Statistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc. Chap -Simple Linear RegressionChapter 11Statistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc!"ap ##-2!"apter $oalsAfter completing this chapter, you should be able to: Explain t"e simple linear regression mo%el &'tain an% interpret t"e simple linear regression e()ation for a set of %ataE*al)ate regression resi%)als for aptness of t"e fitte% mo%elUn%erstan% t"e ass)mptions 'e"in% regression anal+sisExplain meas)res of *ariation an% %etermine ,"et"er t"e in%epen%ent *aria'le is significantStatistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc!"ap ##--!"apter $oalsAfter completing this chapter, you should be able to: !alc)late an% interpret confi%ence inter*als for t"e regression coefficientsUse t"e .)r'in-/atson statistic to c"ec0 for a)tocorrelation1orm confi%ence an% pre%iction inter*als aro)n% an estimate% 2 *al)e for a gi*en 3Recogni4e some potential pro'lems if regression anal+sis is )se% incorrectl+(continued)Statistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc!"ap ##-4!orrelation *s Regression5 scatter plot 6or scatter %iagram7 can 'e )se% to s"o, t"e relations"ip 'et,een t,o *aria'les!orrelation anal+sis is )se% to meas)re strengt" of t"e association 6linear relations"ip7 'et,een t,o *aria'les!orrelation is onl+ concerne% ,it" strengt" of t"e relations"ip 8o ca)sal effect is implie% ,it" correlation!orrelation ,as first presente% in !"apter -Statistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc!"ap ##-9### Intro%)ction to Regression 5nal+sisRegression anal+sis is )se% to:Pre%ict t"e *al)e of a %epen%ent *aria'le 'ase% on t"e *al)e of at least one in%epen%ent *aria'leExplain t"e impact of c"anges in an in%epen%ent *aria'le on t"e %epen%ent *aria'le.epen%ent *aria'le:t"e *aria'le ,e ,is" to explainIn%epen%ent *aria'le:t"e *aria'le )se% to explain t"e %epen%ent *aria'leStatistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc!"ap ##-;##2 Simple Linear Regression Mo%el&nl+ one in%epen%ent *aria'le, 3Relations"ip 'et,een3an%2is %escri'e% '+ a linear f)nction!"anges in2are ass)me% to 'e ca)se% '+ c"anges in3Statistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc!"ap ##-=+pes of Relations"ipsYXYXYYXXStrong relationships Wea relationships(continued)Statistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc!"ap ##-?=+pes of Relations"ipsYXYX!o relationship(continued)Statistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc!"ap ##-#0i i # 0 i@ 3 A A 2 + + =Linear componentSimple Linear Regression Mo%el="e pop)lation regression mo%el:Pop)lation 2intercept Pop)lation Slope!oefficient Ran%om Error term.epen%ent Baria'leIn%epen%ent Baria'leRan%om Error componentStatistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc!"ap ##-##(continued)Ran%om Error for t"is 3i *al)e23&'ser*e% Bal)e of 2 for 3iPre%icte% Bal)e of 2 for 3i i i # 0 i@ 3 A A 2 + + =3iSlope C A#Intercept C A0

@iSimple Linear Regression Mo%elStatistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc!"ap ##-#2i ibX Y+ = a="e simple linear regression e()ation pro*i%es an estimate of t"e pop)lation regression lineSimple Linear Regression E()ationEstimate of t"e regression interceptEstimate of t"e regression slopeEstimate%6or pre%icte%7 2 *al)e for o'ser*ation iBal)e of 3 for o'ser*ation i="e in%i*i%)al ran%om error termsei"a*e a mean of 4eroStatistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc!"ap ##-#-##- Least S()ares Met"o%aan%bare o'taine% '+ fin%ing t"e *al)es of A0an%A#t"at minimi4e t"e s)m of t"e s()are% 6SSE D res)%ial7 %ifferences 'et,een 2 an% :=o minimi4e, %ifferentiate ,it" respect to a an% b, an% set eac" res)lt to 0 ="is generates t,o sim)ltaneo)s e()ations 6calle% normal e()ations7 E t,o )n0no,ns Sol*ing for a an% b, ,e get = == = ==niiniininiinii i ix x ny x y x nb12121 1 1) () )( (2i i2i i)) X ( (Y min ) Y(Y min b a SSE + = = 2Fx b y a =Statistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc!"ap ##-#4Simple Linear Regression Example5 real estate agent ,is"es to examine t"e relations"ip 'et,een t"e selling price of a "ome an% its si4e 6meas)re% in s()are feet75 ran%om sample of #0 "o)ses is selecte%.epen%ent *aria'le 627 C "o)se price in G#000sIn%epen%ent *aria'le 637 C s()are feetStatistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc!"ap ##-#9Sample .ata for Ho)se Price Mo%el"ouse #rice in $1%%%s&Y'S(uare )eet &X'*+, 1+%%-1* 1.%%*/0 1/%%-%1 11/,100 11%%*10 1,,%+%, *-,%-*+ *+,%-10 1+*,*,, 1/%%Statistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc!"ap ##-#;Excel &)tp)tRegression Statistics2ultiple 3 %4/.*113 S(uare %4,1%1*Ad5usted 3 S(uare %4,*1+*Standard 6rror +14--%-*7bservations 1%A!78Adf SS MS F Significance F3egression 1 110-+40-+1 110-+40-+1 114%1+1 %4%1%-03esidual 1 1-..,4,.,* 1/%1410,/9otal 0 -*.