chapter 10 elementary statistics

8/18/2019 Chapter 10 Elementary Statistics

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Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Resered. 10.1 - 1

!ecture "lides

Elementary Statistics Eleenth Edition

and the #riola "tatistics "eries

$y %ario &. #riola


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Chapter 10Correlation and Regression

10-1 Review and Preview

10-2 Correlation

10-3 Regression

10-4 Variation and Prediction Intervals

10-5 Multiple Regression10- Modeling


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!ection 10-1

Review and Preview


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ReviewIn Chapter " we presented #ethods $or #a%ing

in$erences $ro# two sa#ples& In !ection "-4 weconsidered two dependent sa#ples' with eachvalue o$ one sa#ple so#ehow paired with avalue $ro# the other sa#ple& In !ection "-4 we

considered the di$$erences (etween the pairedvalues' and we illustrated the use o$ h)pothesistests $or clai#s a(out the population o$di$$erences& *e also illustrated the construction

o$ con$idence interval esti#ates o$ the #ean o$all such di$$erences& In this chapter we againconsider paired sa#ple data' (ut the o(+ective is$unda#entall) di$$erent $ro# that o$ !ection "-4&


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Preview

In this chapter we introduce #ethods $ordeter#ining whether a correlation' orassociation' (etween two varia(les e,ists andwhether the correlation is linear& or linear

correlations' we can identi$) an e.uation that(est $its the data and we can use that e.uationto predict the value o$ one varia(le given thevalue o$ the other varia(le& In this chapter' we

also present #ethods $or anal)/ingdi$$erences (etween predicted values andactual values&


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Preview

In addition' we consider #ethods $oridenti$)ing linear e.uations $or correlationsa#ong three or #ore varia(les& *e concludethe chapter with so#e (asic #ethods $or

developing a #athe#atical #odel that can (eused to descri(e nonlinear correlations(etween two varia(les&


7/138Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Resered. 10.1 - 7

!ection 10-2Correlation



e) Concept

In part 1 o$ this section introduces the linear

correlation coe$$icient r ' which is a nu#erical#easure o$ the strength o$ the relationship

(etween two varia(les representing.uantitative data&

sing paired sa#ple data so#eti#es called

(ivariate data' we $ind the value o$ r usuall)using technolog)' then we use that value toconclude that there is or is not a linearcorrelation (etween the two varia(les&



e) Concept

In this section we consider onl) linearrelationships' which #eans that whengraphed' the points appro,i#ate a straight-

line pattern&In Part 2' we discuss #ethods o$ h)pothesistesting $or correlation&



Part 1 asic Concepts o$ Correlation



6e$inition

7 correlation e,ists (etween twovaria(les when the values o$ one

are so#ehow associated with thevalues o$ the other in so#e wa)&



6e$inition

8he linear correlation coe$$icient r #easures the strength o$ the linearrelationship (etween the paired

.uantitative x- and y-values in a sa#ple&



9,ploring the 6ata

*e can o$ten see a relationship (etween twovaria(les () constructing a scatterplot&

igure 10-2 $ollowing shows scatterplots withdi$$erent characteristics&



!catterplots o$ Paired 6ata

igure 10-2




igure 10-2


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igure 10-2


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Re.uire#ents

1& 8he sa#ple o$ paired x, y data is a si#plerando# sa#ple o$ .uantitative data&

2& Visual e,a#ination o$ the scatterplot #ust

con$ir# that the points appro,i#ate a straight-line pattern&

3& 8he outliers #ust (e re#oved i$ the) are%nown to (e errors& 8he e$$ects o$ an) other

outliers should (e considered () calculating r with and without the outliers included&


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:otation $or the;inear Correlation Coe$$icient

n < nu#(er o$ pairs o$ sa#ple data

Σ denotes the addition o$ the ite#sindicated&

Σ x denotes the su# o$ all x -values&

Σ x 2 indicates that each x -value should (es.uared and then those s.uares added&

Σ x 2 indicates that the x -values should (eadded and then the total s.uared&


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:otation $or the;inear Correlation Coe$$icient

