+AA,S

• Bhe( the *ata are collecte* )( pa)rs the *)8ere(ces co-p,te* for each pa)r are

the *ata that are ,se* )( the for-,las.






• A co(+*e(ce )(ter4al for the *)8ere(ce )( two pop,lat)o( -ea(s ,s)(g pa)re*

sa-pl)(g )s co-p,te* ,s)(g a for-,la )( the sa-e fash)o( as was *o(e for a

s)(gle pop,lat)o( -ea(.

•


pop,lat)o( -ea( )s ,se* to test hypotheses co(cer()(g the *)8ere(ce betwee(

two pop,lat)o( -ea(s ,s)(g pa)r sa-pl)(g. &he o(ly *)8ere(ce )s )( the for-,la

for the sta(*ar*)Ve* test stat)st)c.











4ouse Count1 )overnment rivate Compan1

1 217 219

2 350 338

3 296 291

4 237 237

5 237 235

6 272 269






4ouse Count1 )overnment rivate Compan1

7 257 239

8 277 275

9 312 32010 335 335

a !)4e a po)(t est)-ate for the *)8ere(ce betwee( the -ea( pr)4ate appra)sal of

all s,ch ho-es a(* the go4er(-e(t appra)sal of all s,ch ho-es.

b %o(str,ct the 99G co(+*e(ce )(ter4al base* o( these *ata for the *)8ere(ce.

c &est at the 1G le4el of s)g()+ca(ce the hypothes)s that appra)se* 4al,es by

the co,(ty go4er(-e(t of all s,ch ho,ses )s greater tha( the appra)se* 4al,es

by the pr)4ate appra)sal co-pa(y.

6 ( or*er to c,t costs a w)(e pro*,cer )s co(s)*er)(g ,s)(g *,o or 1 1 corks )( place

of f,ll (at,ral woo* corks b,t )s co(cer(e* that )t co,l* a8ect b,yersWs percept)o( of

the <,al)ty of the w)(e. &he w)(e pro*,cer sh)ppe* e)ght pa)rs of bottles of )ts best

yo,(g w)(es to e)ght w)(e e7perts. Each pa)r )(cl,*es o(e bottle w)th a (at,ral woo*

cork a(* o(e w)th a *,o cork. &he e7perts are aske* to rate the w)(es o( a o(e to te(

scale h)gher (,-bers correspo(*)(g to h)gher <,al)ty. &he res,lts are:

9ine $:pert %uo Cork 9ood Cork

1 8.5 8.5

2 8.0 8.5

3 6.5 8.0

4 7.5 8.5

5 8.0 7.5

6 8.0 8.0

7 9.0 9.0

8 7.0 7.5

a !)4e a po)(t est)-ate for the *)8ere(ce betwee( the -ea( rat)(gs of the w)(e

whe( bottle* are seale* w)th *)8ere(t k)(*s of corks.

b %o(str,ct the 9;G co(+*e(ce )(ter4al base* o( these *ata for the *)8ere(ce.

c &est at the 1;G le4el of s)g()+ca(ce the hypothes)s that o( the a4erage *,o

corks *ecrease the rat)(g of the w)(e.

9 E(g)(eers at a t)re -a(,fact,r)(g corporat)o( w)sh to test a (ew t)re -ater)al for

)(crease* *,rab)l)ty. &o test the t)res ,(*er real)st)c roa* co(*)t)o(s (ew fro(t t)res are






-o,(te* o( each of 11 co-pa(y cars o(e t)re -a*e w)th a pro*,ct)o( -ater)al a(* the

other w)th the e7per)-e(tal -ater)al. After a +7e* per)o* the 11 pa)rs were -eas,re*

for wear. &he a-o,(t of wear for each t)re )( --5 )s show( )( the table:

Car roduction $:perimental

1 5.1 5.0

2 6.5 6.5

3 3.6 3.1

4 3.5 3.7

5 5.7 4.5

6 5.0 4.1

7 6.4 5.3

8 4.7 2.6

9 3.2 3.0

10 3.5 3.5

11 6.4 5.1

a !)4e a po)(t est)-ate for the *)8ere(ce )( -ea( wear.

b %o(str,ct the 99G co(+*e(ce )(ter4al for the *)8ere(ce base* o( these *ata.

c &est at the 1G le4el of s)g()+ca(ce the hypothes)s that the -ea( wear w)th

the e7per)-e(tal -ater)al )s less tha( that for the pro*,ct)o( -ater)al.

1; A -arr)age co,(selor a*-)()stere* a test *es)g(e* to -eas,re o4erall co(te(t-e(t to

3; ra(*o-ly selecte* -arr)e* co,ples. &he scores for each co,ple are g)4e( below. A

h)gher (,-ber correspo(*s to greater co(te(t-e(t or happ)(ess.

Couple 4usband 9ife

1 47 44

2 44 46

3 49 44

4 53 44

5 42 43

6 45 45

7 48 47

8 45 44






Couple 4usband 9ife

9 52 44

10 47 42

11 40 3412 45 42

13 40 43

14 46 41

15 47 45

16 46 45

17 46 41

18 46 41

19 44 45

20 45 43

21 48 38

22 42 46

23 50 44

24 46 51

25 43 45

26 50 40

27 46 46

28 42 41

29 51 41

30 46 47

a &est at the 1G le4el of s)g()+ca(ce the hypothes)s that o( a4erage -e( a(*

wo-e( are (ot e<,ally happy )( -arr)age.

b &est at the 1G le4el of s)g()+ca(ce the hypothes)s that o( a4erage -e( are

happ)er tha( wo-e( )( -arr)age.







11 Large ata Set 0 l)sts the scores for 20 ra(*o-ly selecte* st,*e(ts o( pract)ce SA&

rea*)(g tests before a(* after tak)(g a two=week SA& preparat)o( co,rse. e(ote the

pop,lat)o( of all st,*e(ts who ha4e take( the co,rse as Pop,lat)o( 1 a(* the pop,lat)o(

of all st,*e(ts who ha4e (ot take( the co,rse as Pop,lat)o( 2.

http://www.0.7ls

a %o-p,te the 20 *)8ere(ces )( the or*er after− before the)r -ea( d− a(* the)r

sa-ple sta(*ar* *e4)at)o( sd.

b !)4e a po)(t est)-ate for µd=µ1−µ2 the *)8ere(ce )( the -ea( score of all

st,*e(ts who ha4e take( the co,rse a(* the -ea( score of all who ha4e (ot.

c %o(str,ct a 96G co(+*e(ce )(ter4al for µd.

* &est at the 1G le4el of s)g()+ca(ce the hypothes)s that the -ea( SA& score

)(creases by at least te( po)(ts by tak)(g the two=week preparat)o( co,rse.

12 Large ata Set 12 l)sts the scores o( o(e ro,(* for 0 ra(*o-ly selecte* -e-bers at a

golf co,rse +rst ,s)(g the)r ow( or)g)(al cl,bs the( two -o(ths later after ,s)(g (ew

cl,bs w)th a( e7per)-e(tal *es)g(. e(ote the pop,lat)o( of all golfers ,s)(g the)r ow(

or)g)(al cl,bs as Pop,lat)o( 1 a(* the pop,lat)o( of all golfers ,s)(g the (ew style cl,bs

as Pop,lat)o( 2.

http://www.12.7ls

a %o-p,te the 0 *)8ere(ces )( the or*er original clubs− new clubs the)r -ea( d− a(*

the)r sa-ple sta(*ar* *e4)at)o( sd.

b !)4e a po)(t est)-ate for µd=µ1−µ2 the *)8ere(ce )( the -ea( score of all

golfers ,s)(g the)r or)g)(al cl,bs a(* the -ea( score of all golfers ,s)(g the

(ew k)(* of cl,bs.

c %o(str,ct a 9;G co(+*e(ce )(ter4al for µd.

* &est at the 1G le4el of s)g()+ca(ce the hypothes)s that the -ea( golf score

*ecreases by at least o(e stroke by ,s)(g the (ew k)(* of cl,bs.






13 %o(s)*er the pre4)o,s proble- aga)(. S)(ce the *ata set )s so large )t )s reaso(able to

,se the sta(*ar* (or-al *)str)b,t)o( )(stea* of St,*e(tWs t =*)str)b,t)o( w)th *egrees

of free*o-.

a %o(str,ct a 9;G co(+*e(ce )(ter4al for µd,s)(g the sta(*ar* (or-al

*)str)b,t)o( -ea()(g that the for-,la )s d−±zα/2sdn−−√.&he co-p,tat)o(s *o(e

)( part a5 of the pre4)o,s proble- st)ll apply a(* (ee* (ot be re*o(e.5 @ow

*oes the res,lt obta)(e* here co-pare to the res,lt obta)(e* )( part c5 of the

pre4)o,s proble-C

b &est at the 1G le4el of s)g()+ca(ce the hypothes)s that the -ea( golf score

*ecreases by at least o(e stroke by ,s)(g the (ew k)(* of cl,bs ,s)(g the

sta(*ar* (or-al *)str)b,t)o(. All the work *o(e )( part *5 of the pre4)o,s

proble- appl)es e7cept the cr)t)cal 4al,e )s (ow zα )(stea* of tα or the p=4al,e

ca( be co-p,te* e7actly )(stea* of o(ly appro7)-ate* )f yo, ,se* the p=

4al,e approach5.5 @ow *oes the res,lt obta)(e* here co-pare to the res,lt

obta)(e* )( part c5 of the pre4)o,s proble-C

c %o(str,ct the 99G co(+*e(ce )(ter4als for µd,s)(g both the t-a(* z-

*)str)b,t)o(s. @ow -,ch *)8ere(ce )s there )( the res,lts (owC











6.7 Comparison o Two 2opulation2roportions

!+A/N&N: 1';+CT&<+S

1 &o lear( how to co(str,ct a co(+*e(ce )(ter4al for the *)8ere(ce )( the

proport)o(s of two *)st)(ct pop,lat)o(s that ha4e a part)c,lar character)st)c of

)(terest.

2 &o lear( how to perfor- a test of hypotheses co(cer()(g the *)8ere(ce )( the

proport)o(s of two *)st)(ct pop,lat)o(s that ha4e a part)c,lar character)st)c of

)(terest.

+uppose we wish to compare the proportions of two populations that have a specific characteristic$

such as the proportion of men who are left-handed compared to the proportion of women who are

left-handed. /igure F.@ 0,ndependent +ampling from Two opulations ,n Order to 7ompare

roportions0 illustrates the conceptual framework of our investigation. ach population is divided

into two groups$ the group of elements that have the characteristic of interest *for example$ being

left-handed) and the group of elements that do not. (e arbitrarily label one population as

opulation 1 and the other as opulation !$ and subscript the proportion of each population that

possesses the characteristic with the number 1 or ! to tell them apart. (e draw a random sample

from opulation 1 and label the sample statistic it yields with the subscript 1. (ithout reference to

the first sample we draw a sample from opulation ! and label its sample statistic with the subscript

!.

!igure 4.1 >ndependent %ampling from Two $opulations >n )rder to <ompare $roportions






Our goal is to use the information in the samples to estimate the difference p1− p2in the

two population proportions and to make statistically valid inferences about it.

Con)dence &ntervals

+ince the sample proportion p1computed using the sample drawn from opulation 1 is a good estimator

of population proportion p1 of opulation 1 and the sample proportion p2computed using the sample

drawn from opulation ! is a good estimator of population proportion p! of opulation !$ a reasonable

point estimate of the difference p1− p2 is p 1− p 2.,n order to widen this point estimate into a confidence

interval we suppose that both samples are large$ as described in +ection @.3 0arge +ample stimation of a

opulation roportion0 in 7hapter @ 0stimation0 and repeated below. ,f so$ then the following formula

for a confidence interval for p1− p2is valid.





















The three forms of the alternative hypothesis$ with the terminology for each case$ are:

Form ofHa Terminolog1

Ha: p1− p2<D0 ?e'-'ai"ed

Ha: p1− p2>D0 igh'-'ai"ed

Ha: p1− p2≠D0 ,/o-'ai"ed






%s long as the samples are independent and both are large the following formula for the standardi'ed

test statistic is valid$ and it has the standard normal distribution.

EKAPLE 11

Us)(g the *ata of Note 9.20 QE7a-ple 1;Q test whether there )s s,?c)e(t

e4)*e(ce to co(cl,*e that p,bl)c web access to the )(spect)o( recor*s has

)(crease* the proport)o( of pro>ects that passe* o( the +rst )(spect)o( by -ore

tha( 0 perce(tage po)(ts. Use the cr)t)cal 4al,e approach at the 1;G le4el of

s)g()+ca(ce.

Sol,t)o(:






• Step 1. &ak)(g )(to acco,(t the label)(g of the pop,lat)o(s a( )(crease )(

pass)(g rate at the +rst )(spect)o( by -ore tha( 0 perce(tage po)(ts after

p,bl)c access o( the web -ay be e7presse* as p2> p1+0.05 wh)ch by algebra )s

the sa-e as p1− p2<−0.05. &h)s )s the alter(at)4e hypothes)s. S)(ce the (,ll

hypothes)s )s always e7presse* as a( e<,al)ty w)th the sa-e (,-ber o( the

r)ght as )s )( the alter(at)4e hypothes)s the test )s






• &he *ata pro4)*e s,?c)e(t e4)*e(ce at the 1;G le4el of s)g()+ca(ce to

co(cl,*e that the rate of pass)(g o( the +rst )(spect)o( has )(crease* by

-ore tha( 0 perce(tage po)(ts s)(ce recor*s were p,bl)cly poste* o( the web.

Figure ."6eEection 6egion and )est %tatistic for 'ote .27 >(ample 11>

+A>2!+ %0


Sol,t)o(:

&he +rst three steps are )*e(t)cal to those )( Note 9.2 QE7a-ple 11Q.

• Step . #eca,se the test )s left=ta)le* the obser4e* s)g()+ca(ce or p=4al,e of the

test )s >,st the area of the left ta)l of the sta(*ar* (or-al *)str)b,t)o( that )s c,t

o8 by the test stat)st)c Z=−1.770.ro- )g,re 12.2 Q%,-,lat)4e Nor-al

Probab)l)tyQ the area of the left ta)l *eter-)(e* by X1. )s ;.;36. &he p=4al,e )s

;.;36.






• Step 0. S)(ce the p=4al,e ;.;36 )s less tha( α=0.10 the *ec)s)o( )s to re>ect the

(,ll hypothes)s: &he *ata pro4)*e s,?c)e(t e4)*e(ce at the 1;G le4el of

s)g()+ca(ce to co(cl,*e that the rate of pass)(g o( the +rst )(spect)o( has

)(crease* by -ore tha( 0 perce(tage po)(ts s)(ce recor*s were p,bl)cly poste*

o( the web.

/inally a common misuse of the formulas given in this section must be mentioned. +uppose a large

pre-election survey of potential voters is conducted. ach person surveyed is asked to express a

preference between$ say$ 7andidate % and 7andidate 4. *erhaps Gno preferenceH or GotherH are also

choices$ but that is not important.) ,n such a survey$ estimators p A and p B of p Aand p B can be

calculated. ,t is important to reali'e$ however$ that these two estimators were not calculated from two

independent samples. (hile p A− p B may be a reasonable estimator of pA− pB the formulas for

confidence intervals and for the standardi'ed test statistic given in this section are not valid for data

obtained in this manner.

*+, TA*+AA,S

• A co(+*e(ce )(ter4al for the *)8ere(ce )( two pop,lat)o( proport)o(s )s co-p,te*



pop,lat)o( proport)o( )s ,se* to test hypotheses co(cer()(g the *)8ere(ce

betwee( two pop,lat)o( proport)o(s. &he o(ly *)8ere(ce )s )( the for-,la for the

sta(*ar*)Ve* test stat)st)c.




























b Test H 0: p1− p2=0.30vs. H a: p1− p2≠0.30I α=0.10@

n1=7500@ p1=0.664

n2=1000@ p2=0.319






A22!&CAT&1NS

( all the re-a)()(g e7ercs)ses the sa-ples are s,?c)e(tly large so th)s (ee* (ot

be checke*5.

13 'oters )( a part)c,lar c)ty who )*e(t)fy the-sel4es w)th o(e or the other of two pol)t)cal

part)es were ra(*o-ly selecte* a(* aske* )f they fa4or a proposal to allow c)t)Ve(s w)th

proper l)ce(se to carry a co(ceale* ha(*g,( )( c)ty parks. &he res,lts are:

art1 A art1 6

a%$"e +iBe n 150 200

N&%ber in avor x 90 140

a !)4e a po)(t est)-ate for the *)8ere(ce )( the proport)o( of all -e-bers of

Party A a(* all -e-bers of Party # who fa4or the proposal.


c &est at the 0G le4el of s)g()+ca(ce the hypothes)s that the proport)o( of all

-e-bers of Party A who fa4or the proposal )s less tha( the proport)o( of all

-e-bers of Party # who *o.

* %o-p,te the p=4al,e of the test.

1 &o )(4est)gate a poss)ble relat)o( betwee( ge(*er a(* ha(*e*(ess a ra(*o- sa-ple of

32; a*,lts was take( w)th the follow)(g res,lts:

#en 9omen

a%$"e +iBe n 168 152

N&%ber o "e'-handed x 24 9

a !)4e a po)(t est)-ate for the *)8ere(ce )( the proport)o( of all -e( who are

left=ha(*e* a(* the proport)o( of all wo-e( who are left=ha(*e*.


c &est at the 0G le4el of s)g()+ca(ce the hypothes)s that the proport)o( of -e(

who are left=ha(*e* )s greater tha( the proport)o( of wo-e( who are.


10 A local school boar* -e-ber ra(*o-ly sa-ple* pr)4ate a(* p,bl)c h)gh school teachers

)( h)s *)str)ct to co-pare the proport)o(s of Nat)o(al #oar* %ert)+e* N#%5 teachers )(

the fac,lty. &he res,lts were:

rivate Schools ublic Schools

a%$"e +iBe n 80 520






rivate Schools ublic Schools

*ro$or'ion o N: 'ea!her+ p 0.175 0.150

a !)4e a po)(t est)-ate for the *)8ere(ce )( the proport)o( of all teachers )(

area p,bl)c schools a(* the proport)o( of all teachers )( pr)4ate schools whoare Nat)o(al #oar* %ert)+e*.

b %o(str,ct the 9;G co(+*e(ce )(ter4al for the *)8ere(ce base* o( these *ata.

c &est at the 1;G le4el of s)g()+ca(ce the hypothes)s that the proport)o( of all

p,bl)c school teachers who are Nat)o(al #oar* %ert)+e* )s less tha( the

proport)o( of pr)4ate school teachers who are.


1 ( profess)o(al basketball ga-es the fa(s of the ho-e tea- always try to *)stract free

throw shooters o( the 4)s)t)(g tea-. &o )(4est)gate whether th)s tact)c )s act,ally

e8ect)4e the free throw stat)st)cs of a profess)o(al basketball player w)th a h)gh free

throw perce(tage were e7a-)(e*. ,r)(g the e(t)re last seaso( th)s player ha* 0

free throws 2; )( ho-e ga-es a(* 23 )( away ga-es. &he res,lts are s,--ar)Ve*

below.

4ome A3a1

a%$"e +iBe n 420 236

ree 'hro/ $er!en' p 81.5D 78.8D

a !)4e a po)(t est)-ate for the *)8ere(ce )( the proport)o( of free throws -a*eat ho-e a(* away.

b %o(str,ct the 9;G co(+*e(ce )(ter4al for the *)8ere(ce base* o( these *ata.

c i&est at the 1;G le4el of s)g()+ca(ce the hypothes)s that there e7)sts a ho-e

a*4a(tage )( free throws.


1 Ra(*o-ly selecte* -)**le=age* people )( both %h)(a a(* the U()te* States were aske*

)f they bel)e4e* that a*,lts ha4e a( obl)gat)o( to +(a(c)ally s,pport the)r age* pare(ts.

&he res,lts are s,--ar)Ve* below.

China USA

a%$"e +iBe n 1300 150

N&%ber o e+ x 1170 110

&est at the 1G le4el of s)g()+ca(ce whether the *ata pro4)*e s,?c)e(t e4)*e(ce to

co(cl,*e that there e7)sts a c,lt,ral *)8ere(ce )( att)t,*e regar*)(g th)s <,est)o(.






16 A -a(,fact,rer of walk=beh)(* p,sh -owers rece)4es ref,rb)she* s-all e(g)(es fro-

two (ew s,ppl)ers 0 a(* -. t )s (ot ,(co--o( that so-e of the ref,rb)she* e(g)(es

(ee* to be l)ghtly ser4)ce* before they ca( be +tte* )(to -owers. &he -ower

-a(,fact,rer rece(tly rece)4e* 1;; e(g)(es fro- each s,ppl)er. ( the sh)p-e(t fro- 0

13 (ee*e* f,rther ser4)ce. ( the sh)p-e(t fro- - 1; (ee*e* f,rther ser4)ce. &est atthe 1;G le4el of s)g()+ca(ce whether the *ata pro4)*e s,?c)e(t e4)*e(ce to co(cl,*e

that there e7)sts a *)8ere(ce )( the proport)o(s of e(g)(es fro- the two s,ppl)ers

(ee*)(g ser4)ce.


19 Large ata Sets A a(* # recor* res,lts of a ra(*o- s,r4ey of 2;; 4oters )( each of

two reg)o(s )( wh)ch they were aske* to e7press whether they prefer %a(*)*ate 0for a

U.S. Se(ate seat or prefer so-e other ca(*)*ate. Let the pop,lat)o( of all 4oters )(

reg)o( 1 be *e(ote* Pop,lat)o( 1 a(* the pop,lat)o( of all 4oters )( reg)o( 2 be *e(ote*

Pop,lat)o( 2. Let p1 be the proport)o( of 4oters )( Pop,lat)o( 1 who prefer %a(*)*ate 0

a(* p2 the proport)o( )( Pop,lat)o( 2 who *o.

http://www.A.7ls

http://www.#.7ls

a )(* the rele4a(t sa-ple proport)o(s p1a(* p2.

b %o(str,ct a po)(t est)-ate for p1− p2.

c %o(str,ct a 90G co(+*e(ce )(ter4al for p1− p2.

* &est at the 0G le4el of s)g()+ca(ce the hypothes)s that the sa-e proport)o(

of 4oters )( the two reg)o(s fa4or %a(*)*ate 0 aga)(st the alter(at)4e that a

larger proport)o( )( Pop,lat)o( 2 *o.