%%4,%%%Coefficients Standard Error t Stat P-value Lower 95% Upper 95%:ntercept 014*+1-- ,14%--+1 14.0*0. %41*10* ;-,4,//*% *-*4%/-1.S(uare )eet %41%0// %4%-*0/ -4-*0-1 %4%1%-0 %4%--/+ %411,1%="e regression e()ation is:feet7 6s()are 0#0?24>-- price "o)se + =Statistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc!"ap ##-# Statistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc!"ap ##-#>Interpretation of t"e Intercept,aa is t"e estimate% a*erage *al)e of 2 ,"en t"e *al)e of 3 is 4ero 6if 3 C 0 is in t"e range of o'ser*e% 3 *al)es7Here, no "o)ses "a% 0 s()are feet, so a C ?>24>-- H)st in%icates t"at, for "o)ses ,it"in t"e range of si4es o'ser*e%, G?>,24>-- is t"e portion of t"e "o)se price not explaine% '+ s()are feetfeet7 6s()are 0#0?24>-- price "o)se + =Statistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc!"ap ##-#?Interpretation of t"e Slope !oefficient, 'b meas)res t"e estimate% c"ange in t"e a*erage *al)e of 2 as a res)lt of a one-)nit c"ange in 3Here, b C #0?-- price "o)se + =Statistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc!"ap ##-20-#907 0#0?>6200 ?>296s(ft7 0#0?> ?>29 price "o)se=+ =+ =Pre%ict t"e price for a "o)se ,it" 2000 s()are feet:="e pre%icte% price for a "o)se ,it" 2000 s()are feet is -#96G#,000s7 C G-#90Pre%ictions )sing Regression 5nal+sisStatistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc!"ap ##-2#Interpolation *s Extrapolation/"en )sing a regression mo%el for pre%iction, onl+ pre%ict ,it"in t"e rele*ant range of %ataRele*ant range for interpolation.o not tr+ to extrapolate 'e+on% t"e range of o'ser*e% 3IsStatistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc!"ap ##-22##4 Meas)res of Bariation=otal *ariation is ma%e )p of t,o parts:SSESSRSS= + ==otal S)m of S()aresRegression S)m of S()aresError S)m of S()ares =2i7 2 2 6 SS= =2i i7 2F2 6 SSE =2i7 2 2F6 SSR,"ere:C 5*erage *al)e of t"e %epen%ent *aria'le2i C &'ser*e% *al)es of t"e %epen%ent *aria'le

iC Pre%icte% *al)e of 2 for t"e gi*en 3i *al)e2F2Statistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc!"ap ##-2-SS= C total s)m of s()ares Meas)res t"e *ariation of t"e 2i *al)es aro)n% t"eir mean 2SSR C regression s)m of s()ares Explaine% *ariation attri')ta'le to t"e relations"ip 'et,een 3 an% 2SSE C error s)m of s()ares Bariation attri')ta'le to factors ot"er t"an t"e relations"ip 'et,een 3 an% 2(continued)Meas)res of BariationStatistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc!"ap ##-24(continued)XiYXYiSS9 < &Yi ; Y'*SS6 < &Yi ; Yi '* SS3 < &Yi ; Y'* ===YYY=YMeas)res of BariationStatistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc!"ap ##-29="e coefficient of %etermination is t"e portion of t"e total *ariation in t"e %epen%ent *aria'le t"at is explaine% '+ *ariation in t"e in%epen%ent *aria'le="e coefficient of %etermination is also calle% R-s()are% an% is %enote% as R2##9 !oefficient of .etermination, R21 02 Rnote:squares of sumsquares of regression12totalsumSSTSSRSSTSSER = = =Statistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc!"ap ##-2;Excel &)tp)tRegression Statistics2ultiple 3 %4/.*113 S(uare %4,1%1*Ad5usted 3 S(uare %4,*1+*Standard 6rror +14--%-*7bservations 1%A!78Adf SS MS F Significance F3egression 1 110-+40-+1 110-+40-+1 114%1+1 %4%1%-03esidual 1 1-..,4,.,* 1/%1410,/9otal 0 -*.%%4,%%%Coefficients Standard Error t Stat P-value Lower 95% Upper 95%:ntercept 014*+1-- ,14%--+1 14.0*0. %41*10* ;-,4,//*% *-*4%/-1.S(uare )eet %41%0// %4%-*0/ -4-*0-1 %4%1%-0 %4%--/+ %411,1%9>0>J of t"e *ariation in "o)se prices is explaine% '+ *ariation in s()are feet09>0>2-2;009000#>?-4?-4>SS=SSRr2= = =Statistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc!"ap ##-2 >Y>>Yresi%)alsresi%)alsStatistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc!"ap ##--4Resi%)al 5nal+sis for In%epen%ence!ot :ndependent:ndependentXXresi%)alsresi%)alsXresi%)alsStatistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc!"ap ##--9Inferences 5'o)t t"e Slope="e stan%ar% error of t"e regression slope coefficient 6b7 is estimate% '+= =2iYX YXb) (XS SSXSSX,"ere:C Estimate of t"e stan%ar% error of t"e least s()ares slope C Stan%ar% error of t"e estimatebS2 nSSES23=Statistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc!"ap ##--;Excel &)tp)tRegression Statistics2ultiple 3 %4/.*113 S(uare %4,1%1*Ad5usted 3 S(uare %4,*1+*Standard 6rror +14--%-*7bservations 1%A!78Adf SS MS F Significance F3egression 1 110-+40-+1 110-+40-+1 114%1+1 %4%1%-03esidual 1 1-..,4,.,* 1/%1410,/9otal 0 -*.%%4,%%%Coefficients Standard Error t Stat P-value Lower 95% Upper 95%:ntercept 014*+1-- ,14%--+1 14.0*0. %41*10* ;-,4,//*% *-*4%/-1.S(uare )eet %41%0// %4%-*0/ -4-*0-1 %4%1%-0 %4%--/+ %411,1%0.03297 Sb =Statistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc!"ap ##--