Σ xy indicates that each x -value should (e $irst#ultiplied () its corresponding y-value&7$ter o(taining all such products' $indtheir su#&

r < linear correlation coe$$icient $or sa#pledata&

ρ < linear correlation coe$$icient $orpopulation data&


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or#ula 10-1

nΣ

xy – (Σ x )(Σ y)

n(Σ

x 2) – (Σ

x )2 n(Σ

y2) – (Σ

y)2r '

8he linear correlation coe$$icient r #easures thestrength o$ a linear relationship (etween thepaired values in a sa#ple&

Co#puter so$tware or calculators can co#pute r

or#ula


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Interpreting r

sing 8a(le 7- I$ the a(solute value o$ theco#puted value o$ r ' denoted =r =' e,ceeds thevalue in 8a(le 7-' conclude that there is a linearcorrelation& >therwise' there is not su$$icient

evidence to support the conclusion o$ a linearcorrelation&

sing !o$tware I$ the co#puted P -value is less

than or e.ual to the signi$icance level' concludethat there is a linear correlation& >therwise' thereis not su$$icient evidence to support theconclusion o$ a linear correlation&


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Caution

now that the #ethods o$ this sectionappl) to a linear correlation& I$ )ouconclude that there does not appear to

(e linear correlation' %now that it ispossi(le that there #ight (e so#e otherassociation that is not linear&


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Round to three deci#al placesso that it can (e co#pared to

critical values in 8a(le 7-&

se calculator or co#puter i$

possi(le&

Rounding the ;inearCorrelation Coe$$icient r


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Properties o$ the;inear Correlation Coe$$icient r

1& ?1 ≤ r ≤ 1

2& i$ all values o$ either varia(le are converted to adi$$erent scale' the value o$ r does not change&

3& 8he value o$ r is not a$$ected () the choice o$ x and y& Interchange all x- and y-values and thevalue o$ r will not change&

4& r #easures strength o$ a linear relationship&5& r is ver) sensitive to outliers' the) can

dra#aticall) a$$ect its value&


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9,a#ple

8he paired pi//a@su(wa) $are costs $ro#

8a(le 10-1 are shown here in 8a(le 10-2& seco#puter so$tware with these paired sa#plevalues to $ind the value o$ the linear

correlation coe$$icient r $or the paired

sa#ple data&

Re.uire#ents are satis$ied si#ple rando#

sa#ple o$ .uantitative dataA Minita(scatterplot appro,i#ates a straight lineAscatterplot shows no outliers - see ne,t slide


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9,a#ple

sing so$tware or a calculator' r is

auto#aticall) calculated


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Interpreting the ;inearCorrelation Coe$$icient r

*e can (ase our interpretation andconclusion a(out correlation on a P -valueo(tained $ro# co#puter so$tware or a criticalvalue $ro# 8a(le 7-&


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sing Co#puter !o$tware to Interpret r

I$ the co#puted P -value is less than or e.ualto the signi$icance level' conclude that thereis a linear correlation&>therwise' there is not su$$icient evidence to

support the conclusion o$ a linear correlation&


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sing 8a(le 7- to Interpret r

I$ =r = e,ceeds the value in 8a(le 7-' concludethat there is a linear correlation&>therwise' there is not su$$icient evidence tosupport the conclusion o$ a linear correlation&


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Critical Values $ro# 8a(le 7- and the

Co#puted Value o$ r


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sing a 0&05 signi$icance level' interpret the

value o$ r < 0&11B $ound using the 2 pairs o$weights o$ discarded paper and glass listedin 6ata !et 22 in 7ppendi, & *hen the

paired data are used with co#puterso$tware' the P -value is $ound to (e 0&34& Isthere su$$icient evidence to support a clai#o$ a linear correlation (etween the weights

o$ discarded paper and glass

9,a#ple


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Re.uire#ents are satis$ied si#ple rando#sa#ple o$ .uantitative dataA scatterplotappro,i#ates a straight lineA no outliers

9,a#ple

sing !o$tware to Interpret r

8he P -value o(tained $ro# so$tware is 0&34&ecause the P -value is not less than or

e.ual to 0&05' we conclude that there is notsu$$icient evidence to support a clai# o$ alinear correlation (etween weights o$discarded paper and glass&