2; Large ata Set 11 recor*s the res,lts of sa-ples of real estate sales )( a certa)( reg)o(

)( the year 2;;6 l)(es 2 thro,gh 035 a(* )( the year 2;1; l)(es 03 thro,gh 11;5.

oreclos,re sales are )*e(t)+e* w)th a 1 )( the seco(* col,-(. Let all real estate sales )(

the reg)o( )( 2;;6 be Pop,lat)o( 1 a(* all real estate sales )( the reg)o( )( 2;1; be

Pop,lat)o( 2.






http://www.11.7ls

a Use the sa-ple *ata to co(str,ct po)(t est)-ates p1a(* p2of the

proport)o(s p1 a(* p2 of all real estate sales )( th)s reg)o( )( 2;;6 a(* 2;1;

that were foreclos,re sales. %o(str,ct a po)(t est)-ate of p1− p2.

b Use the sa-ple *ata to co(str,ct a 9;G co(+*e(ce for p1− p2.

c &est at the 1;G le4el of s)g()+ca(ce the hypothes)s that the proport)o( of

real estate sales )( the reg)o( )( 2;1; that were foreclos,re sales was greater

tha( the proport)o( of real estate sales )( the reg)o( )( 2;;6 that were

foreclos,re sales. &he *efa,lt )s that the proport)o(s were the sa-e.5











6.8 Sample Sie Considerations

!+A/N&N: 1';+CT&<+

1 &o lear( how to apply for-,las for est)-at)(g the s)Ve sa-ples that w)ll be

(ee*e* )( or*er to co(str,ct a co(+*e(ce )(ter4al for the *)8ere(ce )( two

pop,lat)o( -ea(s or proport)o(s that -eets g)4e( cr)ter)a.

%s was pointed out at the beginning of +ection @.6 0+ample +i'e 7onsiderations0in 7hapter @

0stimation0$ sampling is typically done with definite ob#ectives in mind. /or example$ a physician

might wish to estimate the difference in the average amount of sleep gotten by patients suffering a

certain condition with the average amount of sleep got by healthy adults$ at F5B confidence and to

within half an hour. +ince sampling costs time$ effort$ and money$ it would be useful to be able to

estimate the smallest si'e samples that are likely to meet these criteria.




































*+, TA*+AA,S

• f the pop,lat)o( sta(*ar* *e4)at)o(s σ1a(* σ2are k(ow( or ca( be est)-ate*

the( the -)()-,- e<,al s)Ves of )(*epe(*e(t sa-ples (ee*e* to obta)( a

co(+*e(ce )(ter4al for the *)8ere(ce µ1−µ2 )( two pop,lat)o( -ea(s w)th a g)4e(

-a7)-,- error of the est)-ate ( a(* a g)4e( le4el of co(+*e(ce ca( be

est)-ate*.

• f the sta(*ar* *e4)at)o( σdof the pop,lat)o( of *)8ere(ces )( pa)rs *raw( fro-

two pop,lat)o(s )s k(ow( or ca( be est)-ate* the( the -)()-,- (,-ber of

sa-ple pa)rs (ee*e* ,(*er pa)re* *)8ere(ce sa-pl)(g to obta)( a co(+*e(ce

)(ter4al for the *)8ere(ce µd=µ1−µ2 )( two pop,lat)o( -ea(s w)th a g)4e( -a7)-,-

error of the est)-ate ( a(* a g)4e( le4el of co(+*e(ce ca( be est)-ate*.

• &he -)()-,- e<,al sa-ple s)Ves (ee*e* to obta)( a co(+*e(ce )(ter4al for the

*)8ere(ce )( two pop,lat)o( proport)o(s w)th a g)4e( -a7)-,- error of theest)-ate a(* a g)4e( le4el of co(+*e(ce ca( always be est)-ate*. f there )s pr)or

k(owle*ge of the pop,lat)o( proport)o(s p1 a(* p2 the( the est)-ate ca( be

sharpe(e*.

++/C&S+S

'AS&C

1 Est)-ate the co--o( sa-ple s)Ve n of e<,ally s)Ve* )(*epe(*e(t sa-ples (ee*e* to

est)-ate µ1−µ2as spec)+e* whe( the pop,lat)o( sta(*ar* *e4)at)o(s are as show(.

a 9;G co(+*e(ce to w)th)( 3 ,()ts σ1=10a(* σ2=7

b 99G co(+*e(ce to w)th)( ,()ts σ1=6.8a(* σ2=9.3

c 90G co(+*e(ce to w)th)( 0 ,()ts σ1=22.6a(* σ2=31.8

2 Est)-ate the co--o( sa-ple s)Ve n of e<,ally s)Ve* )(*epe(*e(t sa-ples (ee*e* to

est)-ate µ1−µ2as spec)+e* whe( the pop,lat)o( sta(*ar* *e4)at)o(s are as show(.

a 6;G co(+*e(ce to w)th)( 2 ,()ts σ1=14a(* σ2=23

b 9;G co(+*e(ce to w)th)( ;.3 ,()ts σ1=1.3a(* σ2=0.8

c 99G co(+*e(ce to w)th)( 11 ,()ts σ1=42a(* σ2=37

3 Est)-ate the (,-ber n of pa)rs that -,st be sa-ple* )( or*er to est)-ate µd=µ1−µ2as

spec)+e* whe( the sta(*ar* *e4)at)o( sd of the pop,lat)o( of *)8ere(ces )s as show(.

a 6;G co(+*e(ce to w)th)( ,()ts σd=26.5

b 90G co(+*e(ce to w)th)( ,()ts σd=12






c 9;G co(+*e(ce to w)th)( 0.2 ,()ts σd=11.3

Est)-ate the (,-ber n of pa)rs that -,st be sa-ple* )( or*er to est)-ate µd=µ1−µ2as

spec)+e* whe( the sta(*ar* *e4)at)o( sd of the pop,lat)o( of *)8ere(ces )s as show(.

a 9;G co(+*e(ce to w)th)( 2; ,()ts σd=75.5

b 90G co(+*e(ce to w)th)( 11 ,()ts σd=31.4

c 99G co(+*e(ce to w)th)( 1.6 ,()ts σd=4

0 Est)-ate the -)()-,- e<,al sa-ple s)Ves n1=n2(ecessary )( or*er to est)-ate p1− p2as

spec)+e*.

a 6;G co(+*e(ce to w)th)( ;.;0 +4e perce(tage po)(ts5

1 whe( (o pr)or k(owle*ge of p1 or p2 )s a4a)lable

2 whe( pr)or st,*)es )(*)cate that p1≈0.20a(* p2≈0.65

b 9;G co(+*e(ce to w)th)( ;.;2 two perce(tage po)(ts5



c 90G co(+*e(ce to w)th)( ;.1; te( perce(tage po)(ts5



Est)-ate the -)()-,- e<,al sa-ple s)Ves n1=n2(ecessary )( or*er to

est)-ate p1− p2as spec)+e*.

a 6;G co(+*e(ce to w)th)( ;.;2 two perce(tage po)(ts5

a whe( (o pr)or k(owle*ge of p1 or p2 )s a4a)lable

b whe( pr)or st,*)es )(*)cate that p1≈0.78a(* p2≈0.65

b 9;G co(+*e(ce to w)th)( ;.;0 two perce(tage po)(ts5



c 90G co(+*e(ce to w)th)( ;.1; te( perce(tage po)(ts5



A22!&CAT&1NS

A( e*,cat)o(al researcher w)shes to est)-ate the *)8ere(ce )( a4erage scores of

ele-e(tary school ch)l*re( o( two 4ers)o(s of a 1;;=po)(t sta(*ar*)Ve* test at 99G

co(+*e(ce a(* to w)th)( two po)(ts. Est)-ate the -)()-,- e<,al sa-ple s)Ves






(ecessary )f )t )s k(ow( that the sta(*ar* *e4)at)o( of scores o( *)8ere(t 4ers)o(s of

s,ch tests )s .9.

6 A ,()4ers)ty a*-)()strator w)shes to est)-ate the *)8ere(ce )( -ea( gra*e po)(t

a4erages a-o(g all -e( a?l)ate* w)th frater()t)es a(* all ,(a?l)ate* -e( w)th 90G

co(+*e(ce a(* to w)th)( ;.10. t )s k(ow( fro- pr)or st,*)es that the sta(*ar**e4)at)o(s of gra*e po)(t a4erages )( the two gro,ps ha4e co--o( 4al,e ;.. Est)-ate

the -)()-,- e<,al sa-ple s)Ves (ecessary to -eet these cr)ter)a.

9 A( a,to-ot)4e t)re -a(,fact,rer w)shes to est)-ate the *)8ere(ce )( -ea( wear of t)res

-a(,fact,re* w)th a( e7per)-e(tal -ater)al a(* or*)(ary pro*,ct)o( t)re w)th 9;G

co(+*e(ce a(* to w)th)( ;.0 --. &o el)-)(ate e7tra(eo,s factors ar)s)(g fro- *)8ere(t

*r)4)(g co(*)t)o(s the t)res w)ll be teste* )( pa)rs o( the sa-e 4eh)cles. t )s k(ow( fro-

pr)or st,*)es that the sta(*ar* *e4)at)o(s of the *)8ere(ces of wear of t)res co(str,cte*

w)th the two k)(*s of -ater)als )s 1.0 --. Est)-ate the -)()-,- (,-ber of pa)rs )(

the sa-ple (ecessary to -eet these cr)ter)a.

1; &o assess to the relat)4e happ)(ess of -e( a(* wo-e( )( the)r -arr)ages a -arr)age

co,(selor pla(s to a*-)()ster a test -eas,r)(g happ)(ess )( -arr)age ton ra(*o-ly

selecte* -arr)e* co,ples recor* the the)r test scores +(* the *)8ere(ces a(* the(

*raw )(fere(ces o( the poss)ble *)8ere(ce. Let µ1a(* µ2be the tr,e a4erage le4els of

happ)(ess )( -arr)age for -e( a(* wo-e( respect)4ely as -eas,re* by th)s test.

S,ppose )t )s *es)re* to +(* a 9;G co(+*e(ce )(ter4al for est)-at)(g µd=µ1−µ2to w)th)(

two test po)(ts. S,ppose f,rther that fro- pr)or st,*)es )t )s k(ow( that the sta(*ar*

*e4)at)o( of the *)8ere(ces )( test scores )s σd≈10.Bhat )s the -)()-,- (,-ber of

-arr)e* co,ples that -,st be )(cl,*e* )( th)s st,*yC

11 A >o,r(al)st pla(s to )(ter4)ew a( e<,al (,-ber of -e-bers of two pol)t)cal part)es to

co-pare the proport)o(s )( each party who fa4or a proposal to allow c)t)Ve(s w)th a

proper l)ce(se to carry a co(ceale* ha(*g,( )( p,bl)c parks. Let p1 a(* p2 be the tr,e

proport)o(s of -e-bers of the two part)es who are )( fa4or of the proposal. S,ppose )t )s

*es)re* to +(* a 90G co(+*e(ce )(ter4al for est)-at)(g p1− p2to w)th)( ;.;0. Est)-ate the

-)()-,- e<,al (,-ber of -e-bers of each party that -,st be sa-ple* to -eet these

cr)ter)a.

12 A -e-ber of the state boar* of e*,cat)o( wa(ts to co-pare the proport)o(s of Nat)o(al#oar* %ert)+e* N#%5 teachers )( pr)4ate h)gh schools a(* )( p,bl)c h)gh schools )( the

state. @)s st,*y pla( calls for a( e<,al (,-ber of pr)4ate school teachers a(* p,bl)c

school teachers to be )(cl,*e* )( the st,*y. Let p1 a(* p2 be these proport)o(s. S,ppose

)t )s *es)re* to +(* a 99G co(+*e(ce )(ter4al that est)-ates p1− p2to w)th)( ;.;0.






a S,ppos)(g that both proport)o(s are k(ow( fro- a pr)or st,*y to be

appro7)-ately ;.10 co-p,te the -)()-,- co--o( sa-ple s)Ve (ee*e*.

b %o-p,te the -)()-,- co--o( sa-ple s)Ve (ee*e* o( the s,ppos)t)o( that

(oth)(g )s k(ow( abo,t the 4al,es of p1 a(* p2.






Chapter %

Correlation and /e$ression

Our interest in this chapter is in situations in which we can associate to each element of a population

or sample two measurements x and y$ particularly in the case that it is of interest to use the value

of x to predict the value of y. /or example$ the population could be the air in automobile

garages$ x could be the electrical current produced by an electrochemical reaction taking place in a

carbon monoxide meter$ and y the concentration of carbon monoxide in the air. ,n this chapter we

will learn statistical methods for analy'ing the relationship between variables x and y in this context.

% list of all the formulas that appear anywhere in this chapter are collected in the last section for ease

of reference.






%.% !inear /elationships 'etween <ariables

LEARNN! "#$E%&'E

1 &o lear( what )t -ea(s for two 4ar)ables to e7h)b)t a relat)o(sh)p that )s close to

l)(ear b,t wh)ch co(ta)(s a( ele-e(t of ra(*o-(ess.

The following table gives examples of the kinds of pairs of variables which could be of interest from a

statistical point of view.

x y

Pre*)ctor or )(*epe(*e(t 4ar)able Respo(se or *epe(*e(t 4ar)able

&e-perat,re )( *egrees %els),s &e-perat,re )( *egrees ahre(he)t

Area of a ho,se s<.ft.5 'al,e of the ho,se

Age of a part)c,lar -ake a(* -o*el car Resale 4al,e of the car

A-o,(t spe(t by a b,s)(ess o( a*4ert)s)(g )( a

year Re4e(,e rece)4e* that year

@e)ght of a 20=year=ol* -a( Be)ght of the -a(

The first line in the table is different from all the rest because in that case and no other the

relationship between the variables is deterministic: once the value of x is known the value of y is






completely determined. ,n fact there is a formula for y in terms of x :y=95x+32.7hoosing several

values for x and computing the corresponding value for y for each one using the formula gives the

table






!igure "5." $lot of <elsius and !ahrenheit Temperature $airs

The relationship between x and y in the temperature example is deterministic because once the value

of x is known$ the value of y is completely determined. ,n contrast$ all the other relationships listed in

the table above have an element of randomness in them. 7onsider the relationship described in the

last line of the table$ the height x of a man aged !8 and his weight y. ,f we were to randomly select

several !8-year-old men and measure the height and weight of each one$ we might obtain a collection

of(x,y)pairs something like this:

(68,151) (69,146) (70,157) (70,164) (71,171) (72,160)

(72,163)(72,180)(73,170)(73,175)(74,178)(75,188)

% plot of these data is shown in /igure 15.! 0lot of Eeight and (eight airs0. +uch a plot is called

a scatter diagram or scatter plot. ooking at the plot it is evident that there exists a linear

relationship between height x and weight y$ but not a perfect one. The points appear to be following a

line$ but not exactly. There is an element of randomness present.

!igure "5.& $lot of +eight and Feight $airs






,n this chapter we will analy'e situations in which variables x and y exhibit such a linear relationship

with randomness. The level of randomness will vary from situation to situation. ,n the introductory

example connecting an electric current and the level of carbon monoxide in air$ the relationship is

almost perfect. ,n other situations$ such as the height and weights of individuals$ the connection

between the two variables involves a high degree of randomness. ,n the next section we will see how

to "uantify the strength of the linear relationship between two variables.

*+, TA*+AA,S

• &wo 4ar)ables a(* / ha4e a *eter-)()st)c l)(ear relat)o(sh)p )f po)(ts plotte*

fro- (x,y)pa)rs l)e e7actly alo(g a s)(gle stra)ght l)(e.

• ( pract)ce )t )s co--o( for two 4ar)ables to e7h)b)t a relat)o(sh)p that )s close to

l)(ear b,t wh)ch co(ta)(s a( ele-e(t poss)bly large of ra(*o-(ess.






++/C&S+S

'AS&C

1 A l)(e has e<,at)o( y=0.5x+2.

a P)ck +4e *)st)(ct =4al,es ,se the e<,at)o( to co-p,te the correspo(*)(g / =

4al,es a(* plot the +4e po)(ts obta)(e*.

b !)4e the 4al,e of the slope of the l)(eD g)4e the 4al,e of the / =)(tercept.

2 A l)(e has e<,at)o( y=x−0.5.




3 A l)(e has e<,at)o( y=−2x+4.


4al,es a(* plot the +4e po)(ts obta)(e*.b !)4e the 4al,e of the slope of the l)(eD g)4e the 4al,e of the / =)(tercept.

A l)(e has e<,at)o( y=−1.5x+1.




0 #ase* o( the )(for-at)o( g)4e( abo,t a l)(e *eter-)(e how / w)ll cha(ge )(crease

*ecrease or stay the sa-e5 whe( )s )(crease* a(* e7pla)(. ( so-e cases )t -)ght

be )-poss)ble to tell fro- the )(for-at)o( g)4e(.

a &he slope )s pos)t)4e.

b &he / =)(tercept )s pos)t)4e.

c &he slope )s Vero.

#ase* o( the )(for-at)o( g)4e( abo,t a l)(e *eter-)(e how / w)ll cha(ge )(crease

*ecrease or stay the sa-e5 whe( )s )(crease* a(* e7pla)(. ( so-e cases )t -)ght

be )-poss)ble to tell fro- the )(for-at)o( g)4e(.

a &he / =)(tercept )s (egat)4e.

b &he / =)(tercept )s Vero.

c &he slope )s (egat)4e.

A *ata set co(s)sts of e)ght (x,y)pa)rs of (,-bers:

(0,12)(2,15)(4,16)(5,14)(8,22)(13,24)(15,28)(20,30)

a Plot the *ata )( a scatter *)agra-.

b #ase* o( the plot e7pla)( whether the relat)o(sh)p betwee( a(* / appears

to be *eter-)()st)c or to )(4ol4e ra(*o-(ess.






c #ase* o( the plot e7pla)( whether the relat)o(sh)p betwee( a(* / appears

to be l)(ear or (ot l)(ear.

6 A *ata set co(s)sts of te( (x,y)pa)rs of (,-bers:

(3,20)(5,13)(6,9)(8,4)(11,0)(12,0)(14,1)(17,6)(18,9)(20,16)

a Plot the *ata )( a scatter *)agra-.b #ase* o( the plot e7pla)( whether the relat)o(sh)p betwee( a(* / appears




9 A *ata set co(s)sts of ()(e (x,y)pa)rs of (,-bers:

(8,16)(9,9)(10,4)(11,1)(12,0)(13,1)(14,4)(15,9)(16,16)






1; A *ata set co(s)sts of +4e (x,y)pa)rs of (,-bers:

(0,1) (2,5) (3,7) (5,11) (8,17)




c #ase* o( the plot e7pla)( whether the relat)o(sh)p betwee( a(* / appearsto be l)(ear or (ot l)(ear.

A22!&CAT&1NS

11 At ; a part)c,lar ble(* of a,to-ot)4e gasol)(e we)ghts .1 lb/gal. &he we)ght / of

gasol)(e o( a ta(k tr,ck that )s loa*e* w)th gallo(s of gasol)(e )s g)4e( by the l)(ear

e<,at)o(

y=6.17x

a E7pla)( whether the relat)o(sh)p betwee( the we)ght / a(* the a-o,(t of

gasol)(e )s *eter-)()st)c or co(ta)(s a( ele-e(t of ra(*o-(ess.

b Pre*)ct the we)ght of gasol)(e o( a ta(k tr,ck that has >,st bee( loa*e* w)th

0; gallo(s of gasol)(e.






12 &he rate for re(t)(g a -otor scooter for o(e *ay at a beach resort area )s 20 pl,s 3;

ce(ts for each -)le the scooter )s *r)4e(. &he total cost / )( *ollars for re(t)(g a scooter

a(* *r)4)(g )t -)les )s

y=0.30x+25

a E7pla)( whether the relat)o(sh)p betwee( the cost / of re(t)(g the scooter for

a *ay a(* the *)sta(ce that the scooter )s *r)4e( that *ay )s *eter-)()st)c or

co(ta)(s a( ele-e(t of ra(*o-(ess.

b A perso( )(te(*s to re(t a scooter o(e *ay for a tr)p to a( attract)o( 1 -)les

away. Ass,-)(g that the total *)sta(ce the scooter )s *r)4e( )s 3 -)les

pre*)ct the cost of the re(tal.

13 &he pr)c)(g sche*,le for labor o( a ser4)ce call by a( ele4ator repa)r co-pa(y )s 10;

pl,s 0; per ho,r o( s)te.

a Br)te *ow( the l)(ear e<,at)o( that relates the labor cost / to the (,-ber of

ho,rs that the repa)r-a( )s o( s)te.

b %alc,late the labor cost for a ser4)ce call that lasts 2.0 ho,rs.

1 &he cost of a telepho(e call -a*e thro,gh a lease* l)(e ser4)ce )s 2.0 ce(ts per -)(,te.

a Br)te *ow( the l)(ear e<,at)o( that relates the cost / )( ce(ts5 of a call to )ts

le(gth .

b %alc,late the cost of a call that lasts 23 -)(,tes.

!A/:+ (ATA S+T ++/C &S+S

10 Large ata Set 1 l)sts the SA& scores a(* !PAs of 1;;; st,*e(ts. Plot the scatter

*)agra- w)th SA& score as the )(*epe(*e(t 4ar)able 5 a(* !PA as the *epe(*e(t

4ar)able / 5. %o--e(t o( the appeara(ce a(* stre(gth of a(y l)(ear tre(*.

http://www.1.7ls

1 Large ata Set 12 l)sts the golf scores o( o(e ro,(* of golf for 0 golfers +rst ,s)(g the)r

ow( or)g)(al cl,bs the( ,s)(g cl,bs of a (ew e7per)-e(tal *es)g( after two -o(ths of

fa-)l)ar)Vat)o( w)th the (ew cl,bs5. Plot the scatter *)agra- w)th golf score ,s)(g the

or)g)(al cl,bs as the )(*epe(*e(t 4ar)able 5 a(* golf score ,s)(g the (ew cl,bs as the

*epe(*e(t 4ar)able / 5. %o--e(t o( the appeara(ce a(* stre(gth of a(y l)(ear tre(*.






http://www.12.7ls

1 Large ata Set 13 recor*s the (,-ber of b)**ers a(* sales pr)ce of a part)c,lar type of

a(t)<,e gra(*father clock at ; a,ct)o(s. Plot the scatter *)agra- w)th the (,-ber of

b)**ers at the a,ct)o( as the )(*epe(*e(t 4ar)able 5 a(* the sales pr)ce as the

*epe(*e(t 4ar)able / 5. %o--e(t o( the appeara(ce a(* stre(gth of a(y l)(ear tre(*.

http://www.13.7ls






%.0 The !inear Correlation CoeJcient

LEARNN! "#$E%&'E

1 &o lear( what the l)(ear correlat)o( coe?c)e(t )s how to co-p,te )t a(* what )t

tells ,s abo,t the relat)o(sh)p betwee( two 4ar)ables a(* / .






/igure 15.3 0inear Celationships of Darying +trengths0 illustrates linear relationships between two

variables x and y of varying strengths. ,t is visually apparent that in the situation in panel *a)$ x could

serve as a useful predictor of y$ it would be less useful in the situation illustrated in panel *b)$ and in

the situation of panel *c) the linear relationship is so weak as to be practically nonexistent. The linear

correlation coefficient is a number computed directly from the data that measures the strength of the

linear relationship between the two variables x and y.

!igure "5.* 'inear -elationships of 0arying %trengths






! ,f r` is near 5 *that is$ if r is near 5 and of either sign) then the linear relationship

between x and y is weak.






!igure "5. 'inear <orrelation <oefficient -

ay particular attention to panel *f) in /igure 15.6 0inear 7orrelation 7oefficient 0. ,t shows a

perfectly deterministic relationship between x and y$ butr=0 because the relationship is not linear.

*,n this particular case the points lie on the top half of a circle.)

+A>2!+ %

%o-p,te the l)(ear correlat)o( coe?c)e(t for the he)ght a(* we)ght pa)rs plotte*

)( )g,re 1;.2 QPlot of @e)ght a(* Be)ght Pa)rsQ.

Sol,t)o(:

E4e( for s-all *ata sets l)ke th)s o(e co-p,tat)o(s are too lo(g to *o co-pletely

by ha(*. ( act,al pract)ce the *ata are e(tere* )(to a calc,lator or co-p,ter a(*

a stat)st)cs progra- )s ,se*. ( or*er to clar)fy the -ea()(g of the for-,las we

w)ll *)splay the *ata a(* relate* <,a(t)t)es )( tab,lar for-. or each (x,y)pa)r we






co-p,te three (,-bers: 2 xy a(* / 2 as show( )( the table pro4)*e*. ( the last

l)(e of the table we ha4e the s,- of the (,-bers )( each col,-(. Us)(g the- we

co-p,te:

x y x ! xy y!

68 151 4624 10268 22801

69 146 4761 10074 21316

70 157 4900 10990 24649

70 164 4900 11480 26896

71 171 5041 12141 29241

72 160 5184 11520 25600

72 163 5184 11736 26569

72 180 5184 12960 32400

73 170 5329 12410 28900

73 175 5329 12775 30625

74 178 5476 13172 31684

75 188 5625 14100 35344

E 859 2003 61537 143626 336025






IEJ &AIE ABAJS

• &he l)(ear correlat)o( coe?c)e(t -eas,res the stre(gth a(* *)rect)o( of the l)(ear

relat)o(sh)p betwee( two 4ar)ables a(* / .

• &he s)g( of the l)(ear correlat)o( coe?c)e(t )(*)cates the *)rect)o( of the l)(ear

relat)o(sh)p betwee( a(* / .

• Bhe( r )s (ear 1 or X1 the l)(ear relat)o(sh)p )s stro(gD whe( )t )s (ear ; the

l)(ear relat)o(sh)p )s weak.

EKER%SES

#AS%

B)th the e7cept)o( of the e7erc)ses at the e(* of Sect)o( 1;.3 Qo*ell)(g L)(ear

Relat)o(sh)ps w)th Ra(*o-(ess Prese(tQ the +rst #as)c e7erc)se )( each of the

follow)(g sect)o(s thro,gh Sect)o( 1;. QEst)-at)o( a(* Pre*)ct)o(Q ,ses the *ata

fro- the +rst e7erc)se here the seco(* #as)c e7erc)se ,ses the *ata fro- the seco(*

e7erc)se here a(* so o( a(* s)-)larly for the Appl)cat)o( e7erc)ses. Sa4e yo,r

co-p,tat)o(s *o(e o( these e7erc)ses so that yo, *o (ot (ee* to repeat the- later.














































http://www.1.7ls

3; Large ata Set 12 l)sts the golf scores o( o(e ro,(* of golf for 0 golfers +rst ,s)(g

the)r ow( or)g)(al cl,bs the( ,s)(g cl,bs of a (ew e7per)-e(tal *es)g( after two

-o(ths of fa-)l)ar)Vat)o( w)th the (ew cl,bs5. %o-p,te the l)(ear correlat)o(

coe?c)e(t r . %o-pare )ts 4al,e to yo,r co--e(ts o( the appeara(ce a(* stre(gth of

a(y l)(ear tre(* )( the scatter *)agra- that yo, co(str,cte* )( the seco(* large *ata

set proble- for Sect)o( 1;.1 QL)(ear Relat)o(sh)ps #etwee( 'ar)ablesQ.






http://www.12.7ls

31 Large ata Set 13 recor*s the (,-ber of b)**ers a(* sales pr)ce of a part)c,lar type

of a(t)<,e gra(*father clock at ; a,ct)o(s. %o-p,te the l)(ear correlat)o(

coe?c)e(t r . %o-pare )ts 4al,e to yo,r co--e(ts o( the appeara(ce a(* stre(gth of

a(y l)(ear tre(* )( the scatter *)agra- that yo, co(str,cte* )( the th)r* large *ata

set proble- for Sect)o( 1;.1 QL)(ear Relat)o(sh)ps #etwee( 'ar)ablesQ.

http://www.13.7ls






%.3 >odellin$ !inear /elationships with /andomness 2resent

LEARNN! "#$E%&'E






1 &o lear( the fra-ework )( wh)ch the stat)st)cal a(alys)s of the l)(ear relat)o(sh)p

betwee( two 4ar)ables a(* / w)ll be *o(e.

,n this chapter we are dealing with a population for which we can associate to each element two

measurements$ x and y. (e are interested in situations in which the value of x can be used to draw

conclusions about the value of y$ such as predicting the resale value y of a residential house based on

its si'e x . +ince the relationship between x and y is not deterministic$ statistical procedures must be

applied. /or any statistical procedures$ given in this book or elsewhere$ the associated formulas are

valid only under specific assumptions. The set of assumptions in simple linear regression are a

mathematical description of the relationship between x and y. +uch a set of assumptions is known as

a model.

/or each fixed value of x a sub-population of the full population is determined$ such as the collection

of all houses with !$155 s"uare feet of living space. /or each element of that sub-population there is a

measurement y$ such as the value of any !$155-s"uare-foot house. etE(y)denote the mean of all

the y-values for each particular value of x .E(y)can change from x -value to x -value$ such as the mean

value of all !$155-s"uare-foot houses$ the *different) mean value for all !$855-s"uare foot-houses$

and so on.

Our first assumption is that the relationship between x and the mean of they-values in the sub-

population determined by x is linear. This means that there exist numbersβ1andβ0such that

E(y)=β1x+β0

This linear relationship is the reason for the word GlinearH in Gsimple linear regressionH below. *The

word GsimpleH means that y depends on only one other variable and not two or more.)

Our next assumption is that for each value of x the y-values scatter about the meanE(y)according to

a normal distribution centered atE(y)and with a standard deviation 6 that is the same for every






value of x . This is the same as saying that there exists a normally distributed random variable L with

mean 5 and standard deviation 6 so that the relationship between x and y in the whole population is

y=β1x+β0+ε

Our last assumption is that the random deviations associated with different observations are

independent.

,n summary$ the model is:

S)-ple L)(ear Regress)o( o*el

/or each point(x,y)in data set the y-value is an independent observation of

y=β1x+β0+ε

whereβ1andβ0are fixed parameters and L is a normally distributed random variable with mean 5 and an

unknown standard deviation 6 .

The line with e"uationy=β1x+β0is called the population regression line.

/igure 15.8 0The +imple inear odel 7oncept0 illustrates the model. The symbols N(µ,σ2)denote a

normal distribution with mean and varianceσ2$ hence standard deviation 6 .

!igure "5. The %imple 'inear 2odel <oncept






,t is conceptually important to view the model as a sum of two parts:

y=β1x+β0+ε

1 Deterministic &art. The first partβ1x+β0is the e"uation that describes the trend

in y as x increases. The line that we seem to see when we look at the scatter diagram is an

approximation of the liney=β1x+β0.There is nothing random in this part$ and therefore it is calledthe deterministic part of the model.

! -andom &art. The second part L is a random variable$ often called the error term or the noise. This

part explains why the actual observed values of y are not exactly on but fluctuate near a line.

,nformation about this term is important since only when one knows how much noise there is in the

data can one know how trustworthy the detected trend is.

There are three parameters in this model: β0$β1$ and 6 . ach has an important interpretation$

particularlyβ1and 6 . The slope parameterβ1represents the expected change in y brought about by a

unit increase in x . The standard deviation 6 represents the magnitude of the noise in the data.

There are procedures for checking the validity of the three assumptions$ but for us it will be sufficient

to visually verify the linear trend in the data. ,f the data set is large then the points in the scatter






diagram will form a band about an apparent straight line. The normality of L with a constant

standard deviation corresponds graphically to the band being of roughly constant width$ and with

most points concentrated near the middle of the band.

/ortunately$ the three assumptions do not need to hold exactly in order for the procedures and

analysis developed in this chapter to be useful.

IEJ &AIEABAJ

• Stat)st)cal proce*,res are 4al)* o(ly whe( certa)( ass,-pt)o(s are 4al)*. &he

ass,-pt)o(s ,(*erly)(g the a(alyses *o(e )( th)s chapter are graph)cally

s,--ar)Ve* )( )g,re 1;.0 Q&he S)-ple L)(ear o*el %o(ceptQ.

++/C&S+S

1 State the three ass,-pt)o(s that are the bas)s for the S)-ple L)(ear Regress)o( o*el.

2 &he S)-ple L)(ear Regress)o( o*el )s s,--ar)Ve* by the e<,at)o(

y=β1x+β0+ε

*e(t)fy the *eter-)()st)c part a(* the ra(*o- part.

3 s the (,-ber β1)( the e<,at)o( y=β1x+β0a stat)st)c or a pop,lat)o( para-eterC E7pla)(.

s the (,-ber σ )( the S)-ple L)(ear Regress)o( o*el a stat)st)c or a pop,lat)o(

para-eterC E7pla)(.

0 escr)be what to look for )( a scatter *)agra- )( or*er to check that the ass,-pt)o(s of

the S)-ple L)(ear Regress)o( o*el are tr,e.

&r,e or false: the ass,-pt)o(s of the S)-ple L)(ear Regress)o( o*el -,st hol* e7actly

)( or*er for the proce*,res a(* a(alys)s *e4elope* )( th)s chapter to be ,sef,l.

ANS+/S

1

a &he -ea( of / )s l)(early relate* to .

b or each g)4e( / )s a (or-al ra(*o- 4ar)able w)th -ea( β1x+β0a(*

sta(*ar* *e4)at)o( σ .

c All the obser4at)o(s of / )( the sa-ple are )(*epe(*e(t.

3 β1)s a pop,lat)o( para-eter.

0 A l)(ear tre(*.






%.7 The !east S"uares /e$ression !ine

LEARNN! "#$E%&'ES

1 &o lear( how to -eas,re how well a stra)ght l)(e +ts a collect)o( of *ata.

2 &o lear( how to co(str,ct the least s<,ares regress)o( l)(e the stra)ght l)(e that

best +ts a collect)o( of *ata.

3 &o lear( the -ea()(g of the slope of the least s<,ares regress)o( l)(e.

&o lear( how to ,se the least s<,ares regress)o( l)(e to est)-ate the respo(se

4ar)able / )( ter-s of the pre*)ctor 4ar)able .

:oodness o Fit o a Strai$ht !ine to (ata

Once the scatter diagram of the data has been drawn and the model assumptions described in the

previous sections at least visually verified *and perhaps the correlation coefficient r computed to

"uantitatively verify the linear trend)$ the next step in the analysis is to find the straight line that best fits

the data. (e will explain how to measure how well a straight line fits a collection of points by examining

how well the liney=12x−1fits the data set






To each point in the data set there is associated an Gerror$H the positive or negative vertical distance

from the point to the line: positive if the point is above the line and negative if it is below the line.

The error can be computed as the actual y-value of the point minus the y-value y that is GpredictedH

by inserting the x -value of the data point into the formula for the line:

error at data point (x,y)=(truey)−(predictedy)=y−y






The computation of the error for each of the five points in the data set is shown in Table 15.1 0The

rrors in /itting 2ata with a +traight ine0.

Table 15.1 The rrors in /itting 2ata with a +traight ine

x y y=12x−1 y−y (y−y)2

2 ; ; ; ;

2 1 ; 1 1

2 2 ; ;

6 3 3 ; ;

1; 3 X1 1

j = = = ; 2

% first thought for a measure of the goodness of fit of the line to the data would be simply to add the

errors at every point$ but the example shows that this cannot work well in general. The line does not

fit the data perfectly *no line can)$ yet because of cancellation of positive and negative errors the sum

of the errors *the fourth column of numbers) is 'ero. ,nstead goodness of fit is measured by the sum

of the s"uares of the errors. +"uaring eliminates the minus signs$ so no cancellation can occur. /or

the data and line in /igure 15.; 0lot of the /ive-oint 2ata and the ine 0 the sum of the s"uared

errors *the last column of numbers) is !. This number measures the goodness of fit of the line to the

data.

e+()t)o(

The goodness of fit of a line y=mx+bto a set of n pairs (x,y)of numbers in a sample is the sum of the

squared errors

Σ(y−y)2

Dn terms in the sum, one for each data pairE.






The !east S"uares /e$ression !ine

iven any collection of pairs of numbers *except when all the x -values are the same) and the

corresponding scatter diagram$ there always exists exactly one straight line that fits the data better

than any other$ in the sense of minimi'ing the sum of the s"uared errors. ,t is called the least squares

regression line. oreover there are formulas for its slope and y-intercept.











&A#LE 1;.2 &@E ERR"RS N &&N! A&A B&@ &@ E

LEAS& SHUARES RE!RESS"N LNE

x y y=0.34375x−0.125 y−y (y−y)2

2 ; ;.020 X;.020 ;.31;20

2 1 ;.020 ;.30 ;.191;20

2 1.930 ;.;20 ;.;;39;20

6 3 2.20; ;.30; ;.1;20;;

1; 3 3.3120 X;.3120 ;.;9020

EKAPLE 3






&able 1;.3 Qata o( Age a(* 'al,e of Use* A,to-ob)les of a Spec)+c ake a(*

o*elQ shows the age )( years a(* the reta)l 4al,e )( tho,sa(*s of *ollars of a

ra(*o- sa-ple of te( a,to-ob)les of the sa-e -ake a(* -o*el.

a. %o(str,ct the scatter *)agra-.

b. %o-p,te the l)(ear correlat)o( coe?c)e(t r . (terpret )ts 4al,e )( the co(te7t of the

proble-.

c. %o-p,te the least s<,ares regress)o( l)(e. Plot )t o( the scatter *)agra-.

*. (terpret the -ea()(g of the slope of the least s<,ares regress)o( l)(e )( the co(te7t

of the proble-.

e. S,ppose a fo,r=year=ol* a,to-ob)le of th)s -ake a(* -o*el )s selecte* at ra(*o-.

Use the regress)o( e<,at)o( to pre*)ct )ts reta)l 4al,e.

f. S,ppose a 2;=year=ol* a,to-ob)le of th)s -ake a(* -o*el )s selecte* at ra(*o-.

Use the regress)o( e<,at)o( to pre*)ct )ts reta)l 4al,e. (terpret the res,lt.g. %o--e(t o( the 4al)*)ty of ,s)(g the regress)o( e<,at)o( to pre*)ct the pr)ce of a

bra(* (ew a,to-ob)le of th)s -ake a(* -o*el.

&A#LE 1;.3 A&A "N A!E AN 'A LU E " USE

AU&""#LES " A SPE%% AIE AN "EL

2 3 3 3 0 0 0

/ 26. 2.6 2.; 3;.0 23.6 2. 23.6 2;. 21. 22.1

Sol,t)o(:

a. &he scatter *)agra- )s show( )( )g,re 1;. QScatter )agra- for Age a(* 'al,e

of Use* A,to-ob)lesQ.

Figure 1A.7%catter Diagram for 0ge and <alue of Used 0utomobiles











* S)(ce we k(ow (oth)(g abo,t the a,to-ob)le other tha( )ts age we ass,-e

that )t )s of abo,t a4erage 4al,e a(* ,se the a4erage 4al,e of all fo,r=year=ol*

4eh)cles of th)s -ake a(* -o*el as o,r est)-ate. &he a4erage 4al,e )s s)-ply

the 4al,e of y obta)(e* whe( the (,-ber )s )(serte* for )( the least

s<,ares regress)o( e<,at)o(:

e

y=−2.05(4)+32.83=24.63

wh)ch correspo(*s to 23;.






* Now we )(sert x=20)(to the least s<,ares regress)o( e<,at)o( to obta)(

y=−2.05(20)+32.83=−8.17

wh)ch correspo(*s to X61;. So-eth)(g )s wro(g here s)(ce a (egat)4e-akes (o se(se. &he error arose fro- apply)(g the regress)o( e<,at)o( to a

4al,e of (ot )( the ra(ge of =4al,es )( the or)g)(al *ata fro- two to s)7

years.

Apply)(g the regress)o( e<,at)o( y=β1x+β0to a 4al,e of o,ts)*e the ra(ge

of =4al,es )( the *ata set )s calle* etrapolation. t )s a( )(4al)* ,se of the

regress)o( e<,at)o( a(* sho,l* be a4o)*e*.

e &he pr)ce of a bra(* (ew 4eh)cle of th)s -ake a(* -o*el )s the 4al,e of the

a,to-ob)le at age ;. f the 4al,e x=0)s )(serte* )(to the regress)o( e<,at)o( the

res,lt )s always β0 the / =)(tercept )( th)s case 32.63 wh)ch correspo(*s to

3263;. #,t th)s )s a case of e7trapolat)o( >,st as part f5 was he(ce th)s res,lt

)s )(4al)* altho,gh (ot ob4)o,sly so. ( the co(te7t of the proble- s)(ce

a,to-ob)les te(* to lose 4al,e -,ch -ore <,)ckly )--e*)ately after they are

p,rchase* tha( they *o after they are se4eral years ol* the (,-ber 3263; )s

probably a( ,(*erest)-ate of the pr)ce of a (ew a,to-ob)le of th)s -ake a(*

-o*el.

/or emphasis we highlight the points raised by parts *f) and *g) of the example.

e+()t)o(

The process of using the least squares regression equation to estimate the value of y at a value of x that

does not lie in the range of the x:values in the data set that was used to form the regression line is

called extrapolation. >t is an invalid use of the regression equation that can lead to errors, hence should

be avoided.

The Sum o the S"uared +rrors SSE

,n general$ in order to measure the goodness of fit of a line to a set of data$ we must compute the

predicted y-value y at every point in the data set$ compute each error$ s"uare it$ and then add up all the






s"uares. ,n the case of the least s"uares regression line$ however$ the line that best fits the data$ the sum

of the s"uared errors can be computed directly from the data using the following formula.

The sum of the s"uared errors for the least s"uares regression line is denoted by SSE.,t can be computed

using the formula

SSE=SSyy−β1Ssxy











SSE=SSyy−β 1SSxy=87.781−(−2.05)(−28.7)=28.946

*+, TA*+AA,S

• @ow well a stra)ght l)(e +ts a *ata set )s -eas,re* by the s,- of the s<,are*

errors.

• &he least s<,ares regress)o( l)(e )s the l)(e that best +ts the *ata. ts slope a(* / =

)(tercept are co-p,te* fro- the *ata ,s)(g for-,las.






• &he slope β1of the least s<,ares regress)o( l)(e est)-ates the s)Ve a(* *)rect)o(

of the -ea( cha(ge )( the *epe(*e(t 4ar)able / whe( the )(*epe(*e(t

4ar)able )s )(crease* by o(e ,()t.

• &he s,- of the s<,are* errors SSE of the least s<,ares regress)o( l)(e ca( be

co-p,te* ,s)(g a for-,la w)tho,t ha4)(g to co-p,te all the )(*)4)*,al errors.

EKER%SES

#AS%

or the #as)c a(* Appl)cat)o( e7erc)ses )( th)s sect)o( ,se the co-p,tat)o(s that

were *o(e for the e7erc)ses w)th the sa-e (,-ber )(Sect)o( 1;.2 Q&he L)(ear

%orrelat)o( %oe?c)e(tQ.

1 %o-p,te the least s<,ares regress)o( l)(e for the *ata )( E7erc)se 1 of Sect)o( 1;.2 Q&he

L)(ear %orrelat)o( %oe?c)e(tQ.





%o-p,te the least s<,ares regress)o( l)(e for the *ata )( E7erc)se of Sect)o( 1;.2 Q&heL)(ear %orrelat)o( %oe?c)e(tQ.

0 or the *ata )( E7erc)se 0 of Sect)o( 1;.2 Q&he L)(ear %orrelat)o( %oe?c)e(tQ

a %o-p,te the least s<,ares regress)o( l)(e.

b %o-p,te the s,- of the s<,are* errors SSE ,s)(g the *e+()t)o( Σ(y−y)2.

c %o-p,te the s,- of the s<,are* errors SSE ,s)(g the for-,la SSE=SSyy−β1SSxy.

or the *ata )( E7erc)se of Sect)o( 1;.2 Q&he L)(ear %orrelat)o( %oe?c)e(tQ


b %o-p,te the s,- of the s<,are* errors SSE ,s)(g the *e+()t)o( Σ(y−y)2.







%o-p,te the least s<,ares regress)o( l)(e for the *ata )( E7erc)se of Sect)o( 1;.2

Q&he L)(ear %orrelat)o( %oe?c)e(tQ.

6 %o-p,te the least s<,ares regress)o( l)(e for the *ata )( E7erc)se 6 of Sect)o( 1;.2

Q&he L)(ear %orrelat)o( %oe?c)e(tQ.



b %a( yo, co-p,te the s,- of the s<,are* errorsSSE ,s)(g the

*e+()t)o(Σ(y−y)2C E7pla)(.


1; or the *ata )( E7erc)se 1; of Sect)o( 1;.2 Q&he L)(ear %orrelat)o( %oe?c)e(tQ


b %a( yo, co-p,te the s,- of the s<,are* errorsSSE ,s)(g the *e+()t)o(

Σ(y−y)2C E7pla)(.