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9,a#ple

sing 8a(le 7- to Interpret r

I$ we re$er to 8a(le 7- with n < 2 pairs o$sa#ple data' we o(tain the critical value o$0&254 appro,i#atel) $or α < 0&05& ecause =0&11B= does not e,ceed the value o$ 0&254$ro# 8a(le 7-' we conclude that there is notsu$$icient evidence to support a clai# o$ a

linear correlation (etween weights o$discarded paper and glass&


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sing the pi//a su(wa) $are costs in 8a(le10-2' we have $ound that the linear

correlation coe$$icient is r < 0&"DD& *hatproportion o$ the variation in the su(wa)$are can (e e,plained () the variation in the

costs o$ a slice o$ pi//a*ith r < 0&"DD' we get r 2 < 0&"B&

*e conclude that 0&"B or a(out "DE o$ the

variation in the cost o$ a su(wa) $ares can (ee,plained () the linear relationship (etween thecosts o$ pi//a and su(wa) $ares& 8his i#plies thata(out 2E o$ the variation in costs o$ su(wa) $ares

cannot (e e,plained () the costs o$ pi//a&

9,a#ple


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Co##on 9rrorsInvolving Correlation

1& Causation It is wrong to conclude that

correlation i#plies causalit)&

2& 7verages 7verages suppress individualvariation and #a) in$late the correlation coe$$icient&

3& ;inearit) 8here #a) (e so#e relationship

(etween x and y even when there is no linearcorrelation&


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Caution

now that correlation does not

i#pl) causalit)&


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Part 2 or#al F)pothesis 8est


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or#al F)pothesis 8est

*e wish to deter#ine whether thereis a signi$icant linear correlation(etween two varia(les&

F th i 8 t $ C l ti


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F)pothesis 8est $or Correlation:otation

n < nu#(er o$ pairs o$ sa#ple data

r < linear correlation coe$$icient $or a sample o$ paired data

ρ < linear correlation coe$$icient $or a population o$ paired data

F th i 8 t $ C l ti


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F)pothesis 8est $or CorrelationRe.uire#ents

1& 8he sa#ple o$ paired x, y data is a si#plerando# sa#ple o$ .uantitative data&

2& Visual e,a#ination o$ the scatterplot #ust

con$ir# that the points appro,i#ate astraight-line pattern&

3& 8he outliers #ust (e re#oved i$ the) are

%nown to (e errors& 8he e$$ects o$ an)other outliers should (e considered ()calculating r with and without the outliersincluded&


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F)pothesis 8est $or CorrelationF)potheses

H 0 ρ


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9,a#ple


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9,a#ple

se the paired pi//a su(wa) $are data in 8a(le

10-2 to test the clai# that there is a linearcorrelation (etween the costs o$ a slice o$pi//a and the su(wa) $ares& se a 0&05signi$icance level&

Re.uire#ents are satis$ied as in the earliere,a#ple&

H 0 ρ


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9,a#ple

8he test statistic is r < 0&"DD $ro# an earlier

9,a#ple& 8he critical value o$ r < 0&D11 is$ound in 8a(le 7- with n < and α < 0&05&ecause =0&"DD= G 0&D11' we re+ect H 0 r < 0&

Re+ecting no linear correlationJ indicates

that there is a linear correlation&

*e conclude that there is su$$icient evidenceto support the clai# o$ a linear correlation(etween costs o$ a slice o$ pi//a and su(wa)$ares&


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F)pothesis 8est $or CorrelationP -Value $ro# a t 8est

H 0 ρ


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F)pothesis 8est $or CorrelationConclusion

I$ the P -value is less than or e.ual to thesigni$icance level' re+ect H 0 and conclude that thereis su$$icient evidence to support the clai# o$ alinear correlation&

I$ the P -value is greater than the signi$icancelevel' $ail to re+ect H 0 and conclude that there

is not su$$icient evidence to support the clai#

o$ a linear correlation&

P -value se co#puter so$tware or use 8a(le7-3 with n ? 2 degrees o$ $reedo# to $ind theP -value corresponding to the test statistic t &