APPL%A&"NS



b "( a4erage how -a(y (ew wor*s *oes a ch)l* fro- 13 to 16 -o(ths ol* lear(

each -o(thC E7pla)(.

c Est)-ate the a4erage 4ocab,lary of all 1=-o(th=ol* ch)l*re(.



b "( a4erage how -a(y a**)t)o(al feet are a**e* to the brak)(g *)sta(ce for

each a**)t)o(al 1;; po,(*s of we)ghtC E7pla)(.

c Est)-ate the a4erage brak)(g *)sta(ce of all cars we)gh)(g 3;;; po,(*s.








b Est)-ate the a4erage rest)(g heart rate of all ;=year=ol* -e(.

c Est)-ate the a4erage rest)(g heart rate of all (ewbor( baby boys. %o--e(t

o( the 4al)*)ty of the est)-ate.



b Est)-ate the a4erage wa4e he)ght whe( the w)(* )s blow)(g at 1; -)les per

ho,r.

c Est)-ate the a4erage wa4e he)ght whe( there )s (o w)(* blow)(g. %o--e(t

o( the 4al)*)ty of the est)-ate.



b "( a4erage for each a**)t)o(al tho,sa(* *ollars spe(t o( a*4ert)s)(g how

*oes re4e(,e cha(geC E7pla)(.

c Est)-ate the re4e(,e )f 20;; )s spe(t o( a*4ert)s)(g (e7t year.



b "( a4erage for each a**)t)o(al )(ch of he)ght of two=year=ol* g)rl what )s the

cha(ge )( the a*,lt he)ghtC E7pla)(.

c Pre*)ct the a*,lt he)ght of a two=year=ol* g)rl who )s 33 )(ches tall.



b %o-p,te SSE ,s)(g the for-,la SSE=SSyy−β1SSxy.

c Est)-ate the a4erage +(al e7a- score of all st,*e(ts whose co,rse a4erage

>,st before the e7a- )s 60.



b %o-p,te SSE ,s)(g the for-,la SSE=SSyy−β1SSxy.






c Est)-ate the (,-ber of acres that wo,l* be har4este* )f 9; -)ll)o( acres of

cor( were pla(te*.



b (terpret the 4al,e of the slope of the least s<,ares regress)o( l)(e )( the


c Est)-ate the a4erage co(ce(trat)o( of the act)4e )(gre*)e(t )( the bloo* )(

-e( after co(s,-)(g 1 o,(ce of the -e*)cat)o(.



b (terpret the 4al,e of the slope of the least s<,ares regress)o( l)(e )( the


c Est)-ate the age of a( oak tree whose g)rth +4e feet o8 the gro,(* )s 92

)(ches.



b &he 26=*ay stre(gth of co(crete ,se* o( a certa)( >ob -,st be at least 32;;

ps). f the 3=*ay stre(gth )s 13;; ps) wo,l* we a(t)c)pate that the co(crete

w)ll be s,?c)e(tly stro(g o( the 26th *ayC E7pla)( f,lly.



b f the power fac)l)ty )s calle* ,po( to pro4)*e -ore tha( 90 -)ll)o( watt=ho,rs

to-orrow the( e(ergy w)ll ha4e to be p,rchase* fro- elsewhere at a

pre-),-. &he forecast )s for a( a4erage te-perat,re of 2 *egrees. Sho,l*

the co-pa(y pla( o( p,rchas)(g power at a pre-),-C








http://www.1.7ls

a %o-p,te the least s<,ares regress)o( l)(e w)th SA& score as the )(*epe(*e(t

4ar)able 5 a(* !PA as the *epe(*e(t 4ar)able / 5.

b (terpret the -ea()(g of the slope β1of regress)o( l)(e )( the co(te7t of

proble-.

c %o-p,te SSE the -eas,re of the goo*(ess of +t of the regress)o( l)(e to the

sa-ple *ata.






* Est)-ate the !PA of a st,*e(t whose SA& score )s 130;.



fa-)l)ar)Vat)o( w)th the (ew cl,bs5.

http://www.12.7ls

a %o-p,te the least s<,ares regress)o( l)(e w)th scores ,s)(g the or)g)(al cl,bs

as the )(*epe(*e(t 4ar)able 5 a(* scores ,s)(g the (ew cl,bs as the

*epe(*e(t 4ar)able / 5.


proble-.


sa-ple *ata.

* Est)-ate the score w)th the (ew cl,bs of a golfer whose score w)th the ol*

cl,bs )s 3.


a(t)<,e gra(*father clock at ; a,ct)o(s.

http://www.13.7ls

a %o-p,te the least s<,ares regress)o( l)(e w)th the (,-ber of b)**ers prese(t

at the a,ct)o( as the )(*epe(*e(t 4ar)able 5 a(* sales pr)ce as the

*epe(*e(t 4ar)able / 5.


proble-.


sa-ple *ata.

* Est)-ate the sales pr)ce of a clock at a( a,ct)o( at wh)ch the (,-ber of

b)**ers )s se4e(.











%.8 Statistical &nerences About β%

!+A/N&N: 1';+CT&<+S

1 &o lear( how to co(str,ct a co(+*e(ce )(ter4al forβ1 the slope of the pop,lat)o(

regress)o( l)(e.

2 &o lear( how to test hypotheses regar*)(g β1.

The parameterβ1$ the slope of the population regression line$ is of primary importance in regression

analysis because it gives the true rate of change in the mean E(y)in response to a unit increase in the

predictor variable x . /or every unit increase in x the mean of the response variable y changes

byβ1units$ increasing ifβ1>0and decreasing ifβ1<0. (e wish to construct confidence intervals

forβ1and test hypotheses about it.

Con)dence &ntervals or β%

The slope β1of the least s"uares regression line is a point estimate ofβ1. % confidence interval forβ1is

given by the following formula.
















years ol* we are 9;G co(+*e(t that for each a**)t)o(al year of age the a4erage

4al,e of s,ch a 4eh)cle *ecreases by betwee( 11;; a(* 3;;;.






Testin$ =ypotheses About β%

Eypotheses regardingβ1can be tested using the same five-step procedures$ either the critical value

approach or the p-value approach$ that were introduced in +ection ?.1 0The lements of Eypothesis

Testing0 and +ection ?.3 0The Observed +ignificance of a Test0 of 7hapter ? 0Testing Eypotheses0. The

null hypothesis always has the formH0:β1=B0 where 95 is a number determined from the statement of the

problem. The three forms of the alternative hypothesis$ with the terminology for each case$ are:


Ha:β1<B0 Left=ta)le*

Ha:β1>B0 R)ght=ta)le*

Ha:β1≠B0 &wo=ta)le*

The value 'ero for 95 is of particular importance since in that case the null hypothesis is H0:β1=0$ which

corresponds to the situation in which x is not useful for predicting y. /or ifβ1=0then the population

regression line is hori'ontal$ so the mean E(y)is the same for every value of x and we are #ust as well off inignoring x completely and approximating y by its average value. iven two variables x and y$ the burden

of proof is that x is useful for predicting y$ not that it is not. Thus the phrase Gtest whether x is useful for

prediction of y$H or words to that effect$ means to perform the test

H 0:β1=0 vs.H a:β1≠0











• Step 0. As show( )( )g,re 1;.9 QRe>ect)o( Reg)o( a(* &est Stat)st)c for Q the




co(cl,*e that the slope of the pop,lat)o( regress)o( l)(e )s (o(Vero so

that )s ,sef,l as a pre*)ctor of / .






Figure 1A.6eEection 6egion and )est %tatistic for 'ote 1A.** >(ample ">

+A>2!+ 6

A car sales-a( cla)-s that a,to-ob)les betwee( two a(* s)7 years ol* of the

-ake a(* -o*el *)sc,sse* )( Note 1;.19 QE7a-ple 3Q )( Sect)o( 1;. Q&he Least

S<,ares Regress)o( L)(eQ lose -ore tha( 11;; )( 4al,e each year. &est th)s

cla)- at the 0G le4el of s)g()+ca(ce.

Sol,t)o(:

Be w)ll perfor- the test ,s)(g the cr)t)cal 4al,e approach.

• Step 1. ( ter-s of the 4ar)ables a(* / the sales-a(Ws cla)- )s that )f )s

)(crease* by 1 ,()t o(e a**)t)o(al year )( age5 the( / *ecreases by -ore

tha( 1.1 ,()ts -ore tha( 11;;5. &h,s h)s assert)o( )s that the slope of the

pop,lat)o( regress)o( l)(e )s (egat)4e a(* that )t )s -ore (egat)4e tha( X1.1.

( sy-bols β1<−1.1.S)(ce )t co(ta)(s a( )(e<,al)ty th)s has to be the alter(at)4e






hypotheses. &he (,ll hypothes)s has to be a( e<,al)ty a(* ha4e the sa-e

(,-ber o( the r)ght ha(* s)*e so the rele4a(t test )s


co(cl,*e that 4eh)cles of th)s -ake a(* -o*el a(* )( th)s age ra(ge lose -ore

tha( 11;; per year )( 4al,e o( a4erage.






Figure 1A.1A6eEection 6egion and )est %tatistic for 'ote 1A.*8 >(ample >

IEJ &AIE ABAJS

• &he para-eter β1 the slope of the pop,lat)o( regress)o( l)(e )s of pr)-ary

)(terest beca,se )t *escr)bes the a4erage cha(ge )( / w)th respect to ,()t

)(crease )( .

• &he stat)st)c β 1 the slope of the least s<,ares regress)o( l)(e )s a po)(t est)-ate

of β1.%o(+*e(ce )(ter4als for β1ca( be co-p,te* ,s)(g a for-,la.

• @ypotheses regar*)(g β1are teste* ,s)(g the sa-e +4e=step proce*,res

)(tro*,ce* )( %hapter 6 Q&est)(g @ypothesesQ.

EKER%SES

#AS%

or the #as)c a(* Appl)cat)o( e7erc)ses )( th)s sect)o( ,se the co-p,tat)o(s that were

*o(e for the e7erc)ses w)th the sa-e (,-ber )( Sect)o( 1;.2 Q&he L)(ear %orrelat)o(

%oe?c)e(tQ a(* Sect)o( 1;. Q&he Least S<,ares Regress)o( L)(eQ.






1 %o(str,ct the 90G co(+*e(ce )(ter4al for the slope β1of the pop,lat)o( regress)o( l)(e

base* o( the sa-ple *ata set of E7erc)se 1 of Sect)o( 1;.2 Q&he L)(ear %orrelat)o(

%oe?c)e(tQ.

2 %o(str,ct the 9;G co(+*e(ce )(ter4al for the slope β1of the pop,lat)o( regress)o( l)(e


%oe?c)e(tQ.

3 %o(str,ct the 9;G co(+*e(ce )(ter4al for the slope β1of the pop,lat)o( regress)o( l)(e


%oe?c)e(tQ.

%o(str,ct the 99G co(+*e(ce )(ter4al for the slope β1of the pop,lat)o( regress)o(

E7erc)se of Sect)o( 1;.2 Q&he L)(ear %orrelat)o( %oe?c)e(tQ.

0 or the *ata )( E7erc)se 0 of Sect)o( 1;.2 Q&he L)(ear %orrelat)o( %oe?c)e(tQ test at the

1;G le4el of s)g()+ca(ce whether )s ,sef,l for pre*)ct)(g / that )s whether β1≠05.

or the *ata )( E7erc)se of Sect)o( 1;.2 Q&he L)(ear %orrelat)o( %oe?c)e(tQ test at the

0G le4el of s)g()+ca(ce whether )s ,sef,l for pre*)ct)(g / that )s whether β1≠05.

%o(str,ct the 9;G co(+*e(ce )(ter4al for the slope β1of the pop,lat)o( regress)o( l)(e

base* o( the sa-ple *ata set of E7erc)se of Sect)o( 1;.2 Q&he L)(ear %orrelat)o(

%oe?c)e(tQ.

6 %o(str,ct the 90G co(+*e(ce )(ter4al for the slope β1of the pop,lat)o( regress)o( l)(e


%oe?c)e(tQ.

9 or the *ata )( E7erc)se 9 of Sect)o( 1;.2 Q&he L)(ear %orrelat)o( %oe?c)e(tQ test at the

1G le4el of s)g()+ca(ce whether )s ,sef,l for pre*)ct)(g / that )s whether β1≠05.

1; or the *ata )( E7erc)se 1; of Sect)o( 1;.2 Q&he L)(ear %orrelat)o( %oe?c)e(tQ test at

the 1G le4el of s)g()+ca(ce whether )s ,sef,l for pre*)ct)(g / that )s whether β1≠05.

A22!&CAT&1NS






11 or the *ata )( E7erc)se 11 of Sect)o( 1;.2 Q&he L)(ear %orrelat)o( %oe?c)e(tQco(str,ct

a 9;G co(+*e(ce )(ter4al for the -ea( (,-ber of (ew wor*s ac<,)re* per -o(th by

ch)l*re( betwee( 13 a(* 16 -o(ths of age.

12 or the *ata )( E7erc)se 12 of Sect)o( 1;.2 Q&he L)(ear %orrelat)o( %oe?c)e(tQco(str,ct

a 9;G co(+*e(ce )(ter4al for the -ea( )(crease* brak)(g *)sta(ce for each a**)t)o(al1;; po,(*s of 4eh)cle we)ght.

13 or the *ata )( E7erc)se 13 of Sect)o( 1;.2 Q&he L)(ear %orrelat)o( %oe?c)e(tQ test at

the 1;G le4el of s)g()+ca(ce whether age )s ,sef,l for pre*)ct)(g rest)(g heart rate.


the 1;G le4el of s)g()+ca(ce whether w)(* spee* )s ,sef,l for pre*)ct)(g wa4e he)ght.

10 or the s)t,at)o( *escr)be* )( E7erc)se 10 of Sect)o( 1;.2 Q&he L)(ear %orrelat)o(

%oe?c)e(tQ

a %o(str,ct the 90G co(+*e(ce )(ter4al for the -ea( )(crease )( re4e(,e per

a**)t)o(al tho,sa(* *ollars spe(t o( a*4ert)s)(g.

b A( a*4ert)s)(g age(cy tells the b,s)(ess ow(er that for e4ery a**)t)o(al

tho,sa(* *ollars spe(t o( a*4ert)s)(g re4e(,e w)ll )(crease by o4er 20;;;.

&est th)s cla)- wh)ch )s the alter(at)4e hypothes)s5 at the 0G le4el of

s)g()+ca(ce.

c Perfor- the test of part b5 at the 1;G le4el of s)g()+ca(ce.

* #ase* o( the res,lts )( b5 a(* c5 how bel)e4able )s the a* age(cyWs cla)-C

&h)s )s a s,b>ect)4e >,*ge-e(t.5


%oe?c)e(tQ

a %o(str,ct the 9;G co(+*e(ce )(ter4al for the -ea( )(crease )( he)ght per

a**)t)o(al )(ch of le(gth at age two.

b t )s cla)-e* that for g)rls each a**)t)o(al )(ch of le(gth at age two -ea(s

-ore tha( a( a**)t)o(al )(ch of he)ght at -at,r)ty. &est th)s cla)- wh)ch )s the

alter(at)4e hypothes)s5 at the 1;G le4el of s)g()+ca(ce.


the 1;G le4el of s)g()+ca(ce whether co,rse a4erage before the +(al e7a- )s ,sef,l for

pre*)ct)(g the +(al e7a- gra*e.


%oe?c)e(tQ a( agro(o-)st cla)-s that each a**)t)o(al -)ll)o( acres pla(te* res,lts )(

-ore tha( 0;;;; a**)t)o(al acres har4este*. &est th)s cla)- at the 1G le4el of

s)g()+ca(ce.


the 1/1;th of 1G le4el of s)g()+ca(ce whether )g(or)(g all other facts s,ch as age a(*






bo*y -ass the a-o,(t of the -e*)cat)o( co(s,-e* )s a ,sef,l pre*)ctor of bloo*

co(ce(trat)o( of the act)4e )(gre*)e(t.

2; or the *ata )( E7erc)se 2; of Sect)o( 1;.2 Q&he L)(ear %orrelat)o( %oe?c)e(tQ test at

the 1G le4el of s)g()+ca(ce whether for each a**)t)o(al )(ch of g)rth the age of the tree

)(creases by at least two a(* o(e=half years.21 or the *ata )( E7erc)se 21 of Sect)o( 1;.2 Q&he L)(ear %orrelat)o( %oe?c)e(tQ

a %o(str,ct the 90G co(+*e(ce )(ter4al for the -ea( )(crease )( stre(gth at 26

*ays for each a**)t)o(al h,(*re* ps) )(crease )( stre(gth at 3 *ays.

b &est at the 1/1;th of 1G le4el of s)g()+ca(ce whether the 3=*ay stre(gth )s

,sef,l for pre*)ct)(g 26=*ay stre(gth.


%oe?c)e(tQ

a %o(str,ct the 99G co(+*e(ce )(ter4al for the -ea( *ecrease )( e(ergy

*e-a(* for each o(e=*egree *rop )( te-perat,re.

b A( e(g)(eer w)th the power co-pa(y bel)e4es that for each o(e=*egree

)(crease )( te-perat,re *a)ly e(ergy *e-a(* w)ll *ecrease by -ore tha( 3.

-)ll)o( watt=ho,rs. &est th)s cla)- at the 1G le4el of s)g()+ca(ce.

!A/:+ (ATA S+T ++/C &S+S


http://www.1.7ls

a %o-p,te the 9;G co(+*e(ce )(ter4al for the slope β1of the pop,lat)o(

regress)o( l)(e w)th SA& score as the )(*epe(*e(t 4ar)able 5 a(* !PA as the*epe(*e(t 4ar)able / 5.

b &est at the 1;G le4el of s)g()+ca(ce the hypothes)s that the slope of the

pop,lat)o( regress)o( l)(e )s greater tha( ;.;;1 aga)(st the (,ll hypothes)s

that )t )s e7actly ;.;;1.



fa-)l)ar)Vat)o( w)th the (ew cl,bs5.

http://www.12.7ls

a %o-p,te the 90G co(+*e(ce )(ter4al for the slope β1of the pop,lat)o(

regress)o( l)(e w)th scores ,s)(g the or)g)(al cl,bs as the )(*epe(*e(t

4ar)able 5 a(* scores ,s)(g the (ew cl,bs as the *epe(*e(t 4ar)able / 5.

b &est at the 1;G le4el of s)g()+ca(ce the hypothes)s that the slope of the

pop,lat)o( regress)o( l)(e )s *)8ere(t fro- 1 aga)(st the (,ll hypothes)s that

)t )s e7actly 1.








http://www.13.7ls

a %o-p,te the 90G co(+*e(ce )(ter4al for the slopeβ1

of the pop,lat)o(regress)o( l)(e w)th the (,-ber of b)**ers prese(t at the a,ct)o( as the

)(*epe(*e(t 4ar)able 5 a(* sales pr)ce as the *epe(*e(t 4ar)able / 5.

b &est at the 1;G le4el of s)g()+ca(ce the hypothes)s that the a4erage sales

pr)ce )(creases by -ore tha( 9; for each a**)t)o(al b)**er at a( a,ct)o(

aga)(st the *efa,lt that )t )(creases by e7actly 9;.











.9 The CoeJcient o (etermination

LEARNN! "#$E%&'E

1 &o lear( what the coe?c)e(t of *eter-)(at)o( )s how to co-p,te )t a(* what )t

tells ,s abo,t the relat)o(sh)p betwee( two 4ar)ables a(* / .

,f the scatter diagram of a set of (x,y)pairs shows neither an upward or downward trend$ then the

hori'ontal line y=y−fits it well$ as illustrated in /igure 15.11. The lack of any upward or downward

trend means that when an element of the population is selected at random$ knowing the value of the

measurement x for that element is not helpful in predicting the value of the measurement y.

!igure "5.""

y=y−

,f the scatter diagram shows a linear trend upward or downward then it is useful to compute the least

s"uares regression line y=β1x+β0and use it in predicting y. /igure 15.1! 0+ame +catter 2iagram






with Two %pproximating ines0 illustrates this. ,n each panel we have plotted the height and weight

data of +ection 15.1 0inear Celationships 4etween Dariables0. This is the same scatter plot as /igure

15.! 0lot of Eeight and (eight airs0$ with the average value line y=y−superimposed on it in the

left panel and the least s"uares regression line imposed on it in the right panel. The errors are

indicated graphically by the vertical line segments.

!igure "5."& %ame %catter (iagram with Two Approximating 'ines






EKAPLE 1;






&he

4al,e of

,se*

4eh)cles

of the-ake

a(*

-o*el

*)sc,sse* )( Note 1;.19 QE7a-ple 3Q )( Sect)o( 1;. Q&he Least S<,ares Regress)o(

L)(eQ4ar)es w)*ely. &he -ost e7pe(s)4e a,to-ob)le )( the sa-ple )( &able 1;.3 Qata

o( Age a(* 'al,e of Use* A,to-ob)les of a Spec)+c ake a(* o*elQ has 4al,e

3;0;; wh)ch )s (early half aga)( as -,ch as the least e7pe(s)4e o(e wh)ch )s

worth 2;;;. )(* the proport)o( of the 4ar)ab)l)ty )( 4al,e that )s acco,(te* for by

the l)(ear relat)o(sh)p betwee( age a(* 4al,e.

Sol,t)o(:

&he proport)o( of the 4ar)ab)l)ty )( 4al,e / that )s acco,(te* for by the l)(ear

relat)o(sh)p betwee( )t a(* age )s g)4e( by the coe?c)e(t of *eter-)(at)o( r 2. S)(ce

the correlat)o( coe?c)e(t r was alrea*y co-p,te* )( Note 1;.19 QE7a-ple

3Q as r=−0.819 r2=(−0.819)2=0.671.Abo,t G of the 4ar)ab)l)ty )( the 4al,e of th)s 4eh)cle

ca( be e7pla)(e* by )ts age.

EKAPLE 11

Use each of the three for-,las for the coe?c)e(t of *eter-)(at)o( to co-p,te )ts

4al,e for the e7a-ple of ages a(* 4al,es of 4eh)cles.

Sol,t)o(:






( Note 1;.19 QE7a-ple 3Q )( Sect)o( 1;. Q&he Least S<,ares Regress)o( L)(eQ we

co-p,te* the e7act 4al,es

The coefficient of determination r! can always be computed by s"uaring the correlation coefficient r if

it is known. %ny one of the defining formulas can also be used. Typically one would make the choice

based on which "uantities have already been computed. (hat should be avoided is trying to

compute r by taking the s"uare root of r!$ if it is already known$ since it is easy to make a sign error

this way. To see what can go wrong$ supposer2=0.64.Taking the s"uare root of a positive number

with any calculating device will always return a positive result. The s"uare root of 5.;6 is 5.?.

Eowever$ the actual value of r might be the negative number Z5.?.






*+, TA*+AA,S

• &he coe?c)e(t of *eter-)(at)o( r 2 est)-ates the proport)o( of the 4ar)ab)l)ty )(

the 4ar)able / that )s e7pla)(e* by the l)(ear relat)o(sh)p betwee( / a(* the

4ar)able .

• &here are se4eral for-,las for co-p,t)(g r 2. &he cho)ce of wh)ch o(e to ,se ca(

be base* o( wh)ch <,a(t)t)es ha4e alrea*y bee( co-p,te* so far.