9,a#ple


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9,a#ple

se the paired pi//a su(wa) $are data in 8a(le

10-2 and use the P -value #ethod to test theclai# that there is a linear correlation (etweenthe costs o$ a slice o$ pi//a and the su(wa)$ares& se a 0&05 signi$icance level&

Re.uire#ents are satis$ied as in the earliere,a#ple&

H 0 ρ


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9,a#ple

8he linear correlation coe$$icient is r < 0&"DD

$ro# an earlier 9,a#ple and n < si, pairso$ data' so the test statistic is

*ith d$ < 4' 8a(le 7- )ields a P -value that isless than 0&01&

Co#puter so$tware generates a test statistic o$t < 12&"2 and P -value o$ 0&00022&

t =r

1− r 2

n − 2

=0.988

1− 0.9882

6 − 2

= 12.793

9,a#ple


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9,a#ple

sing either #ethod' the P -value is less

than the signi$icance level o$ 0&05 so were+ect H 0 ρ < 0&

*e conclude that there is su$$icient evidenceto support the clai# o$ a linear correlation(etween costs o$ a slice o$ pi//a and su(wa)$ares&

> 8 il d 8 t


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>ne-8ailed 8ests

>ne-tailed tests can occur with a clai# o$ a

positive linear correlation or a clai# o$ a negativelinear correlation& In such cases' the h)potheseswill (e as shown here&

or these one-tailed tests' the P -value #ethodcan (e used as in earlier chapters&

R


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Recap

In this section' we have discussed Correlation&

8he linear correlation coe$$icient r &

Re.uire#ents' notation and $or#ula $or r &

Interpreting r &

or#al h)pothesis testing&


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!ection 10-3

Regression

C t


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e) Concept

In part 1 o$ this section we $ind the e.uation o$the straight line that (est $its the paired sa#pledata& 8hat e.uation alge(raicall) descri(es therelationship (etween two varia(les&

8he (est-$itting straight line is called aregression line and its e.uation is called theregression e.uation&

In part 2' we discuss #arginal change'in$luential points' and residual plots as tools$or anal)/ing correlation and regressionresults&


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Part 1 asic Concepts o$ Regression

R i


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Regression

8he t)pical e.uation o$ a straight line

y < mx K b is e,pressed in the $or# y < b0 K b1 x ' where b0 is the y-intercept and b1

is the slope&

L

8he regression e.uation e,presses arelationship (etween x called the

e,planator) varia(le' predictor varia(le or

independent varia(le' and y called the

response varia(le or dependent varia(le&

L

6 $i iti


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6e$initions

Regression 9.uationiven a collection o$ paired data' the regressione.uation

Regression ;ine

8he graph o$ the regression e.uation is calledthe regression line or line o$ (est $it' or least s.uares line&

y < b0 K b1 x L

alge(raicall) descri(es the relationship (etween the two varia(les&

: t ti $


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:otation $orRegression 9.uation

y-intercept o$regression e.uation

!lope o$ regressione.uation

9.uation o$ theregression line

PopulationPara#eter

!a#ple!tatistic

β0 b0

β1 b1

y < β 0 K β1 x y < b0 K b1 x L

R i t


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Re.uire#ents

1& 8he sa#ple o$ paired x , y data is arando# sa#ple o$ .uantitative data&

2& Visual e,a#ination o$ the scatterplot

shows that the points appro,i#ate astraight-line pattern&

3& 7n) outliers #ust (e re#oved i$ the) are%nown to (e errors& Consider the e$$ectso$ an) outliers that are not %nown errors&

or#ulas $or b and b


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or#ulas $or b0 and b1

or#ula 10-3 slope

y-interceptor#ula 10-4

calculators or co#puters canco#pute these values

b0 = y − b1 x

b1

= r s

y

s x

!pecial Propert)


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8he regression line $its the

sa#ple points (est&

!pecial Propert)

Rounding the y intercept b


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Rounding the y-intercept b0 and the !lope b1

Round to three signi$icant digits&

I$ )ou use the $or#ulas 10-3 and 10-4'do not round inter#ediate values&


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9,a#ple


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9,a#ple

Re.uire#ents are satis$ied si#ple rando#

sa#pleA scatterplot appro,i#ates a straightlineA no outliers

Fere are results $ro# $our di$$erent technologiestechnologies

9,a#ple


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9,a#ple

7ll o$ these technologies show that the

regression e.uation can (e e,pressed asy < 0&034 K0&"45 x ' where y is the predictedcost o$ a su(wa) $are and x is the cost o$ aslice o$ pi//a&

*e should %now that the regression e.uation isan esti#ate o$ the true regression e.uation&8his esti#ate is (ased on one particular set o$sa#ple data' (ut another sa#ple drawn $ro#

the sa#e population would pro(a(l) lead to aslightl) di$$erent e.uation&

L L

9,a#ple


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9,a#ple

raph the regression e.uation

$ro# the preceding 9,a#ple on thescatterplot o$ the pi//a@su(wa) $are data ande,a#ine the graph to su(+ectivel) deter#inehow well the regression line $its the data&

ˆ 0.0346 0.945= + y x

>n the ne,t slide is the Minita( displa) o$ the

scatterplot with the graph o$ the regression lineincluded& *e can see that the regression line$its the data .uite well&

9,a#ple


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9,a#ple

sing the Regression 9.uation


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1& se the regression e.uation $or predictionsonl) i$ the graph o$ the regression line on thescatterplot con$ir#s that the regression line

$its the points reasona(l) well&

sing the Regression 9.uation$or Predictions

2& se the regression e.uation $or predictionsonl) i$ the linear correlation coe$$icient r indicates that there is a linear correlation

(etween the two varia(les as descri(ed in!ection 10-2&



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3& se the regression line $or predictions onl) i$the data do not go #uch (e)ond the scope o$the availa(le sa#ple data& Predicting too $ar(e)ond the scope o$ the availa(le sa#pledata is called extrapolation' and it couldresult in (ad predictions&


4& I$ the regression e.uation does not appear to(e use$ul $or #a%ing predictions' the (estpredicted value o$ a varia(le is its pointesti#ate' which is its sa#ple #ean&

!trateg) $or Predicting Values o$ Y


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!trateg) $or Predicting Values o$ Y



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I$ the regression e.uation is not a good#odel' the (est predicted value o$ y is si#pl)y ' the #ean o$ the y values&

Re#e#(er' this strateg) applies to linearpatterns o$ points in a scatterplot&

I$ the scatterplot shows a pattern that is not a

straight-line pattern' other #ethods appl)' asdescri(ed in !ection 10-&


L


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Part 2 e)ond the asics o$ Regression

6e$initions


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6e$initions

In wor%ing with two varia(les related ()a regression e.uation' the #arginalchange in a varia(le is the a#ount that

it changes when the other varia(lechanges () e,actl) one unit& 8he slopeb1 in the regression e.uation representsthe #arginal change in y that occurs

when x changes () one unit&

6e$initions


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6e$initions

In a scatterplot' an outlier is a pointl)ing $ar awa) $ro# the other datapoints&

Paired sa#ple data #a) include one or#ore in$luential points' which are

points that strongl) a$$ect the graph o$the regression line&

9,a#ple


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9,a#ple

Consider the pi//a su(wa) $are data $ro# the

Chapter Pro(le#& 8he scatterplot located tothe le$t on the ne,t slide shows theregression line& I$ we include this additionalpair o$ data x < 2&00'y < ?20&00 pi//a is still

N2&00 per slice' (ut the su(wa) $are is N?20&00which #eans that people are paid N20 to ridethe su(wa)' this additional point would (e anin$luential point (ecause the graph o$ the

regression line would change considera(l)'as shown () the regression line located tothe right&

9,a#ple


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9,a#ple

9,a#ple


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9,a#ple

Co#pare the two graphs and )ou will see

clearl) that the addition o$ that one pair o$values has a ver) dra#atic e$$ect on theregression line' so that additional point is anin$luential point& 8he additional point is also

an outlier (ecause it is $ar $ro# the otherpoints&

6e$inition


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or a pair o$ sa#ple x and y values' theresidual is the di$$erence (etween theobserved sa#ple value o$ y and the y -value that is predicted () using theregression e.uation& 8hat is'