EKER%SES

#AS%

or the #as)c a(* Appl)cat)o( e7erc)ses )( th)s sect)o( ,se the co-p,tat)o(s that

were *o(e for the e7erc)ses w)th the sa-e (,-ber )(Sect)o( 1;.2 Q&he L)(ear

%orrelat)o( %oe?c)e(tQ Sect)o( 1;. Q&he Least S<,ares Regress)o( L)(eQ

a(* Sect)o( 1;.0 QStat)st)cal (fere(ces Abo,t Q.

1 or the sa-ple *ata set of E7erc)se 1 of Sect)o( 1;.2 Q&he L)(ear %orrelat)o(

%oe?c)e(tQ +(* the coe?c)e(t of *eter-)(at)o( ,s)(g the

for-,la r2=β1SSxy/SSyy.%o(+r- yo,r a(swer by s<,ar)(g r as co-p,te* )( that e7erc)se.












or the sa-ple *ata set of E7erc)se of Sect)o( 1;.2 Q&he L)(ear %orrelat)o(











for-,la r2=(SSyy−SSE)/SSyy.%o(+r- yo,r a(swer by s<,ar)(g r as co-p,te* )( that

e7erc)se.




e7erc)se.




e7erc)se.

1; or the sa-ple *ata set of E7erc)se 9 of Sect)o( 1;.2 Q&he L)(ear %orrelat)o(



e7erc)se.

APPL%A&"NS






11 or the *ata )( E7erc)se 11 of Sect)o( 1;.2 Q&he L)(ear %orrelat)o( %oe?c)e(tQ co-p,te

the coe?c)e(t of *eter-)(at)o( a(* )(terpret )ts 4al,e )( the co(te7t of age a(*

4ocab,lary.12 or the *ata )( E7erc)se 12 of Sect)o( 1;.2 Q&he L)(ear %orrelat)o( %oe?c)e(tQ co-p,te

the coe?c)e(t of *eter-)(at)o( a(* )(terpret )ts 4al,e )( the co(te7t of 4eh)cle we)ght

a(* brak)(g *)sta(ce.13 or the *ata )( E7erc)se 13 of Sect)o( 1;.2 Q&he L)(ear %orrelat)o( %oe?c)e(tQ co-p,te

the coe?c)e(t of *eter-)(at)o( a(* )(terpret )ts 4al,e )( the co(te7t of age a(* rest)(g

heart rate. ( the age ra(ge of the *ata *oes age see- to be a 4ery )-porta(t factor

w)th regar* to heart rateC1 or the *ata )( E7erc)se 1 of Sect)o( 1;.2 Q&he L)(ear %orrelat)o( %oe?c)e(tQ co-p,te

the coe?c)e(t of *eter-)(at)o( a(* )(terpret )ts 4al,e )( the co(te7t of w)(* spee* a(*

wa4e he)ght. oes w)(* spee* see- to be a 4ery )-porta(t factor w)th regar* to wa4e

he)ghtC10 or the *ata )( E7erc)se 10 of Sect)o( 1;.2 Q&he L)(ear %orrelat)o( %oe?c)e(tQ +(* the

proport)o( of the 4ar)ab)l)ty )( re4e(,e that )s e7pla)(e* by le4el of a*4ert)s)(g.1 or the *ata )( E7erc)se 1 of Sect)o( 1;.2 Q&he L)(ear %orrelat)o( %oe?c)e(tQ +(* the

proport)o( of the 4ar)ab)l)ty )( a*,lt he)ght that )s e7pla)(e* by the 4ar)at)o( )( le(gth at

age two.1 or the *ata )( E7erc)se 1 of Sect)o( 1;.2 Q&he L)(ear %orrelat)o( %oe?c)e(tQ co-p,te

the coe?c)e(t of *eter-)(at)o( a(* )(terpret )ts 4al,e )( the co(te7t of co,rse a4erage

before the +(al e7a- a(* score o( the +(al e7a-.16 or the *ata )( E7erc)se 16 of Sect)o( 1;.2 Q&he L)(ear %orrelat)o( %oe?c)e(tQ co-p,te

the coe?c)e(t of *eter-)(at)o( a(* )(terpret )ts 4al,e )( the co(te7t of acres pla(te*

a(* acres har4este*.19 or the *ata )( E7erc)se 19 of Sect)o( 1;.2 Q&he L)(ear %orrelat)o( %oe?c)e(tQ co-p,te

the coe?c)e(t of *eter-)(at)o( a(* )(terpret )ts 4al,e )( the co(te7t of the a-o,(t of

the -e*)cat)o( co(s,-e* a(* bloo* co(ce(trat)o( of the act)4e )(gre*)e(t.2; or the *ata )( E7erc)se 2; of Sect)o( 1;.2 Q&he L)(ear %orrelat)o( %oe?c)e(tQ co-p,te

the coe?c)e(t of *eter-)(at)o( a(* )(terpret )ts 4al,e )( the co(te7t of tree s)Ve a(*

age.21 or the *ata )( E7erc)se 21 of Sect)o( 1;.2 Q&he L)(ear %orrelat)o( %oe?c)e(tQ +(* the

proport)o( of the 4ar)ab)l)ty )( 26=*ay stre(gth of co(crete that )s acco,(te* for by

4ar)at)o( )( 3=*ay stre(gth.

22 or the *ata )( E7erc)se 22 of Sect)o( 1;.2 Q&he L)(ear %orrelat)o( %oe?c)e(tQ +(* theproport)o( of the 4ar)ab)l)ty )( e(ergy *e-a(* that )s acco,(te* for by 4ar)at)o( )(

a4erage te-perat,re.

LAR!E A&A SE & EKE R%SES






23 Large ata Set 1 l)sts the SA& scores a(* !PAs of 1;;; st,*e(ts. %o-p,te the

coe?c)e(t of *eter-)(at)o( a(* )(terpret )ts 4al,e )( the co(te7t of SA& scores a(*

!PAs.

http://www.1.7ls



fa-)l)ar)Vat)o( w)th the (ew cl,bs5. %o-p,te the coe?c)e(t of *eter-)(at)o( a(*

)(terpret )ts 4al,e )( the co(te7t of golf scores w)th the two k)(*s of golf cl,bs.

http://www.12.7ls


a(t)<,e gra(*father clock at ; a,ct)o(s. %o-p,te the coe?c)e(t of *eter-)(at)o( a(*

)(terpret )ts 4al,e )( the co(te7t of the (,-ber of b)**ers at a( a,ct)o( a(* the pr)ce of

th)s type of a(t)<,e gra(*father clock.

http://www.13.7ls






%.4 +stimation and 2rediction

LEARNN! "#$E%&'ES

1 &o lear( the *)st)(ct)o( betwee( est)-at)o( a(* pre*)ct)o(.

2 &o lear( the *)st)(ct)o( betwee( a co(+*e(ce )(ter4al a(* a pre*)ct)o( )(ter4al.

3 &o lear( how to )-ple-e(t for-,las for co-p,t)(g co(+*e(ce )(ter4als a(*

pre*)ct)o( )(ter4als.






7onsider the following pairs of problems$ in the context of 9ote 15.1F 0xample 30 in +ection 15.6

0The east +"uares Cegression ine0$ the automobile age and value example.

1

1 stimate the average value of all four-year-old automobiles of this make and

model.

! 7onstruct a F8B confidence interval for the average value of all four-year-old

automobiles of this make and model.

!

1 +hylock intends to buy a four-year-old automobile of this make and model next

week. redict the value of the first such automobile that he encounters.

! 7onstruct a F8B confidence interval for the value of the first such automobile

that he encounters.

The method of solution and answer to the first "uestion in each pair$ *1a) and *!a)$ are the

same. (hen we set x e"ual to 6 in the least s"uares regression

e"uation y=−2.05x+32.83that was computed in part *c) of 9ote 15.1F 0xample

30 in +ection 15.6 0The east +"uares Cegression ine0$ the number returned$

y=−2.05(4)+32.83=24.63

which corresponds to value >!6$;35$ is an estimate of precisely the number sought in

"uestion *1a): the meanE(y)of all y values when x M 6. +ince nothing is known about the firstfour-year-old automobile of this make and model that +hylock will encounter$ our best guess

as to its value is the mean value E(y)of all such automobiles$ the number !6.;3 or >!6$;35$

computed in the same way.






The answers to the second part of each "uestion differ. ,n "uestion *1b) we are trying to

estimate a population parameter: the mean of the all the y-values in the sub-population

picked out by the value x M 6$ that is$ the average value of all four-year-old automobiles. ,n

"uestion *!b)$ however$ we are not trying to capture a fixed parameter$ but the value of the

random variable y in one trial of an experiment: examine the first four-year-old car +hylockencounters. ,n the first case we seek to construct a confidence interval in the same sense that

we have done before. ,n the second case the situation is different$ and the interval

constructed has a different name$ prediction interval. ,n the second case we are trying to

GpredictH where a the value of a random variable will take its value.






a. x p is a particular value of x that lies in the range of x -values in the data set used to construct the

least s"uares regression line






b. y pis the numerical value obtained when the least s"uare regression e"uation is evaluated atx=x p

and

c. the number of degrees of freedom fortα/2isdf=n−2.

The assumptions listed in +ection 15.3 0odelling inear Celationships with Candomness resent0 must

hold.

EKAPLE 12

Us)(g the sa-ple *ata of Note 1;.19 QE7a-ple 3Q )( Sect)o( 1;. Q&he Least S<,ares

Regress)o( L)(eQ recor*e* )( &able 1;.3 Qata o( Age a(* 'al,e of Use* A,to-ob)les

of a Spec)+c ake a(* o*elQ co(str,ct a 90G co(+*e(ce )(ter4al for the a4erage

4al,e of all three=a(*=o(e=half=year=ol* a,to-ob)les of th)s -ake a(* -o*el.

Sol,t)o(:

Sol4)(g th)s proble- )s -erely a -atter of +(*)(g the 4al,es of y p αa(* tα 2@ sε@ x−

a(* SSxx a(* )(sert)(g the- )(to the co(+*e(ce )(ter4al for-,la g)4e( >,st abo4e.

ost of these <,a(t)t)es are alrea*y k(ow(. ro- Note 1;.19 QE7a-ple 3Q )( Sect)o(

1;. Q&he Least S<,ares Regress)o( L)(eQ SSxx=14a(* x−=4.ro- Note 1;.31

QE7a-ple Q )(Sect)o( 1;.0 QStat)st)cal (fere(ces Abo,t Q sε=1.902169814.











IEJ &AIE ABAJS

• A co(+*e(ce )(ter4al )s ,se* to est)-ate the -ea( 4al,e of / )( the s,b=

pop,lat)o( *eter-)(e* by the co(*)t)o( that ha4e so-e spec)+c 4al,e p.

• &he pre*)ct)o( )(ter4al )s ,se* to pre*)ct the 4al,e that the ra(*o- 4ar)able / w)ll

take whe( has so-e spec)+c 4al,e p.

EKER%SES

#AS%

or the #as)c a(* Appl)cat)o( e7erc)ses )( th)s sect)o( ,se the co-p,tat)o(s that were

*o(e for the e7erc)ses w)th the sa-e (,-ber )( pre4)o,s sect)o(s.






1 or the sa-ple *ata set of E7erc)se 1 of Sect)o( 1;.2 Q&he L)(ear %orrelat)o( %oe?c)e(tQ

a !)4e a po)(t est)-ate for the -ea( 4al,e of / )( the s,b=pop,lat)o(

*eter-)(e* by the co(*)t)o( .

b %o(str,ct the 9;G co(+*e(ce )(ter4al for that -ea( 4al,e.








b %o(str,ct the 90G co(+*e(ce )(ter4al for that -ea( 4al,e.

or the sa-ple *ata set of E7erc)se of Sect)o( 1;.2 Q&he L)(ear %orrelat)o( %oe?c)e(tQ


*eter-)(e* by the co(*)t)o( 2.



















c s )t 4al)* to -ake the sa-e est)-ates for 12C E7pla)(.





c s )t 4al)* to -ake the sa-e est)-ates for ;C E7pla)(.



*eter-)(e* by the co(*)t)o( ;.


c s )t 4al)* to -ake the sa-e est)-ates for x=−1C E7pla)(.

1; or the sa-ple *ata set of E7erc)se 9 of Sect)o( 1;.2 Q&he L)(ear %orrelat)o( %oe?c)e(tQ




c s )t 4al)* to -ake the sa-e est)-ates for ;C E7pla)(.

APPL%A&"NS


a !)4e a po)(t est)-ate for the a4erage (,-ber of wor*s )( the 4ocab,lary of

16=-o(th=ol* ch)l*re(.


c s )t 4al)* to -ake the sa-e est)-ates for two=year=ol*sC E7pla)(.


a !)4e a po)(t est)-ate for the a4erage brak)(g *)sta(ce of a,to-ob)les that

we)gh 320; po,(*s.


c s )t 4al)* to -ake the sa-e est)-ates for 0;;;=po,(* a,to-ob)lesC E7pla)(.







a !)4e a po)(t est)-ate for the rest)(g heart rate of a -a( who )s 30 years ol*.

b "(e of the -e( )( the sa-ple )s 30 years ol* b,t h)s rest)(g heart rate )s (ot

what yo, co-p,te* )( part a5. E7pla)( why th)s )s (ot a co(tra*)ct)o(.

c %o(str,ct the 9;G co(+*e(ce )(ter4al for the -ea( rest)(g heart rate of all

30=year=ol* -e(.


a !)4e a po)(t est)-ate for the wa4e he)ght whe( the w)(* spee* )s 13 -)les per

ho,r.

b "(e of the w)(* spee*s )( the sa-ple )s 13 -)les per ho,r b,t the he)ght of

wa4es that *ay )s (ot what yo, co-p,te* )( part a5. E7pla)( why th)s )s (ot a

co(tra*)ct)o(.

c %o(str,ct the 9;G co(+*e(ce )(ter4al for the -ea( wa4e he)ght o( *ays

whe( the w)(* spee* )s 13 -)les per ho,r.


a &he b,s)(ess ow(er )(te(*s to spe(* 20;; o( a*4ert)s)(g (e7t year. !)4e a(

est)-ate of (e7t yearWs re4e(,e base* o( th)s fact.

b %o(str,ct the 9;G pre*)ct)o( )(ter4al for (e7t yearWs re4e(,e base* o( the

)(te(t to spe(* 20;; o( a*4ert)s)(g.


a A two=year=ol* g)rl )s 32.3 )(ches lo(g. Pre*)ct her a*,lt he)ght.

b %o(str,ct the 90G pre*)ct)o( )(ter4al for the g)rlWs a*,lt he)ght.


a Lo*o4)co has a 6. a4erage )( h)s phys)cs class >,st before the +(al. !)4e a

po)(t est)-ate of what h)s +(al e7a- gra*e w)ll be.

b E7pla)( whether a( )(ter4al est)-ate for th)s proble- )s a co(+*e(ce )(ter4al

or a pre*)ct)o( )(ter4al.

c #ase* o( yo,r a(swer to b5 co(str,ct a( )(ter4al est)-ate for Lo*o4)coWs

+(al e7a- gra*e at the 9;G le4el of co(+*e(ce.







a &h)s year 6.2 -)ll)o( acres of cor( were pla(te*. !)4e a po)(t est)-ate of the

(,-ber of acres that w)ll be har4este* th)s year.



c #ase* o( yo,r a(swer to b5 co(str,ct a( )(ter4al est)-ate for the (,-ber of

acres that w)ll be har4este* th)s year at the 99G le4el of co(+*e(ce.


a !)4e a po)(t est)-ate for the bloo* co(ce(trat)o( of the act)4e )(gre*)e(t of

th)s -e*)cat)o( )( a -a( who has co(s,-e* 1.0 o,(ces of the -e*)cat)o( >,st

rece(tly.

b !rat)a(o >,st co(s,-e* 1.0 o,(ces of th)s -e*)cat)o( 3; -)(,tes ago.

%o(str,ct a 90G pre*)ct)o( )(ter4al for the co(ce(trat)o( of the act)4e

)(gre*)e(t )( h)s bloo* r)ght (ow.


a Jo, -eas,re the g)rth of a free=sta(*)(g oak tree +4e feet o8 the gro,(* a(*

obta)( the 4al,e 12 )(ches. @ow ol* *o yo, est)-ate the tree to beC

b %o(str,ct a 9;G pre*)ct)o( )(ter4al for the age of th)s tree.


a A test cyl)(*er of co(crete three *ays ol* fa)ls at 10; ps). Pre*)ct what the

26=*ay stre(gth of the co(crete w)ll be.

b %o(str,ct a 99G pre*)ct)o( )(ter4al for the 26=*ay stre(gth of th)s co(crete.

c #ase* o( yo,r a(swer to b5 what wo,l* be the -)()-,- 26=*ay stre(gth

yo, co,l* e7pect th)s co(crete to e7h)b)tC


a &o-orrowWs a4erage te-perat,re )s forecast to be 03 *egrees. Est)-ate the

e(ergy *e-a(* to-orrow.

b %o(str,ct a 99G pre*)ct)o( )(ter4al for the e(ergy *e-a(* to-orrow.

c #ase* o( yo,r a(swer to b5 what wo,l* be the -)()-,- *e-a(* yo, co,l*

e7pectC








http://www.1.7ls

a !)4e a po)(t est)-ate of the -ea( !PA of all st,*e(ts who score 130; o( the

SA&.

b %o(str,ct a 9;G co(+*e(ce )(ter4al for the -ea( !PA of all st,*e(ts who

score 130; o( the SA&.


ow( or)g)(al cl,bs the( ,s)(g cl,bs of a (ew e7per)-e(tal *es)g( after two -o(ths offa-)l)ar)Vat)o( w)th the (ew cl,bs5.

http://www.12.7ls

a &h,r)o a4erages 2 strokes per ro,(* w)th h)s ow( cl,bs. !)4e a po)(t

est)-ate for h)s score o( o(e ro,(* )f he sw)tches to the (ew cl,bs.



c #ase* o( yo,r a(swer to b5 co(str,ct a( )(ter4al est)-ate for &h,r)oWs score

o( o(e ro,(* )f he sw)tches to the (ew cl,bs at 9;G co(+*e(ce.



http://www.13.7ls

a &here are se4e( l)kely b)**ers at the 'ero(a a,ct)o( to*ay. !)4e a po)(t

est)-ate for the pr)ce of s,ch a clock at to*ayWs a,ct)o(.








c #ase* o( yo,r a(swer to b5 co(str,ct a( )(ter4al est)-ate for the l)kely sale

pr)ce of s,ch a clock at to*ayWs sale at 90G co(+*e(ce.






%.5 A Complete +xample

LEARNN! "#$E%&'E

1 &o see a co-plete l)(ear correlat)o( a(* regress)o( a(alys)s )( a pract)cal sett)(g

as a cohes)4e whole.

,n the preceding sections numerous concepts were introduced and illustrated$ but the analysis was

broken into dis#oint pieces by sections. ,n this section we will go through a complete example of the

use of correlation and regression analysis of data from start to finish$ touching on all the topics ofthis chapter in se"uence.

,n general educators are convinced that$ all other factors being e"ual$ class attendance has a

significant bearing on course performance. To investigate the relationship between attendance and






performance$ an education researcher selects for study a multiple section introductory statistics

course at a large university. ,nstructors in the course agree to keep an accurate record of attendance

throughout one semester. %t the end of the semester !; students are selected a random. /or each

student in the sample two measurements are taken: x $ the number of days the student was absent$

andy$ the student&s score on the common final exam in the course. The data are summari'ed in Table15.6 0%bsence and +core 2ata0.

Table 15.6 %bsence and +core 2ata

Absences Score Absences Score

x y x y

2 1

29 0 3

2 9 66

3 ; 96

2 9 1 99

1 ; 69

; 66 1 9

; 92 3 9;

00 1 9;

; 3 6

2 6; 1 6

2 0 3 6;

1 3 1 6






% scatter plot of the data is given in /igure 15.13 0lot of the %bsence and xam +core airs0. There

is a downward trend in the plot which indicates that on average students with more absences tend to

do worse on the final examination.

!igure "5."* $lot of the Absence and ;xam %core $airs

The trend observed in /igure 15.13 0lot of the %bsence and xam +core airs0 as well as the fairly

constant width of the apparent band of points in the plot makes it reasonable to assume a

relationship between x and y of the form

y=β1x+β0+ε

whereβ1andβ0are unknown parameters and L is a normal random variable with mean 'ero and

unknown standard deviation 6 . 9ote carefully that this model is being proposed for the population of

all students taking this course$ not #ust those taking it this semester$ and certainly not #ust those in

the sample. The numbersβ1$β0$ and 6 are parameters relating to this large population.

/irst we perform preliminary computations that will be needed later. The data are processed in Table

15.8 0rocessed %bsence and +core 2ata0.











The statisticsεestimates the standard deviation 6 of the normal random variable L in the model. ,ts

meaning is that among all students with the same number of absences$ the standard deviation of

their scores on the final exam is about 1!.1 points. +uch a large value on a 155-point exam means

that the final exam scores of each sub-population of students$ based on the number of absences$ are

highly variable.

The si'e and sign of the slope β1=−5.23indicate that$ for every class missed$ students tend to score

about 8.!3 fewer points lower on the final exam on average. +imilarly for every two classes missed






students tend to score on average2×5.23=10.46fewer points on the final exam$ or about a letter

grade worse on average.

+ince 5 is in the range of x -values in the data set$ the y-intercept also has meaning in this problem. ,t

is an estimate of the average grade on the final exam of all students who have perfect attendance. The

predicted average of such students is β0=91.24.

4efore we use the regression e"uation further$ or perform other analyses$ it would be a good idea to

examine the utility of the linear regression model. (e can do this in two ways: 1) by computing the

correlation coefficient r to see how strongly the number of absences x and the score y on the final

exam are correlated$ and !) by testing the null hypothesisH0:β1=0*the slope of

the population regression line is 'ero$ so x is not a good predictor of y) against the natural

alternativeHa:β1<0*the slope of the population regression line is negative$ so final exam scores y go

down as absences x go up).
















or about 6FB. Thus although there is a significant correlation between attendance and performance

on the final exam$ and we can estimate with fair accuracy the average score of students who miss a

certain number of classes$ nevertheless less than half the total variation of the exam scores in the

sample is explained by the number of absences. This should not come as a surprise$ since there are

many factors besides attendance that bear on student performance on exams.

*+, TA*+AA,

• t )s a goo* )*ea to atte(* class.

++/C&S+S






&he e7erc)ses )( th)s sect)o( are ,(relate* to those )( pre4)o,s sect)o(s.