6e$inition

residual < o(served y ? predicted y = y – y L

Residuals


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Residuals


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6e$initions


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7 residual plot is a scatterplot o$ the x ' y values a$ter each o$ they -coordinate values has (een replaced

() the residual value y ? y where y denotes the predicted value o$ y & 8hatis' a residual plot is a graph o$ thepoints x ' y ? y &

6e$initions

L L

L

Residual Plot 7nal)sis


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Residual Plot 7nal)sis

*hen anal)/ing a residual plot' loo% $or apattern in the wa) the points are con$igured'and use these criteria

8he residual plot should not have an o(viouspattern that is not a straight-line pattern&

8he residual plot should not (eco#e thic%er

or thinner when viewed $ro# le$t to right&

Residuals Plot - Pi//a@!u(wa)


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Residuals Plot Pi//a@!u(wa)

Residual Plots


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Residual Plots

Residual Plots


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Residual Plots

Residual Plots


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Residual Plots

Co#plete Regression 7nal)sis


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1& Construct a scatterplot and veri$) that thepattern o$ the points is appro,i#atel) astraight-line pattern without outliers& I$there are outliers' consider their e$$ects ()

co#paring results that include the outliersto results that e,clude the outliers&

2& Construct a residual plot and veri$) that

there is no pattern other than a straight-line pattern and also veri$) that theresidual plot does not (eco#e thic%er orthinner&



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3& se a histogra# and@or nor#al .uantileplot to con$ir# that the values o$ theresiduals have a distri(ution that isappro,i#atel) nor#al&

4& Consider an) e$$ects o$ a pattern over ti#e&

Recap


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Recap

In this section we have discussed

8he (asic concepts o$ regression&

Rounding rules&

sing the regression e.uation $orpredictions&

Interpreting the regression e.uation&

>utliers Residuals and least-s.uares&

Residual plots&


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!ection 10-4

Variation and PredictionIntervals

e) Concept


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e) Concept

In this section we present a #ethod $orconstructing a prediction interval' which is an

interval esti#ate o$ a predicted value o$ y&


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6e$inition


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8he total deviation o$ x , y is thevertical distance y – y' which is thedistance (etween the point x , y andthe hori/ontal line passing through thesa#ple #ean y&

6e$inition


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8he e,plained deviation is the verticaldistance y ? y' which is the distance

(etween the predicted y-value and thehori/ontal line passing through thesa#ple #ean y.

L

6e$inition


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8he une,plained deviation is thevertical distance y ? y' which is thevertical distance (etween the point

x , y and the regression line& 8hedistance y ? y is also called a residual' as de$ined in !ection 10-3&

L

L

ne,plained' 9,plained' and 8otal 6eviation


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igure 10-B

p ' p '

Relationships


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total deviation < e,plained deviation K une,plained deviation

y - y < y - y K ( y - yL L

total variation < e,plained variation K une,plained variation

Σ

y - y 2

<Σ

y - y2

KΣ

( y - y)2L L

or#ula 10-5

p

6e$inition


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r 2 <e,plained variation&

total variation

8he value o$ r 2

is the proportion o$ thevariation in y that is e,plained () the linearrelationship (etween x and y&

Coe$$icient o$ deter#inationis the a#ount o$ the variation in y thatis e,plained () the regression line&

6e$inition


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7 prediction interval' is an interval

esti#ate o$ a predicted value o$ y &

6e$inition


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8he standard error o$ esti#ate' denoted

() se is a #easure o$ the di$$erences or

distances (etween the o(served

sa#ple y-values and the predicted

values y that are o(tained using the

regression e.uation&

(


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9,a#ple


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se or#ula 10- to $ind the standard error o$esti#ate se $or the paired pi//a@su(wa) $are

data listed in 8a(le 10-1in the Chapter Pro(le#&n = 6

Σ

y2 = 9.2175

Σ y = 6.35

Σ

xy = 9.4575

b0 = 0.034560171

b1 = 0.94502138

se <n - 2

Σ

y

2

- b0 Σ

y - b1 Σ

xy

se <

6 – 2

9.2175 – (0.034560171)(6.35) – (0.94502138)(9.4575)

se < 0&122"DB00 < 0&123

Prediction Interval $or an


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y - E O y O y K E L L

Individual y

where

E < tα

2 se n(Σ x 2) ? (Σ x )