1 &he *ata g)4e the a-o,(t of s)l)co[,or)*e )( the water -g/L5 a(* the a-o,(t / of

lea* )( the bloo*strea- μg/*L5 of te( ch)l*re( )( 4ar)o,s co--,()t)es w)th a(* w)tho,t

-,()c)pal water. Perfor- a co-plete a(alys)s of the *ata )( a(alogy w)th the *)sc,ss)o(

)( th)s sect)o( that )s -ake a scatter plot *o prel)-)(ary co-p,tat)o(s +(* the leasts<,ares regress)o( l)(e +(* SSE sε a(* r a(* so o(5. ( the hypothes)s test ,se as the

alter(at)4e hypothes)s β1>0 a(* test at the 0G le4el of s)g()+ca(ce. Use co(+*e(ce le4el

90G for the co(+*e(ce )(ter4al for β1.%o(str,ct 90G co(+*e(ce a(* pre*)ct)o(s

)(ter4als at x p=2at the e(*.






http://www.3.7ls

http://www.3A.7ls

Separate o,t fro- Large ata Set 3A >,st the *ata o( -e( a(* *o a co-plete

a(alys)s w)th shoe s)Ve as the )(*epe(*e(t 4ar)able 5 a(* he)ght as the *epe(*e(t

4ar)able / 5. Use α=0.05a(* x p=10whe(e4er appropr)ate.

http://www.3A.7ls

0 Separate o,t fro- Large ata Set 3A >,st the *ata o( wo-e( a(* *o a co-plete

a(alys)s w)th shoe s)Ve as the )(*epe(*e(t 4ar)able 5 a(* he)ght as the *epe(*e(t

4ar)able / 5. Use α=0.05a(* x p=10whe(e4er appropr)ate.

http://www.3A.7ls





















Chapter %%

Chi-S"uare Tests and F-Tests

,n previous chapters you saw how to test hypotheses concerning population means and populationproportions. The idea of testing hypotheses can be extended to many other situations that involve

different parameters and use different test statistics. (hereas the standardi'ed test statistics that

appeared in earlier chapters followed either a normal or +tudent t -distribution$ in this chapter the

tests will involve two other very common and useful distributions$ the chi-s"uare and the ! -

distributions. The chi(square distribution arises in tests of hypotheses concerning the

independence of two random variables and concerning whether a discrete random variable follows a

specified distribution. The 7(distribution arises in tests of hypotheses concerning whether or not

two population variances are e"ual and concerning whether or not three or more population means

are e"ual.

%%.% Chi-S"uare Tests or &ndependence






!+A/N&N: 1';+CT&<+S

1 &o ,(*ersta(* what ch)=s<,are *)str)b,t)o(s are.

2 &o ,(*ersta(* how to ,se a ch)=s<,are test to >,*ge whether two factors are

)(*epe(*e(t.

Chi-S"uare (istributions

%s you know$ there is a whole family of t -distributions$ each one specified by a parameter called

the degrees of freedom$ denoteddf.+imilarly$ all the chi-s"uare distributions form a family$ and each of

its members is also specified by a parameterdf$ the number of degrees of freedom. 7hi is a reek letter

denoted by the symbolχand chi-s"uare is often denoted byχ2./igure 11.1 0any 0 shows several chi-

s"uare distributions for different degrees of freedom. % chi-s"uare random variable is a random variable

that assumes only positive values and follows a chi-s"uare distribution.

!igure ""." 2anyχ2 (istributions

e+()t)o(

The value of the chi:square random variable χ2with df=kthat cuts off a right tail of area c is

denoted χ2cand is called a critical value. %ee !igure "".&.






!igure "".&χ2c >llustrated

/igure 1!.6 07ritical Dalues of 7hi-+"uare 2istributions0 gives values ofχ2cfor various values of c and

under several chi-s"uare distributions with various degrees of freedom.

Tests or &ndependence

Eypotheses tests encountered earlier in the book had to do with how the numerical values of two

population parameters compared. ,n this subsection we will investigate hypotheses that have to do with

whether or not two random variables take their values independently$ or whether the value of one has a

relation to the value of the other. Thus the hypotheses will be expressed in words$ not mathematical

symbols. (e build the discussion around the following example.






There is a theory that the gender of a baby in the womb is related to the baby&s heart rate: baby girls tend

to have higher heart rates. +uppose we wish to test this theory. (e examine the heart rate records of 65

babies taken during their mothers& last prenatal checkups before delivery$ and to each of these 65

randomly selected records we compute the values of two random measures: 1) gender and !) heart rate. ,n

this context these two random measures are often called factors. +ince the burden of proof is that heart

rate and gender are related$ not that they are unrelated$ the problem of testing the theory on baby gender

and heart rate can be formulated as a test of the following hypotheses:

HO: Baby gender and baby heart rate are independent

vs H. a: Baby gender and baby heart rate arenot independent

The factor gender has two natural categories or levels: boy and girl. (e divide the second factor$

heart rate$ into two levels$ low and high$ by choosing some heart rate$ say 168 beats per minute$ as

the cutoff between them. % heart rate below 168 beats per minute will be considered low and 168 and

above considered high. The 65 records give rise to a ! !contingency table. 4y ad#oining row totals$

column totals$ and a grand total we obtain the table shown as Table 11.1 04aby ender and Eeart

Cate0. The four entries in boldface type are counts of observations from the sample of nM 65. There

were 11 girls with low heart rate$ 1@ boys with low heart rate$ and so on. They form the core of the

expanded table.

Table 11.1 4aby ender and Eeart Cate

4eart 'ate

2o3 4igh 'o3 Total

(ender

(ir" ; 18

:o ; + 22

o"&%n ,o'a" 28 12 ,o'a" F 40

,n analogy with the fact that the probability of independent events is the product of the probabilities

of each event$ if heart rate and gender were independent then we would expect the number in each

core cell to be close to the product of the row total - and column total < of the row and column

containing it$ divided by the sample si'e n. 2enoting such an expected number of observations ; $

these four expected values are:






• 1st row and 1st column:E=(R×C)/n=18×28/40=12.6

• 1st row and !nd column:E=(R×C)/n=18×12/40=5.4

• !nd row and 1st column:E=(R×C)/n=22×28/40=15.4

• !nd row and !nd column:E=(R×C)/n=22×12/40=6.6

(e update Table 11.1 04aby ender and Eeart Cate0 by placing each expected value in its

corresponding core cell$ right under the observed value in the cell. This gives the updated table Table

11.! 0<pdated 4aby ender and Eeart Cate0.

Table 11.! <pdated 4aby ender and Eeart Cate

=eart /ate

!ow =i$h /ow Total

!e(*er

!)rl O=11E=12.6 O=7E=5.4 6 16

#oy O=17E=15.4 O=5E=6.6 6 22

%ol,-( &otal , 26 , 12 n ;

% measure of how much the data deviate from what we would expect to see if the factors really were

independent is the sum of the s"uares of the difference of the numbers in each core cell$ or$ standardi'ing

by dividing each s"uare by the expected number in the cell$ the sumΣ(O−E)2

E. (e would re#ect the

null hypothesis that the factors are independent only if this number is large$ so the test is right-tailed. ,n






this example the random variableΣ(O−E)2 Ehas the chi-s"uare distribution with one degree of

freedom. ,f we had decided at the outset to test at the 15B level of significance$ the critical value defining

the re#ection region would be$ reading from /igure 1!.6 07ritical Dalues of 7hi-+"uare

2istributions0$χ2α=χ20.10=2.706$ so that the re#ection region would be the interval[2.706,∞). (hen

we compute the value of the standardi'ed test statistic we obtain

%s in the example each factor is divided into a number of categories or levels. These could arise

naturally$ as in the boy-girl division of gender$ or somewhat arbitrarily$ as in the high-low division of

heart rate. +uppose /actor 1 has > levels and /actor ! has M levels. Then the information from a






random sample gives rise to a general > M contingency table$ which with row totals$ column totals$

and a grand total would appear as shown in Table 11.3 0eneral 7ontingency Table0. ach cell may

be labeled by a pair of indices(i, j).Oijstands for the observed count of observations in the cell in

row i and column j $ -i for theithrow total and < j for the jthcolumn total. To simplify the notation we

will drop the indices so Table 11.3 0eneral 7ontingency Table0 becomes Table 11.6 0+implified

eneral 7ontingency Table0. 9evertheless it is important to keep in mind that the )s$ the -s and

the < s$ though denoted by the same symbols$ are in fact different numbers.

Table 11.3 eneral 7ontingency Table

Factor 0 !evels

% Y Y Y j Y Y Y J /ow Total

actor 1 Le4els

1 11 Y Y Y O1 j Y Y Y O1J 61

i Oi1 Y Y Y Oij Y Y Y OiJ 6i

# OI1 Y Y Y OIj Y Y Y OIJ 6#

%ol,-( &otal ,1 Y Y Y , E

Y Y Y , n

Table 11.6 +implified eneral 7ontingency Table

Factor 0 !evels


actor 1 Le4els 1 Y Y Y Y Y Y 6

i Y Y Y Y Y Y 6






Factor 0 !evels


# Y Y Y Y Y Y 6

%ol,-( &otal , Y Y Y , Y Y Y , n

%s in the example$ for each core cell in the table we compute what would be the expected number ; of

observations if the two factors were independent. ; is computed for each core cell *each cell with

an ) in it) of Table 11.6 0+implified eneral 7ontingency Table0 by the rule applied in the example:











EKAPLE 1

A researcher w)shes to )(4est)gate whether st,*e(tsW scores o( a college

e(tra(ce e7a-)(at)o( %EE5 ha4e a(y )(*)cat)4e power for f,t,re college

perfor-a(ce as -eas,re* by !PA. ( other wor*s he w)shes to )(4est)gate

whether the factors %EE a(* !PA are )(*epe(*e(t or (ot. @e ra(*o-ly

selects n 1;; st,*e(ts )( a college a(* (otes each st,*e(tWs score o( the

e(tra(ce e7a-)(at)o( a(* h)s gra*e po)(t a4erage at the e(* of the sopho-ore

year. @e *)4)*es e(tra(ce e7a- scores )(to two le4els a(* gra*e po)(t a4erages

)(to three le4els. Sort)(g the *ata accor*)(g to these *)4)s)o(s he for-s the

co(t)(ge(cy table show( as &able 11. Q%EE 4ers,s !PA %o(t)(ge(cy &ableQ )(

wh)ch the row a(* col,-( totals ha4e alrea*y bee( co-p,te*.






&A#LE 11. %EE 'ERSUS !PA %"N&N!EN%J &A#LE

:2A

K0.4 0.4 to 3.0 L3.0 /ow Total

%EE

<1800 38 %0 8 02

≥1800 9 07 %5 6

%ol,-( &otal 1 3 23 Total=100

&est at the 1G le4el of s)g()+ca(ce whether these *ata pro4)*e s,?c)e(t

e4)*e(ce to co(cl,*e that %EE scores )(*)cate f,t,re perfor-a(ce le4els of

)(co-)(g college fresh-e( as -eas,re* by !PA.

Sol,t)o(:

Be perfor- the test ,s)(g the cr)t)cal 4al,e approach follow)(g the ,s,al +4e=step -etho* o,tl)(e* at the e(* of Sect)o( 6.1 Q&he Ele-e(ts of @ypothes)s

&est)(gQ )( %hapter 6 Q&est)(g @ypothesesQ.

• Step 1. &he hypotheses are

H0: CEE and GPA are independent factors

vs.Ha: CEE and GPA are not independent factors

• Step 2. &he *)str)b,t)o( )s ch)=s<,are.

• Step 3. &o co-p,te the 4al,e of the test stat)st)c we -,st +rst co-p,te* the

e7pecte* (,-ber for each of the s)7 core cells the o(es whose e(tr)es are

bol*face5:

o 1st row a(* 1st col,-(: E=(R×C)/n=41×52/100=21.32

o 1st row a(* 2(* col,-(: E=(R×C)/n=36×52/100=18.72






o 1st row a(* 3r* col,-(: E=(R×C)/n=23×52/100=11.96

o 2(* row a(* 1st col,-(: E=(R×C)/n=41×48/100=19.68

o 2(* row a(* 2(* col,-(: E=(R×C)/n=36×48/100=17.28

o

2(* row a(* 3r* col,-(: E=(R×C)/n=23×48/100=11.04 &able 11. Q%EE 4ers,s !PA %o(t)(ge(cy &ableQ )s ,p*ate* to &able 11.

QUp*ate* %EE 4ers,s !PA %o(t)(ge(cy &ableQ.






• Step 0. S)(ce 31.0 9.21 the *ec)s)o( )s to re>ect the (,ll hypothes)s.

See )g,re 11.. &he *ata pro4)*e s,?c)e(t e4)*e(ce at the 1G le4el

of s)g()+ca(ce to co(cl,*e that %EE score a(* !PA are (ot

)(*epe(*e(t: the e(tra(ce e7a- score has pre*)ct)4e power.

Figure 11.8'ote 11. >(ample 1>

IEJ &AIEABAJS

• %r)t)cal 4al,es of a ch)=s<,are *)str)b,t)o( w)th *egrees of free*o- dfare fo,(*

)( )g,re 12. Q%r)t)cal 'al,es of %h)=S<,are )str)b,t)o(sQ.

• A ch)=s<,are test ca( be ,se* to e4al,ate the hypothes)s that two ra(*o-

4ar)ables or factors are )(*epe(*e(t.











Factor

2evel 2evel ! 'o3 Total

a!'or 2

?eve" 1 20 10 R

?eve" 2 15 5 R

?eve" 3 10 20 R

o"&%n ,o'a" C C n

a )(* the col,-( totals the row totals a(* the gra(* total n of the table.






b )(* the e7pecte* (,-ber ( of obser4at)o(s for each cell base* o( the ass,-pt)o( that

the two factors are )(*epe(*e(t that )s >,st ,se the for-,la E=(R×C)/n5.

c )(* the 4al,e of the ch)=s<,are test stat)st)c χ2.

* )(* the (,-ber of *egrees of free*o- of the ch)=s<,are test stat)st)c.

A22!&CAT&1NS9 A ch)l* psycholog)st bel)e4es that ch)l*re( perfor- better o( tests whe( they are g)4e(

perce)4e* free*o- of cho)ce. &o test th)s bel)ef the psycholog)st carr)e* o,t a(

e7per)-e(t )( wh)ch 2;; th)r* gra*ers were ra(*o-ly ass)g(e* to two gro,ps 0 a(* -.

Each ch)l* was g)4e( the sa-e s)-ple log)c test. @owe4er )( gro,p - each ch)l* was

g)4e( the free*o- to choose a te7t booklet fro- -a(y w)th 4ar)o,s *raw)(gs o( the

co4ers. &he perfor-a(ce of each ch)l* was rate* as 'ery !oo* !oo* a(* a)r. &he

res,lts are s,--ar)Ve* )( the table pro4)*e*. &est at the 0G le4el of s)g()+ca(ce

whether there )s s,?c)e(t e4)*e(ce )( the *ata to s,pport the psycholog)stWs bel)ef.

)roup

A B

*eror%an!e

=er (ood 32 29

(ood 55 61

air 10 13

1; ( regar* to w)(e tast)(g co-pet)t)o(s -a(y e7perts cla)- that the +rst glass of w)(e

ser4e* sets a refere(ce taste a(* that a *)8ere(t refere(ce w)(e -ay alter the relat)4e

ra(k)(g of the other w)(es )( co-pet)t)o(. &o test th)s cla)- three w)(es 0 - a(* ,

were ser4e* at a w)(e tast)(g e4e(t. Each perso( was ser4e* a s)(gle glass of each

w)(e b,t )( *)8ere(t or*ers for *)8ere(t g,ests. At the close each perso( was aske* to

(a-e the best of the three. "(e h,(*re* se4e(ty=two people were at the e4e(t a(*

the)r top p)cks are g)4e( )( the table pro4)*e*. &est at the 1G le4el of s)g()+ca(ce

whether there )s s,?c)e(t e4)*e(ce )( the *ata to s,pport the cla)- that w)(e e7pertsW

prefere(ce )s *epe(*e(t o( the +rst ser4e* w)(e.

Top ick

A B C

ir+' ("a++

A 12 31 27

B 15 40 21

C 10 9 7

11 s be)(g left=ha(*e* here*)taryC &o a(swer th)s <,est)o( 20; a*,lts are ra(*o-ly

selecte* a(* the)r ha(*e*(ess a(* the)r pare(tsW ha(*e*(ess are (ote*. &he res,lts are






s,--ar)Ve* )( the table pro4)*e*. &est at the 1G le4el of s)g()+ca(ce whether there )s

s,?c)e(t e4)*e(ce )( the *ata to co(cl,*e that there )s a here*)tary ele-e(t )(

ha(*e*(ess.

Number of arents 2eft-4anded

0 !

Handedne++

?e' 8 10 12

igh' 178 21 21

12 So-e ge(et)c)sts cla)- that the ge(es that *eter-)(e left=ha(*e*(ess also go4er(

*e4elop-e(t of the la(g,age ce(ters of the bra)(. f th)s cla)- )s tr,e the( )t wo,l* be

reaso(able to e7pect that left=ha(*e* people te(* to ha4e stro(ger la(g,age ab)l)t)es. A

st,*y *es)g(e* to te7t th)s cla)- ra(*o-ly selecte* 6; st,*e(ts who took the

!ra*,ate Recor* E7a-)(at)o( !RE5. &he)r scores o( the la(g,age port)o( of the

e7a-)(at)o( were class)+e* )(to three categor)es: lo& average a(* !ig! a(* the)r

ha(*e*(ess was also (ote*. &he res,lts are g)4e( )( the table pro4)*e*. &est at the 0G

le4el of s)g()+ca(ce whether there )s s,?c)e(t e4)*e(ce )( the *ata to co(cl,*e that

left=ha(*e* people te(* to ha4e stro(ger la(g,age ab)l)t)es.

)'$ $nglish Scores

2o3 Average 4igh

Handedne++

?e' 18 40 22

igh' 201 360 166

13 t )s ge(erally bel)e4e* that ch)l*re( bro,ght ,p )( stable fa-)l)es te(* to *o well )(

school. &o 4er)fy s,ch a bel)ef a soc)al sc)e(t)st e7a-)(e* 29; ra(*o-ly selecte*

st,*e(tsW recor*s )( a p,bl)c h)gh school a(* (ote* each st,*e(tWs fa-)ly str,ct,re a(*

aca*e-)c stat,s fo,r years after e(ter)(g h)gh school. &he *ata were the( sorte* )(to a

2 3 co(t)(ge(cy table w)th two factors. actor 1 has two le4els: graduated a(* did not

graduate. actor 2 has three le4els: no parent one parent a(* t&o parents. &he res,lts

are g)4e( )( the table pro4)*e*. &est at the 1G le4el of s)g()+ca(ce whether there )s

s,?c)e(t e4)*e(ce )( the *ata to co(cl,*e that fa-)ly str,ct,re -atters )( school

perfor-a(ce of the st,*e(ts.

Academic Status

)raduated %id Not )raduate

a%i" No $aren' 18 31

One $aren' 101 44






Academic Status

)raduated %id Not )raduate

,/o $aren'+ 70 26

1 A large -)**le school a*-)()strator w)shes to ,se celebr)ty )([,e(ce to e(co,rage

st,*e(ts to -ake health)er cho)ces )( the school cafeter)a. &he cafeter)a )s s)t,ate* at

the ce(ter of a( ope( space. E4ery*ay at l,(ch t)-e st,*e(ts get the)r l,(ch a(* a

*r)(k )( three separate l)(es lea*)(g to three separate ser4)(g stat)o(s. As a(

e7per)-e(t the school a*-)()strator *)splaye* a poster of a pop,lar tee( pop star

*r)(k)(g -)lk at each of the three areas where *r)(ks are pro4)*e* e7cept the -)lk )(

the poster )s *)8ere(t at each locat)o(: o(e shows wh)te -)lk o(e shows strawberry=

[a4ore* p)(k -)lk a(* o(e shows chocolate -)lk. After the +rst *ay of the e7per)-e(t

the a*-)()strator (ote* the st,*e(tsW -)lk cho)ces separately for the three l)(es. &he

*ata are g)4e( )( the table pro4)*e*. &est at the 1G le4el of s)g()+ca(ce whether there)s s,?c)e(t e4)*e(ce )( the *ata to co(cl,*e that the posters ha* so-e )-pact o( the

st,*e(tsW *r)(k cho)ces.

Student Choice

'egular Stra3berr1 Chocolate

*o+'er hoi!e

eg&"ar 38 28 40

'ra/berr 18 51 24

ho!o"a'e 32 32 53

LAR!E A&A SE& EKER%SE

10 Large ata Set 6 recor*s the res,lt of a s,r4ey of 3;; ra(*o-ly selecte* a*,lts who go

to -o4)e theaters reg,larly. or each perso( the ge(*er a(* preferre* type of -o4)e

were recor*e*. &est at the 0G le4el of s)g()+ca(ce whether there )s s,?c)e(t e4)*e(ce

)( the *ata to co(cl,*e that the factors ge(*er a(* preferre* type of -o4)e are

*epe(*e(t.

http://www.6.7ls











%%.0 Chi-S"uare 1ne-Sample :oodness-o-Fit Tests

LEARNN! "#$E%&'E

1 &o ,(*ersta(* how to ,se a ch)=s<,are test to >,*ge whether a sa-ple +ts a

part)c,lar pop,lat)o( well.