2

n( x 0 ? x )2

1 K K1n

x 0 represents the given value o$ x

tα

2 has n ? 2 degrees o$ $reedo#

9,a#ple


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E < tα

2 seK

n(Σ x 2) – (Σ x )2

n( x 0 ? x )2

1 + 1 n

E = (2.776)(0.12298700)

6(9.77) – (6.50)2

6(2.25 – 1.0833333)2

1 + 1 6

E = (2.776)(0.12298700)(1.2905606) = 0.441

or the paired pi//a@su(wa) $are costs $ro# theChapter Pro(le#' we have $ound that $or a pi//a

cost o$ N2&25' the (est predicted cost o$ a su(wa)$are is N2&1& Construct a "5E prediction interval$or the cost o$ a su(wa) $are' given that a slice o$pi//a costs N2&25 so that , < 2&25&

K

9,a#ple


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Construct the con$idence interval&

y – E < y < y + E

2.16 – 0.441


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9,plained and une,plained variation&

Coe$$icient o$ deter#ination&

!tandard error esti#ate&

Prediction intervals&


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!ection 10-5Multiple Regression

e) Concept


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8his section presents a #ethod $or anal)/ing alinear relationship involving #ore than two varia(les&

*e $ocus on three %e) ele#ents1& 8he #ultiple regression e.uation&

2& 8he values o$ the ad+usted R2.

3& 8he P -value&


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6e$inition


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7 #ultiple regression e.uation e,presses alinear relationship (etween a response varia(le

y and two or #ore predictor varia(les x 1, x 2, x 3 .

. . , x k

8he general $or# o$ the #ultiple regressione.uation o(tained $ro# sa#ple data is

y = b0 + b1 x 1 + b2 x 2 + . . . + bk x k .L

:otation


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y = b0 + b1 x 1+ b2 x 2+ b3 x 3 +. . .+ bk x k eneral $or# o$ the #ultiple regression e.uation

n < sa#ple si/e

k < nu#(er o$ predictor varia(les

y < predicted value o$ y

x 1, x 2, x 3 . . . , x k are the predictorvaria(les

L

L

:otation - cont


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ß 0, ß 1, ß 2, . . . , ß k are the para#eters $or the#ultiple regression e.uation

y = ß 0 + ß 1 x 1+ ß 2 x 2+…+ ß k x k

b0, b1, b2, . . . , bk are the sample estimates

o$ the para#eters ß 0, ß 1, ß 2, . . . , ß k


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9,a#ple


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8a(le 10- includes a rando#

sa#ple o$ heights o$ #others'$athers' and their daughters(ased on data $ro# the :ationalFealth and :utrition

9,a#ination& ind the #ultipleregression e.uation in which theresponse ) varia(le is theheight o$ a daughter and the

predictor , varia(les are theheight o$ the #other and heighto$ the $ather&

9,a#ple


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8he Minita( results are shown here

9,a#ple


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ro# the displa)' we see that the #ultiple

regression e.uation isFeight < B&5 K B&0BMother K 0&14ather

sing our notation presented earlier in this

section' we could write this e.uation as

y < B&5 K 0&B0B x 1 K 0&14 x 2

where ) is the predicted height o$ a daughter' x 1 is the height o$ the #other' and x 2 is the

height o$ the $ather&

L

L

6e$inition


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8he #ultiple coe$$icient o$ deter#ination R2

is a #easure o$ how well the #ultipleregression e.uation $its the sa#ple data&

8he ad+usted coe$$icient o$ deter#ination is the #ultiple coe$$icient o$ deter#ination R2

#odi$ied to account $or the nu#(er o$

varia(les and the sa#ple si/e&

7d+usted Coe$$icient o$6 t i ti


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6eter#ination

7d+usted R2 = 1 – (n – 1)

[n – (k + 1)](1– R

2)

or#ula 10-B

where n < sa#ple si/e

k < nu#(er o$ predictor x varia(les

P -Value


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8he P -value is a #easure o$ theoverall signi$icance o$ the #ultipleregression e.uation& ;i%e thead+usted R 2' this P -value is a good

#easure o$ how well the e.uation $itsthe sa#ple data&

P -Value


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8he displa)ed Minita( P -value o$ 0&000

rounded to three deci#al places is s#all'indicating that the #ultiple regressione.uation has good overall signi$icance and isusa(le $or predictions& 8hat is' it #a%es sense

to predict heights o$ daughters (ased onheights o$ #others and $athers& 8he value o$0&000 results $ro# a test o$ the null h)pothesisthat β 1 < β 2 < 0& Re+ection o$ β 1 < β2 < 0 i#plies

that at least one o$ β1 and β 2 is not 0' indicatingthat this regression e.uation is e$$ective inpredicting heights o$ daughters&

inding the est MultipleR i 9 ti


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Regression 9.uation

1& se co##on sense and practical considerations toinclude or e,clude varia(les&

2& Consider the P -value& !elect an e.uation havingoverall signi$icance' as deter#ined () the P -value$ound in the co#puter displa)&

inding the est MultipleR i 9 ti


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Regression 9.uation

3& Consider e.uations with high values o$ ad+usted R2

and tr) to include onl) a $ew varia(les&

I$ an additional predictor varia(le is included' thevalue o$ ad+usted R2 does not increase () asu(stantial a#ount&

or a given nu#(er o$ predictor x varia(les'select the e.uation with the largest value o$ad+usted R2&

In weeding out predictor x varia(les that donthave #uch o$ an e$$ect on the response yvaria(le' it #ight (e help$ul to $ind the linearcorrelation coe$$icient r $or each o$ the pairedvaria(les (eing considered&


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Part 2 6u##) Varia(les and

;ogistic 9.uations

6u##) Varia(le


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Man) applications involve a dichoto#ousvaria(le which has onl) two possi(le discretevalues such as #ale@$e#ale' dead@alive' etc&&7 co##on procedure is to represent the two

possi(le discrete values () 0 and 1' where 0represents $ailureJ and 1 represents success&

7 dichoto#ous varia(le with the two values 0and 1 is called a du##) varia(le&

;ogistic Regression


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*e can use the #ethods o$ this section i$the du##) varia(le is the predictor varia(le&

I$ the du##) varia(le is the response

varia(le we need to use a #ethod %nown aslogistic regression&

7s the na#e i#plies logistic regression

involves the use o$ natural logarith#s& 8histe,t (oo% does not include detailedprocedures $or using logistic regression&


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!ection 10-Modeling

e) Concept


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8his section introduces so#e (asic conceptso$ developing a #athe#atical #odel' which isa $unction that $itsJ or descri(es real-world

data&

nli%e !ection 10-3' we will not (e restrictedto a #odel that #ust (e linear&

8I-D3@D4 Plus eneric Models


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;inear y = a + bx

Quadratic y = ax 2 + bx + c

;ogarith#ic y = a + b ln x 9,ponential y = ab x

Power y = ax b


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8he slides that $ollow illustrate the graphso$ so#e co##on #odels displa)ed on a

8I-D3@D4 Plus Calculator


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6evelop#ent o$ a oodMathe#atical Model


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Mathe#atical Model

;oo% $or a Pattern in the raph 9,a#inethe graph o$ the plotted points andco#pare the (asic pattern to the %nowngeneric graphs o$ a linear $unction&

ind and Co#pare Values o$ R 2 !elect$unctions that result in larger values o$ R 2'(ecause such larger values correspond to$unctions that (etter $it the o(served

points&8hin% se co##on sense& 6ont use a

#odel that leads to predicted values %nownto (e totall) unrealistic&

I#portant Point


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p

8he (est choice o$ a #odeldepends on the set o$ data (eing

anal)/ed and re.uires an e,ercise in +udg#ent' not +ust co#putation&J

Recap


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8he concept o$ #athe#atical #odeling&

raphs $ro# a 8I-D3@D4 Plus calculator& Rules $or developing a good #athe#atical#odel&

chapter 10 elementary statistics

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