+uppose we wish to determine if an ordinary-looking six-sided die is fair$ or balanced$ meaning that

every face has probability 1I; of landing on top when the die is tossed. (e could toss the die do'ens$

maybe hundreds$ of times and compare the actual number of times each face landed on top to the

expected number$ which would be 1I; of the total number of tosses. (e wouldn&t expect each

number to be exactly 1I; of the total$ but it should be close. To be specific$ suppose the die is

tossed n M ;5 times with the results summari'ed in Table 11.? 02ie 7ontingency Table0. /or ease of

reference we add a column of expected fre"uencies$ which in this simple example is simply a column

of 15s. The result is shown as Table 11.F 0<pdated 2ie 7ontingency Table0. ,n analogy with the

previous section we call this an GupdatedH table. % measure of how much the data deviate from what

we would expect to see if the die really were fair is the sum of the s"uares of the differences between

the observed fre"uency ) and the expected fre"uency ; in each row$ or$ standardi'ing by dividing

each s"uare by the expected number$ the sumΣ(O−E)2

E.,f we formulate the investigation as a

test of hypotheses$ the test is

H0: The die is fair






vs.Ha: The die isnot fair

Table 11.? 2ie 7ontingency Table

%ie alue Assumed %istribution Observed Fre<uenc1

1 1G6 9

2 1G6 15

3 1G6 9

4 1G6 8

5 1G6 6

6 1G6 13

Table 11.F <pdated 2ie 7ontingency Table

%ie alue Assumed %istribution Observed Fre<8 $:pected Fre<8

1 1G6 9 10

2 1G6 15 10

3 1G6 9 10

4 1G6 8 10

5 1G6 6 10

6 1G6 13 10

(e would re#ect the null hypothesis that the die is fair only if the numberΣ(O−E)2/Eis large$ so the test

is right-tailed. ,n this example the random variable Σ(O−E)2/Ehas the chi-s"uare distribution with five

degrees of freedom. ,f we had decided at the outset to test at the 15B level of significance$ the critical

value defining the re#ection region would be$ reading from /igure 1!.6 07ritical Dalues of 7hi-+"uare

2istributions0$χ2α=χ20.10=9.236$ so that the re#ection region would be the interval[9.236,∞). (hen we

compute the value of the standardi'ed test statistic using the numbers in the last two columns of Table

11.F 0<pdated 2ie 7ontingency Table0$ we obtain











Table 11.15 eneral 7ontingency Table

Factor !evels Assumed (istribution 1bserved Fre"uency

1 p1 1

2 p2 2

# p# #

Table 11.15 0eneral 7ontingency Table0 is updated to Table 11.11 0<pdated eneral 7ontingency

Table0 by adding the expected fre"uency for each value of B . To simplify the notation we drop indices






for the observed and expected fre"uencies and represent Table 11.11 0<pdated eneral 7ontingency

Table0 by Table 11.1! 0+implified <pdated eneral 7ontingency Table0.

Table 11.11 <pdated eneral 7ontingency Table

Factor !evels Assumed (istribution 1bserved Fre". +xpected Fre".

1 p1 1 (1

2 p2 2 (2

# p# # (#

Table 11.1! +implified <pdated eneral 7ontingency Table

Factor !evels Assumed (istribution 1bserved Fre". +xpected Fre".

1 p1 (

2 p2 (

# p# (

Eere is the test statistic for the general hypothesis based on Table 11.1! 0+implified <pdated eneral

7ontingency Table0$ together with the conditions that it follow a chi-s"uare distribution.






EKAPLE 2

&able 11.13 QEth()c !ro,ps )( the %e(s,s JearQ shows the *)str)b,t)o( of 4ar)o,s

eth()c gro,ps )( the pop,lat)o( of a part)c,lar state base* o( a *ece(()al U.S.

ce(s,s. )4e years later a ra(*o- sa-ple of 20;; res)*e(ts of the state was

take( w)th the res,lts g)4e( )( &able 11.1 QSa-ple ata )4e Jears After the

%e(s,s JearQ alo(g w)th the probab)l)ty *)str)b,t)o( fro- the ce(s,s year5. &est

at the 1G le4el of s)g()+ca(ce whether there )s s,?c)e(t e4)*e(ce )( the sa-ple

to co(cl,*e that the *)str)b,t)o( of eth()c gro,ps )( th)s state +4e years after the

ce(s,s ha* cha(ge* fro- that )( the ce(s,s year.

&A#LE 11.13 E&@N% !R"UPS N &@E %ENSUS JEAR

+thnicity hite 'lack Amer.-&ndian =ispanic Asian 1thers

Proport)o( ;.3 ;.21 ;.;12 ;.;12 ;.;;6 ;.;;9






&A#LE 11.1 SAPLE A&A 'E JEARS A&ER &@E

%ENSUS JEAR

$thnicit1 Assumed %istribution Observed Fre<uenc1

hi'e 0.743 1732

:"a! 0.216 538

%eri!an-ndian 0.012 32

Hi+$ani! 0.012 42

+ian 0.008 133

O'her+ 0.009 23

Sol,t)o(:

Be test ,s)(g the cr)t)cal 4al,e approach.

• Step 1. &he hypotheses of )(terest )( th)s case ca( be e7presse* as

H0:The distribution of ethnic groups has not changed

vs.Ha: The distribution of ethnic groupshas changed

• Step 2. &he *)str)b,t)o( )s ch)=s<,are.

Step 3. &o co-p,te the 4al,e of the test stat)st)c we -,st +rst co-p,te the

e7pecte* (,-ber for each row of &able 11.1 QSa-ple ata )4e Jears After the

%e(s,s JearQ. S)(ce n 20;; ,s)(g the for-,la Ei=n× pia(* the 4al,es

of pi fro- e)ther &able 11.13 QEth()c !ro,ps )( the %e(s,s JearQ or &able 11.1

QSa-ple ata )4e Jears After the %e(s,s JearQ











*+, TA*+AA,

• &he chi-s"uare $oodness-o-)t test ca( be ,se* to e4al,ate the hypothes)s

that a sa-ple )s take( fro- a pop,lat)o( w)th a( ass,-e* spec)+c probab)l)ty

*)str)b,t)o(.

++/C&S+S

'AS&C

1 A *ata sa-ple )s sorte* )(to +4e categor)es w)th a( ass,-e* probab)l)ty *)str)b,t)o(.






Factor 2evels Assumed %istribution Observed Fre<uenc1

1 p1=0.1 10

2 p2=0.4 35

3 p3=0.4 45

4 p4=0.1 10

a )(* the s)Ve n of the sa-ple.

b )(* the e7pecte* (,-ber ( of obser4at)o(s for each le4el )f the sa-ple*

pop,lat)o( has a probab)l)ty *)str)b,t)o( as ass,-e* that )s >,st ,se the

for-,la Ei=n× pi5.

c )(* the ch)=s<,are test stat)st)c χ2.


2 A *ata sa-ple )s sorte* )(to +4e categor)es w)th a( ass,-e* probab)l)ty *)str)b,t)o(.

Factor 2evels Assumed %istribution Observed Fre<uenc1

1 p1=0.3 23

2 p2=0.3 30

3 p3=0.2 19

4 p4=0.1 8

5 p5=0.1 10

a )(* the s)Ve n of the sa-ple.

b )(* the e7pecte* (,-ber ( of obser4at)o(s for each le4el )f the sa-ple*

pop,lat)o( has a probab)l)ty *)str)b,t)o( as ass,-e* that )s >,st ,se the

for-,la Ei=n× pi5.

c )(* the ch)=s<,are test stat)st)c χ2.


A22!&CAT&1NS

3 Reta)lers of collect)ble postage sta-ps ofte( b,y the)r sta-ps )( large <,a(t)t)es by

we)ght at a,ct)o(s. &he pr)ces the reta)lers are w)ll)(g to pay *epe(* o( how ol* the

postage sta-ps are. a(y collect)ble postage sta-ps at a,ct)o(s are *escr)be* by the

proport)o(s of sta-ps )ss,e* at 4ar)o,s per)o*s )( the past. !e(erally the ol*er the

sta-ps the h)gher the 4al,e. At o(e part)c,lar a,ct)o( a lot of collect)ble sta-ps )s

a*4ert)se* to ha4e the age *)str)b,t)o( g)4e( )( the table pro4)*e*. A reta)l b,yer took a

sa-ple of 3 sta-ps fro- the lot a(* sorte* the- by age. &he res,lts are g)4e( )( the






table pro4)*e*. &est at the 0G le4el of s)g()+ca(ce whether there )s s,?c)e(t e4)*e(ce

)( the *ata to co(cl,*e that the age *)str)b,t)o( of the lot )s *)8ere(t fro- what was

cla)-e* by the seller.

(ear Claimed %istribution Observed Fre<uenc1

:eore 1940 0.10 6

1940 'o 1959 0.25 15

1960 'o 1979 0.45 30

'er 1979 0.20 22

&he l)tter s)Ve of #e(gal t)gers )s typ)cally two or three c,bs b,t )t ca( 4ary betwee( o(e

a(* fo,r. #ase* o( lo(g=ter- obser4at)o(s the l)tter s)Ve of #e(gal t)gers )( the w)l* has

the *)str)b,t)o( g)4e( )( the table pro4)*e*. A Voolog)st bel)e4es that #e(gal t)gers )(

capt)4)ty te(* to ha4e *)8ere(t poss)bly s-aller5 l)tter s)Ves fro- those )( the w)l*. &o

4er)fy th)s bel)ef the Voolog)st searche* all *ata so,rces a(* fo,(* 31 l)tter s)Ve

recor*s of #e(gal t)gers )( capt)4)ty. &he res,lts are g)4e( )( the table pro4)*e*. &est at

the 0G le4el of s)g()+ca(ce whether there )s s,?c)e(t e4)*e(ce )( the *ata to co(cl,*e

that the *)str)b,t)o( of l)tter s)Ves )( capt)4)ty *)8ers fro- that )( the w)l*.

2itter Si=e 9ild 2itter %istribution Observed Fre<uenc1

1 0.11 41

2 0.69 243

3 0.18 274 0.02 5

0 A( o(l)(e shoe reta)ler sells -e(Ws shoes )( s)Ves 6 to 13. ( the past or*ers for the

*)8ere(t shoe s)Ves ha4e followe* the *)str)b,t)o( g)4e( )( the table pro4)*e*. &he

-a(age-e(t bel)e4es that rece(t -arket)(g e8orts -ay ha4e e7pa(*e* the)r

c,sto-er base a(* as a res,lt there -ay be a sh)ft )( the s)Ve *)str)b,t)o( for f,t,re

or*ers. &o ha4e a better ,(*ersta(*)(g of )ts f,t,re sales the shoe seller e7a-)(e*

1;; sales recor*s of rece(t or*ers a(* (ote* the s)Ves of the shoes or*ere*. &he

res,lts are g)4e( )( the table pro4)*e*. &est at the 1G le4el of s)g()+ca(ce whether

there )s s,?c)e(t e4)*e(ce )( the *ata to co(cl,*e that the shoe s)Ve *)str)b,t)o( of

f,t,re sales w)ll *)8er fro- the h)stor)c o(e.

Shoe Si=e ast Si=e %istribution 'ecent Si=e Fre<uenc1

8.0 0.03 25

8.5 0.06 43







9.0 0.09 88

9.5 0.19 221

10.0 0.23 27210.5 0.14 150

11.0 0.10 107

11.5 0.06 51

12.0 0.05 37

12.5 0.03 35

13.0 0.02 11

A( o(l)(e shoe reta)ler sells wo-e(Ws shoes )( s)Ves 0 to 1;. ( the past or*ers for the

*)8ere(t shoe s)Ves ha4e followe* the *)str)b,t)o( g)4e( )( the table pro4)*e*. &he

-a(age-e(t bel)e4es that rece(t -arket)(g e8orts -ay ha4e e7pa(*e* the)r c,sto-er

base a(* as a res,lt there -ay be a sh)ft )( the s)Ve *)str)b,t)o( for f,t,re or*ers. &o

ha4e a better ,(*ersta(*)(g of )ts f,t,re sales the shoe seller e7a-)(e* 11 sales

recor*s of rece(t or*ers a(* (ote* the s)Ves of the shoes or*ere*. &he res,lts are g)4e(

)( the table pro4)*e*. &est at the 1G le4el of s)g()+ca(ce whether there )s s,?c)e(t

e4)*e(ce )( the *ata to co(cl,*e that the shoe s)Ve *)str)b,t)o( of f,t,re sales w)ll *)8er

fro- the h)stor)c o(e.


5.0 0.02 20

5.5 0.03 23

6.0 0.07 88

6.5 0.08 90

7.0 0.20 222

7.5 0.20 258

8.0 0.15 177

8.5 0.11 121

9.0 0.08 91

9.5 0.04 53

10.0 0.02 31






A chess ope()(g )s a se<,e(ce of -o4es at the beg)(()(g of a chess ga-e. &here are

-a(y well=st,*)e* (a-e* ope()(gs )( chess l)terat,re. re(ch efe(se )s o(e of the

-ost pop,lar ope()(gs for black altho,gh )t )s co(s)*ere* a relat)4ely weak ope()(g

s)(ce )t g)4es black probab)l)ty ;.3 of w)(()(g probab)l)ty ;.;0 of los)(g a(*

probab)l)ty ;.201 of *raw)(g. A chess -aster bel)e4es that he has *)sco4ere* a (ew4ar)at)o( of re(ch efe(se that -ay alter the probab)l)ty *)str)b,t)o( of the o,tco-e of

the ga-e. ( h)s -a(y (ter(et chess ga-es )( the last two years he was able to apply

the (ew 4ar)at)o( )( ga-es. &he w)(s losses a(* *raws )( the ga-es are g)4e( )(

the table pro4)*e*. &est at the 0G le4el of s)g()+ca(ce whether there )s s,?c)e(t

e4)*e(ce )( the *ata to co(cl,*e that the (ewly *)sco4ere* 4ar)at)o( of re(ch efe(se

alters the probab)l)ty *)str)b,t)o( of the res,lt of the ga-e.

'esult for 6lack robabilit1 %istribution Ne3 ariation 9ins

in 0.344 31

?o++ 0.405 25

#ra/ 0.251 21

6 &he epart-e(t of Parks a(* B)l*l)fe stocks a large lake w)th +sh e4ery s)7 years. t )s

*eter-)(e* that a healthy *)4ers)ty of +sh )( the lake sho,l* co(s)st of 1;G large-o,th

bass 10G s-all-o,th bass 1;G str)pe* bass 1;G tro,t a(* 2;G cat+sh. &herefore

each t)-e the lake )s stocke* the +sh pop,lat)o( )( the lake )s restore* to -a)(ta)( that

part)c,lar *)str)b,t)o(. E4ery three years the *epart-e(t co(*,cts a st,*y to see

whether the *)str)b,t)o( of the +sh )( the lake has sh)fte* away fro- the target

proport)o(s. ( o(e part)c,lar year a research gro,p fro- the *epart-e(t obser4e* a

sa-ple of 292 +sh fro- the lake w)th the res,lts g)4e( )( the table pro4)*e*. &est at the

0G le4el of s)g()+ca(ce whether there )s s,?c)e(t e4)*e(ce )( the *ata to co(cl,*e

that the +sh pop,lat)o( *)str)b,t)o( has sh)fte* s)(ce the last stock)(g.

Fish Tar$et (istribution Fish in Sample

Large-o,th #ass ;.1; 1

S-all-o,th #ass ;.10 9

Str)pe* #ass ;.1; 21

&ro,t ;.1; 22

%at+sh ;.2; 0

"ther ;.30 111






!A/:+ (ATA S+T ++/ C&S+

9 Large ata Set recor*s the res,lt of 0;; tosses of s)7=s)*e* *)e. &est at the 1;G le4el

of s)g()+ca(ce whether there )s s,?c)e(t e4)*e(ce )( the *ata to co(cl,*e that the *)e

)s (ot fa)r or bala(ce*5 that )s that the probab)l)ty *)str)b,t)o( *)8ers fro-

probab)l)ty 1/ for each of the s)7 faces o( the *)e.

http://www..7ls

%%.3 F-tests or +"uality o Two <ariances

LEARNN! "#$E%&'ES

1 &o ,(*ersta(* what F =*)str)b,t)o(s are.

2 &o ,(*ersta(* how to ,se a( F =test to >,*ge whether two pop,lat)o( 4ar)a(ces

are e<,al.






F-(istributions

%nother important and useful family of distributions in statistics is the family of ! -distributions. ach

member of the ! -distribution family is specified by a pair of parameters called degrees of freedom and

denoteddf1anddf2./igure 11.@ 0any 0 shows several ! -distributions for different pairs of degrees of

freedom. %n 7 random variable is a random variable that assumes only positive values and follows

an ! -distribution.

!igure "".1 2any ! :(istributions

The parameterdf1is often referred to as the numerator degrees of freedom and the parameterdf2as

the denominator degrees of freedom. ,t is important to keep in mind that they are not interchangeable.

/or example$ the ! -distribution with degrees of freedomdf1=3anddf2=8is a different distribution from

the ! -distribution with degrees of freedomdf1=8anddf2=3.






e+()t)o(

The value of the ! random variable ! with degrees of freedom df1 and df!that cuts off a right tail of

area c is denoted ! c and is called a critical value. %ee !igure "".3.

!igure "".3 ! c >llustrated

Tables containing the values of ! c are given in 7hapter 11 07hi-+"uare Tests and 0. ach of the tables

is for a fixed collection of values of c$ either 5.F55$ 5.F85$ 5.F@8$ 5.FF5$ and 5.FF8 *yielding what are

called GlowerH critical values)$ or 5.558$ 5.515$ 5.5!8$ 5.585$ and 5.155 *yielding what are called

GupperH critical values). ,n each table critical values are given for various pairs *df1$df!). (e illustrate

the use of the tables with several examples.

+A>2!+ 3

S,ppose F )s a( F ra(*o- 4ar)able w)th *egrees of free*o- *f10 a(**f2. Use

the tables to +(*

a. F ;.1;






b. F ;.90

Sol,t)o(:

a &he col,-( hea*)(gs of all the tables co(ta)( *f10. Look for the table for

wh)ch ;.1; )s o(e of the e(tr)es o( the e7tre-e left a table of ,pper cr)t)cal 4al,es5

a(* that has a row hea*)(g *f2 )( the left -arg)( of the table. A port)o( of the

rele4a(t table )s pro4)*e*. &he e(try )( the )(tersect)o( of the col,-( w)th

hea*)(g *f10 a(* the row w)th the hea*)(gs ;.1; a(* *f2 wh)ch )s sha*e* )( the

table pro4)*e* )s the a(swer ;.1;.;0.

F Tail Area

df

! > > > + > > > df!

0.005 4 I I I I I I I I I 22.5 I I I

0.01 4 I I I I I I I I I 15.5 I I I

0.025 4 I I I I I I I I I 9.36 I I I

0.05 4 I I I I I I I I I 6.26 I I I

0.10 4 I I I I I I I I I 4.05 I I I

b Look for the table for wh)ch ;.90 )s o(e of the e(tr)es o( the e7tre-e

left a table of lower cr)t)cal 4al,es5 a(* that has a row hea*)(g *f2 )( theleft -arg)( of the table. A port)o( of the rele4a(t table )s pro4)*e*. &he e(try

)( the )(tersect)o( of the col,-( w)th hea*)(g *f10 a(* the row w)th the

hea*)(gs ;.90 a(* *f2 wh)ch )s sha*e* )( the table pro4)*e* )s the

a(swer ;.90;.19.

F Tail Area

d%

% 0 8 d0

;.9; m m m m m m m m m ;.26 m m m

;.90 m m m m m m m m m ;.19 m m m

;.90 m m m m m m m m m ;.1 m m m






F Tail Area

d%

% 0 8 d0

;.99 m m m m m m m m m ;.;9 m m m

;.990 m m m m m m m m m ;.; m m m

+A>2!+ 7

S,ppose )s a( F ra(*o- 4ar)able w)th *egrees of

free*o- *f12 a(**f22;. Let n;.;0. Use the tables to +(*

a. n

b. n2

c. 1Xn

*. 1Xn2

Sol,t)o(:

a. &he col,-( hea*)(gs of all the tables co(ta)( *f12. Look for the table for

wh)ch n;.;0 )s o(e of the e(tr)es o( the e7tre-e left a table of ,pper cr)t)cal

4al,es5 a(* that has a row hea*)(g *f22; )( the left -arg)( of the table. A port)o(of the rele4a(t table )s pro4)*e*. &he sha*e* e(try )( the )(tersect)o( of the col,-(

w)th hea*)(g *f12 a(* the row w)th the hea*)(gs ;.;0 a(* *f22; )s the

a(swer ;.;03.9.

F Tail Area

df

! > > >

df!

0.005 20 I I I 6.99 I I I

0.01 20 I I I 5.85 I I I

0.025 20 I I I 4.46 I I I

0.05 20 I I I 3.49 I I I

0.10 20 I I I 2.59 I I I

b. Look for the table for wh)ch n2;.;20 )s o(e of the e(tr)es o( the e7tre-e

left a table of ,pper cr)t)cal 4al,es5 a(* that has a row hea*)(g *f22; )( the left






-arg)( of the table. A port)o( of the rele4a(t table )s pro4)*e*. &he sha*e* e(try )(

the )(tersect)o( of the col,-( w)th hea*)(g *f12 a(* the row w)th the hea*)(gs

;.;20 a(* *f22; )s the a(swer ;.;20..

F Tail Area

df

! > > >

df!

0.005 20 I I I 6.99 I I I

0.01 20 I I I 5.85 I I I

0.025 20 I I I 4.46 I I I

0.05 20 I I I 3.49 I I I

0.10 20 I I I 2.59 I I I

3 Look for the table for wh)ch 1Xn;.90 )s o(e of the e(tr)es o( the

e7tre-e left a table of lower cr)t)cal 4al,es5 a(* that has a row

hea*)(g *f22; )( the left -arg)( of the table. A port)o( of the

rele4a(t table )s pro4)*e*. &he sha*e* e(try )( the )(tersect)o( of the

col,-( w)th hea*)(g *f12 a(* the row w)th the hea*)(gs ;.90

a(* *f22; )s the a(swer ;.90;.;0.

F Tail Area

df

! > > > df!

0.90 20 I I I 0.11 I I I

0.95 20 I I I 0.05 I I I

0.975 20 I I I 0.03 I I I

0.99 20 I I I 0.01 I I I

0.995 20 I I I 0.01 I I I

* Look for the table for wh)ch 1Xn2;.90 )s o(e of the e(tr)es o( the

e7tre-e left a table of lower cr)t)cal 4al,es5 a(* that has a row

hea*)(g *f22; )( the left -arg)( of the table. A port)o( of the rele4a(t table

)s pro4)*e*. &he sha*e* e(try )( the )(tersect)o( of the col,-( w)th






hea*)(g *f12 a(* the row w)th the hea*)(gs ;.90 a(* *f22; )s the

a(swer;.90;.;3.

F Tail Area

df

! > > >

df!

0.90 20 I I I 0.11 I I I

0.95 20 I I I 0.05 I I I

0.975 20 I I I 0.03 I I I

0.99 20 I I I 0.01 I I I

0.995 20 I I I 0.01 I I I

% fact that sometimes allows us to find a critical value from a table that we could not read otherwise

is:

,f /u*r$s) denotes the value of the ! -distribution with degrees of freedom df1Mr and df!Ms that cuts off a

right tail of area u$ then

ck511Xck5

+A>2!+ 8

Use the tables to +(*

a. F ;.;1 for a( F ra(*o- 4ar)able w)th *f113 a(* *f26

b. F ;.90 for a( F ra(*o- 4ar)able w)th *f1; a(* *f21;

Sol,t)o(:

a. &here )s (o table w)th *f113 b,t there )s o(e w)th *f16.&h,s we ,se the

fact that

;.;113651;.996135






Us)(g the rele4a(t table we +(* that ;.996135;.16

he(ce;.;11365;.16X10.00.

b. &here )s (o table w)th *f1; b,t there )s o(e w)th *f11;.&h,s we ,se the

fact that

;.90;1;51;.;201;;5

Us)(g the rele4a(t table we +(* that ;.;201;;53.31

he(ce;.90;1;53.31X1;.3;2.

F-Tests or +"uality o Two <ariances

,n 7hapter F 0Two-+ample roblems0 we saw how to test hypotheses about the difference between

two population means 1 and !. ,n some practical situations the difference between the population

standard deviations 1 and! is also of interest. +tandard deviation measures the variability of a

random variable. /or example$ if the random variable measures the si'e of a machined part in a

manufacturing process$ the si'e of standard deviation is one indicator of product "uality. % smaller

standard deviation among items produced in the manufacturing process is desirable since it

indicates consistency in product "uality.

/or theoretical reasons it is easier to compare the s"uares of the population standard deviations$ the

population variances 1! and !!. This is not a problem$ since 1M! precisely

when 1!M!!$ 1V! precisely when1!V!!$ and 1X! precisely when 1!X!!.

The null hypothesis always has the form E5:1!M!!. The three forms of the alternative hypothesis$

with the terminology for each case$ are:


@a:1222 R)ght=ta)le*

@a:12_22 Left=ta)le*







@a:12q22 &wo=ta)le*

Rust as when we test hypotheses concerning two population means$ we take a random sample from

each population$ of si'es n1 and n!$ and compute the sample standard deviations s1 and s!. ,n this

context the samples are always independent. The populations themselves must be normally

distributed.

&est Stat)st)c for @ypothes)s &ests %o(cer()(g the)8ere(ce#etwee( &wo Pop,lat)o( 'ar)a(ces

s12s22

,f the two populations are normally distributed and if E5:1!M!! is true then under independent

sampling ! approximately follows an ! -distribution with degrees of freedom df1Mn1Z1 and df!Mn!Z1.

% test based on the test statistic / is called an ! -test.

% most important point is that while the re#ection region for a right-tailed test is exactly as in every

other situation that we have encountered$ because of the asymmetry in the ! -distribution the critical

value for a left-tailed test and the lower critical value for a two-tailed test have the special forms

shown in the following table:

Terminolo$y Alternative =ypothesis /eGection /e$ion

R)ght=ta)le* @a:1222 n






Terminolo$y Alternative =ypothesis /eGection /e$ion

Left=ta)le* @a:12_22 ^1Xn

&wo=ta)le* @a:12q22 ^1Xn2 or n2

/igure 11.F 0Ce#ection Cegions: *a) Cight-Tailed *b) eft-Tailed *c) Two-Tailed0 illustrates these

re#ection regions.

!igure "".4 -ejection -egions8 DaE -ight:TailedN DbE 'eft:TailedN DcE Two:Tailed

The test is performed using the usual five-step procedure described at the end of +ection ?.1 0The

lements of Eypothesis Testing0 in 7hapter ? 0Testing Eypotheses0.

EKAPLE






"(e of the <,al)ty -eas,res of bloo* gl,cose -eter str)ps )s the co(s)ste(cy of

the test res,lts o( the sa-e sa-ple of bloo*. &he co(s)ste(cy )s -eas,re* by the

4ar)a(ce of the rea*)(gs )( repeate* test)(g. S,ppose two types of str)ps 0 a(* -

are co-pare* for the)r respect)4e co(s)ste(c)es. Be arb)trar)ly label the

pop,lat)o( of &ype 0str)ps Pop,lat)o( 1 a(* the pop,lat)o( of &ype - str)psPop,lat)o( 2. S,ppose 10 &ype 0 str)ps were teste* w)th bloo* *rops fro- a well=

shake( 4)al a(* 2; &ype - str)ps were teste* w)th the bloo* fro- the sa-e 4)al.

&he res,lts are s,--ar)Ve* )( &able 11.1 Q&wo &ypes of &est Str)psQ. Ass,-e the

gl,cose rea*)(gs ,s)(g &ype 0 str)ps follow a (or-al *)str)b,t)o( w)th

4ar)a(ce σ21a(* those ,s)(g &ype - str)ps follow a (or-al *)str)b,t)o( w)th

4ar)a(ce w)th σ22. &est at the 1;G le4el of s)g()+ca(ce whether the *ata pro4)*e

s,?c)e(t e4)*e(ce to co(cl,*e that the co(s)ste(c)es of the two types of str)ps

are *)8ere(t.
















IEJ &AIEABAJS

• %r)t)cal 4al,es of a( F =*)str)b,t)o( w)th *egrees of free*o- df1a(* df2are fo,(* )(

tables )( %hapter 12 QAppe(*)7Q.

• A( F =test ca( be ,se* to e4al,ate the hypothes)s of two )*e(t)cal (or-al

pop,lat)o( 4ar)a(ces.





















A22!&CAT&1NS10 $apa(ese st,rgeo( )s a s,bspec)es of the st,rgeo( fa-)ly )(*)ge(o,s to $apa( a(* the

Northwest Pac)+c. ( a part)c,lar +sh hatchery (ewly hatche* baby $apa(ese st,rgeo(

are kept )( ta(ks for se4eral weeks before be)(g tra(sferre* to larger po(*s. )ssol4e*

o7yge( )( ta(k water )s 4ery t)ghtly -o()tore* by a( electro()c syste- a(* r)goro,sly

-a)(ta)(e* at a target le4el of .0 -)ll)gra-s per l)ter -g/l5. &he +sh hatchery looks to






,pgra*e the)r water -o()tor)(g syste-s for t)ghter co(trol of *)ssol4e* o7yge(. A (ew

syste- )s e4al,ate* aga)(st the ol* o(e c,rre(tly be)(g ,se* )( ter-s of the 4ar)a(ce )(

-eas,re* *)ssol4e* o7yge(. &h)rty=o(e water sa-ples fro- a ta(k operate* w)th the

(ew syste- were collecte* a(* 1 water sa-ples fro- a ta(k operate* w)th the ol*

syste- were collecte* all *,r)(g the co,rse of a *ay. &he sa-ples y)el* the follow)(g)(for-at)o(:

New Sample 1 :n1=31s21=0.0121

Old Sample 2:n2=16s22=0.0319

&est at the 1;G le4el of s)g()+ca(ce whether the *ata pro4)*e s,?c)e(t e4)*e(ce to

co(cl,*e that the (ew syste- w)ll pro4)*e a t)ghter co(trol of *)ssol4e* o7yge( )( the

ta(ks.

1 &he r)sk of )(4est)(g )( a stock )s -eas,re* by the 4olat)l)ty or the 4ar)a(ce )( cha(ges

)( the pr)ce of that stock. ,t,al f,(*s are baskets of stocks a(* o8er ge(erally lower

r)sk to )(4estors. )8ere(t -,t,al f,(*s ha4e *)8ere(t foc,ses a(* o8er *)8ere(t le4els

of r)sk. @)ppolyta )s *ec)*)(g betwee( two -,t,al f,(*s 0 a(* - w)th s)-)lar e7pecte*

ret,r(s. &o -ake a +(al *ec)s)o( she e7a-)(e* the a((,al ret,r(s of the two f,(*s

*,r)(g the last te( years a(* obta)(e* the follow)(g )(for-at)o(:






&est at the 1;G le4el of s)g()+ca(ce whether the *ata pro4)*e s,?c)e(t e4)*e(ce to

co(cl,*e that the (ew playl)st has e7pa(*e* the ra(ge of l)ste(er ages.

19 A laptop co-p,ter -aker ,ses battery packs s,ppl)e* by two co-pa()es 0a(* -.

Bh)le both bra(*s ha4e the sa-e a4erage battery l)fe betwee( charges L#%5 the

co-p,ter -aker see-s to rece)4e -ore co-pla)(ts abo,t shorter L#% tha( e7pecte*

for battery packs s,ppl)e* by co-pa(y -. &he co-p,ter -aker s,spects that th)s

co,l* be ca,se* by h)gher 4ar)a(ce )( L#% for #ra(* -. &o check that te( (ew

battery packs fro- each bra(* are selecte* )(stalle* o( the sa-e -o*els of laptops

a(* the laptops are allowe* to r,( ,(t)l the battery packs are co-pletely *)scharge*.

&he follow)(g are the obser4e* L#%s )( ho,rs.






!A/:+ (ATA S+T ++/C &S+S

21 Large ata Sets 1A a(* 1# recor* SA& scores for 19 -ale a(* 061 fe-ale st,*e(ts.

&est at the 1G le4el of s)g()+ca(ce whether the *ata pro4)*e s,?c)e(t e4)*e(ce to

co(cl,*e that the 4ar)a(ces of scores of -ale a(* fe-ale st,*e(ts *)8er.

http://www.1A.7ls

http://www.1#.7ls

22 Large ata Sets A a(* # recor* the s,r4)4al t)-es of 1; laboratory -)ce w)th

thy-)c le,ke-)a. &est at the 1;G le4el of s)g()+ca(ce whether the *ata pro4)*e






s,?c)e(t e4)*e(ce to co(cl,*e that the 4ar)a(ces of s,r4)4al t)-es of -ale -)ce a(*

fe-ale -)ce *)8er.

http://www..7ls

http://www.A.7lshttp://www.#.7ls






%%.7 F-Tests in 1ne-ay AN1<A

LEARNN! "#$E%&'E

1 &o ,(*ersta(* how to ,se a( F =test to >,*ge whether se4eral pop,lat)o( -ea(s

are all e<,al.

,n 7hapter F 0Two-+ample roblems0 we saw how to compare two population meansµ1andµ2.,n

this section we will learn to compare three or more population means at the same time$ which is

often of interest in practical applications. /or example$ an administrator at a university may be

interested in knowing whether student grade point averages are the same for different ma#ors. ,n

another example$ an oncologist may be interested in knowing whether patients with the same type of

cancer have the same average survival times under several different competing cancer treatments.

,n general$ suppose there are J normal populations with possibly different means$µ1,µ2,…,µK$ but

all with the same varianceσ2.The study "uestion is whether all the J population means are the same. (e formulate this "uestion as the test of hypotheses

H0: µ1=µ2=Y Y Y =µK

vs.Ha: not allK population means are equal






To perform the test J independent random samples are taken from the J normal populations.

The J sample means$ the J sample variances$ and the J sample si'es are summari'ed in the table:

opulation Sample Si=e Sample #ean Sample ariance

1 n1 x

−1

s21

2 n2 x−2 s22

K nK x−K s2K

2efine the following "uantities:






EKAPLE 6

&he a4erage of gra*e po)(t a4erages !PAs5 of college co,rses )( a spec)+c -a>or

)s a -eas,re of *)?c,lty of the -a>or. A( e*,cator w)shes to co(*,ct a st,*y to

+(* o,t whether the *)?c,lty le4els of *)8ere(t -a>ors are the sa-e. or s,ch a

st,*y a ra(*o- sa-ple of -a>or gra*e po)(t a4erages !PA5 of 11 gra*,at)(g

se()ors at a large ,()4ers)ty )s selecte* for each of the fo,r -a>ors -athe-at)cs

E(gl)sh e*,cat)o( a(* b)ology. &he *ata are g)4e( )( &able 11.1 Q)?c,lty

Le4els of %ollege a>orsQ. &est at the 0G le4el of s)g()+ca(ce whether the *ata

co(ta)( s,?c)e(t e4)*e(ce to co(cl,*e that there are *)8ere(ces a-o(g the

a4erage -a>or !PAs of these fo,r -a>ors.

&A#LE 11.1 %UL&J LE'ELS " %"LLE!E A$"RS

>athematics +n$lish +ducation 'iolo$y

2.09 3. .;; 2.6

3.13 3.19 3.09 3.01

2.9 3.10 2.6; 2.0

2.0; 3.6 2.39 3.1

2.03 3.;3 3. 2.9






>athematics +n$lish +ducation 'iolo$y

3.29 2.1 3.09 2.32

2.03 3.2; 3. 2.06

3.1 3.3; 3. 3.21

2.; 3.0 3.13 3.23

3.66 3.20 3.;; 3.0

2. .;; 3. 3.22











EKAPLE 9

A research laboratory *e4elope* two treat-e(ts wh)ch are bel)e4e* to ha4e the

pote(t)al of prolo(g)(g the s,r4)4al t)-es of pat)e(ts w)th a( ac,te for- of thy-)cle,ke-)a. &o e4al,ate the pote(t)al treat-e(t e8ects 33 laboratory -)ce w)th

thy-)c le,ke-)a were ra(*o-ly *)4)*e* )(to three gro,ps. "(e gro,p rece)4e*

&reat-e(t 1 o(e rece)4e* &reat-e(t 2 a(* the th)r* was obser4e* as a co(trol

gro,p. &he s,r4)4al t)-es of these -)ce are g)4e( )( &able 11.16 Q)ce S,r4)4al

&)-es )( aysQ. &est at the 1G le4el of s)g()+ca(ce whether these *ata pro4)*e






s,?c)e(t e4)*e(ce to co(+r- the bel)ef that at least o(e of the two treat-e(ts

a8ects the a4erage s,r4)4al t)-e of -)ce w)th thy-)c le,ke-)a.

&A#LE 11.16 %E SUR' 'AL &ES N AJS

Treatment % Treatment 0 Control

1 0 61

2 3 9

0 2 9 3

6; 0 6 1

; 3 61 0

0 9 2 6

3 1

6 1 6

91











*+, TA*+AA,

• A( F =test ca( be ,se* to e4al,ate the hypothes)s that the -ea(s of se4eral

(or-al pop,lat)o(s all w)th the sa-e sta(*ar* *e4)at)o( are )*e(t)cal.

++/C&S+S

'AS&C

1 &he follow)(g three ra(*o- sa-ples are take( fro- three (or-al pop,lat)o(s w)th

respect)4e -ea(s µ1 µ2 a(* µ3 a(* the sa-e 4ar)a(ce σ2.

Sample Sample ! Sample "

2 3 0

2 5 1

3 7 2







5 1

3

a )(* the co-b)(e* sa-ple s)Ve n.

b )(* the co-b)(e* sa-ple -ea( x−.

c )(* the sa-ple -ea( for each of the three sa-ples.

* )(* the sa-ple 4ar)a(ce for each of the three sa-ples.

e )(* MST.

f )(* MSE.

$ )(* F=MST/MSE.

2 &he follow)(g three ra(*o- sa-ples are take( fro- three (or-al pop,lat)o(s w)th

respect)4e -ea(s µ1 µ2 a(* µ3 a(* a sa-e 4ar)a(ce σ2.


0.0 1.3 0.2

0.1 1.5 0.2

0.2 1.7 0.3

0.1 0.5

0.0a )(* the co-b)(e* sa-ple s)Ve n.

b )(* the co-b)(e* sa-ple -ea(x−.

c )(* the sa-ple -ea( for each of the three sa-ples.

* )(* the sa-ple 4ar)a(ce for each of the three sa-ples.

e )(* MST.

f )(* MSE.

g )(* F=MST/MSE.

3 Refer to E7erc)se 1.

a )(* the (,-ber of pop,lat)o(s ,(*er co(s)*erat)o( @ .

b )(* the *egrees of free*o- df1=K−1a(* df2=n−K.

c or α=0.05 +(* Fα w)th the *egrees of free*o- co-p,te* abo4e.

* At α=0.05 test hypotheses






A22!&CAT&1NS

0 &he oVart e8ect refers to a boost of a4erage perfor-a(ce o( tests for ele-e(tary

school st,*e(ts )f the st,*e(ts l)ste( to oVartWs cha-ber -,s)c for a per)o* of t)-e

)--e*)ately before the test. ( or*er to atte-pt to test whether the oVart e8ect

act,ally e7)sts a( ele-e(tary school teacher co(*,cte* a( e7per)-e(t by *)4)*)(g her

th)r*=gra*e class of 10 st,*e(ts )(to three gro,ps of 0. &he +rst gro,p was g)4e( a(

e(*=of=gra*e test w)tho,t -,s)cD the seco(* gro,p l)ste(e* to oVartWs cha-ber -,s)c

for 1; -)(,tesD a(* the th)r* gro,ps l)ste(e* to oVartWs cha-ber -,s)c for 2; -)(,tes

before the test. &he scores of the 10 st,*e(ts are g)4e( below:

)roup )roup ! )roup "

80 79 73

63 73 82

74 74 79

71 77 82

70 81 84Us)(g the AN"'A =test at α=0.10 )s there s,?c)e(t e4)*e(ce )( the *ata to s,ggest that

the oVart e8ect e7)stsC

&he oVart e8ect refers to a boost of a4erage perfor-a(ce o( tests for ele-e(tary

school st,*e(ts )f the st,*e(ts l)ste( to oVartWs cha-ber -,s)c for a per)o* of t)-e

)--e*)ately before the test. a(y e*,cators bel)e4e that s,ch a( e8ect )s (ot






(ecessar)ly *,e to oVartWs -,s)c per se b,t rather a rela7at)o( per)o* before the test.

&o s,pport th)s bel)ef a( ele-e(tary school teacher co(*,cte* a( e7per)-e(t by

*)4)*)(g her th)r*=gra*e class of 10 st,*e(ts )(to three gro,ps of 0. St,*e(ts )( the +rst

gro,p were aske* to g)4e the-sel4es a self=a*-)()stere* fac)al -assageD st,*e(ts )(

the seco(* gro,p l)ste(e* to oVartWs cha-ber -,s)c for 10 -)(,tesD st,*e(ts )( theth)r* gro,p l)ste(e* to Sch,bertWs cha-ber -,s)c for 10 -)(,tes before the test. &he

scores of the 10 st,*e(ts are g)4e( below:

)roup )roup ! )roup "

79 82 80

81 84 81

80 86 71

89 91 90

86 82 86

&est ,s)(g the AN"'A F =test at the 1;G le4el of s)g()+ca(ce whether the *ata pro4)*e

s,?c)e(t e4)*e(ce to co(cl,*e that a(y of the three rela7at)o( -etho* *oes better tha(

the others.

Prec)s)o( we)gh)(g *e4)ces are se(s)t)4e to e(4)ro(-e(tal co(*)t)o(s. &e-perat,re a(*

h,-)*)ty )( a laboratory roo- where s,ch a *e4)ce )s )(stalle* are t)ghtly co(trolle* to

e(s,re h)gh prec)s)o( )( we)gh)(g. A (ewly *es)g(e* we)gh)(g *e4)ce )s cla)-e* to be

-ore rob,st aga)(st s-all 4ar)at)o(s of te-perat,re a(* h,-)*)ty. &o 4er)fy s,ch a

cla)- a laboratory tests the (ew *e4)ce ,(*er fo,r sett)(gs of te-perat,re=h,-)*)tyco(*)t)o(s. )rst two le4els of !ig! a(* lo& te-perat,re a(* two le4els

of !ig! a(* lo& h,-)*)ty are )*e(t)+e*. Let ) sta(* for te-perat,re a(* for h,-)*)ty.

&he fo,r e7per)-e(tal sett)(gs are *e+(e* a(* (ote* as ) 5: h)gh h)gh5 h)gh low5

low h)gh5 a(* low low5. A pre=cal)brate* sta(*ar* we)ght of 1 kg was we)ghe* by the

(ew *e4)ce fo,r t)-es )( each sett)(g. &he res,lts )( ter-s of error )( -)crogra-s -cg5

are g)4e( below:

?high@ high ?high@ lo3 ?lo3@ high ?lo3@ lo3

C1.50 11.47 C14.29 5.54C6.73 9.28 C18.11 10.34

11.69 5.58 C11.16 15.23

C5.72 10.80 C10.41 C5.69






&est ,s)(g the AN"'A F =test at the 1G le4el of s)g()+ca(ce whether the *ata pro4)*e

s,?c)e(t e4)*e(ce to co(cl,*e that the -ea( we)ght rea*)(gs by the (ewly *es)g(e*

*e4)ce 4ary a-o(g the fo,r sett)(gs.

6 &o )(4est)gate the real cost of ow()(g *)8ere(t -akes a(* -o*els of (ew

a,to-ob)les a co(s,-er protect)o( age(cy followe* 1 ow(ers of (ew 4eh)cles offo,r pop,lar -akes a(* -o*els call the- TC HA NA a(* FT a(* kept a recor* of

each of the ow(erWs real cost )( *ollars for the +rst +4e years. &he +4e=year costs of

the 1 car ow(ers are g)4e( below:

TC 4A NA FT

8423 7776 8907 10333

7889 7211 9077 9217

8665 6870 8732 10540

7129 9747

7359 8677

&est ,s)(g the AN"'A F =test at the 0G le4el of s)g()+ca(ce whether the *ata pro4)*e

s,?c)e(t e4)*e(ce to co(cl,*e that there are *)8ere(ces a-o(g the -ea( real costs of

ow(ersh)p for these fo,r -o*els.

9 @elp)(g people to lose we)ght has beco-e a h,ge )(*,stry )( the U()te* States w)th

a((,al re4e(,e )( the h,(*re*s of b)ll)o( *ollars. Rece(tly each of the three -arket=

lea*)(g we)ght re*,c)(g progra-s cla)-e* to be the -ost e8ect)4e. A co(s,-er

research co-pa(y recr,)te* 33 people who w)she* to lose we)ght a(* se(t the- tothe three lea*)(g progra-s. After s)7 -o(ths the)r we)ght losses were recor*e*. &he

res,lts are s,--ar)Ve* below:

Statistic rog8 rog8 ! rog8 "

a%$"e ean x−1=10.65 x−2=8.90 x−3=9.33

a%$"e =arian!e s21=27.20 s22=16.86 s23=32.40

a%$"e iBe n1=11 n2=11 n3=11

&he -ea( we)ght loss of the co-b)(e* sa-ple of all 33 people wasx−=9.63. &est ,s)(g

the AN"'A F =test at the 0G le4el of s)g()+ca(ce whether the *ata pro4)*e s,?c)e(t

e4)*e(ce to co(cl,*e that so-e progra- )s -ore e8ect)4e tha( the others.

1; A lea*)(g phar-ace,t)cal co-pa(y )( the *)sposable co(tact le(ses -arket has always

take( for gra(te* that the sales of certa)( per)pheral pro*,cts s,ch as co(tact le(s

sol,t)o(s wo,l* a,to-at)cally go w)th the establ)she* bra(*s. &he lo(g=sta(*)(g c,lt,re

)( the co-pa(y has bee( that le(s sol,t)o(s wo,l* (ot -ake a s)g()+ca(t *)8ere(ce )(








Chapter %0

Appendix !igure "&." <umulative 9inomial $robability
















!igure "&.& <umulative Iormal $robability











!igure "&.* <ritical 0alues of t











!igure "&. <ritical 0alues of <hi:%quare (istributions






!igure "&. pper <ritical 0alues of !:(istributions
















!igure "&./ 'ower <ritical 0alues of !:(istributions

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