STA134 Complete Note PDF

EDWARD CARES STA134- INTRODUCTI ONTOSTATI STI CALI NFERENCEI COURSEOUTLI NE: Modul e1 Ti meser ies Modul e2 Demo...

EDWARD CARES STA134- INTRODUCTI ONTOSTATI STI CALI NFERENCEI COURSEOUTLI NE: Modul e1 Ti meser ies Modul e2 Demogr aphi cmeasur es Modul e3 I ndexnumber s Modul e4 I nfer ence( Est imat ionandHy pot hesi s) Modul e5 Regr essi onandCor rel ati onAnal ysi s Modul e1 TI MESERI ES Def ini ti on Timeser iesisasequenceofdata( obser vat ions)takenatequal int erv aloft imer ecor dedov era peri odofti mee.ghour l y,dai l y,weekl y,monthl y,annual ly, etc EXAMPLESOFTI MESERI ESDATA i. Dai l ymi l kpr oduct ionbyadi aryi ndust ryi nthemont hofMay ,2020 i i. Hour lyt emper atur edecl aredbyt heweat herf orecastbur eaui nNi ger iai nJanuar y 2021 i i i. Quar ter lypr oduct ionofcocoa( tons)i nOy oSt atebet ween1979- 2020 i v. Theunempl oymentst ati sti cs( fi gur es)i nNi ger ia, 2000–2021 v. Themont hlyr ainf all sinKwar aSt atei n2020 v i. Theannual product ionofst eel inOsogboSt eel rol l ingCompanyov ert hel astt went y year s v ii. Annual product ionoft etr a-oxosul phat e(I V)Oxi debyachemi cal indust ryi nRussi a si nce1970 1  EDWARD CARES COMPONENTS( CHARACTERI STI CS)OFTI MESERI ES Ther ear efourcomponent stoat imeser iesdat a,v iz: i. Secul armov ementorsi mpl yTr end( T) i i. Cy cli cal var iat ion( C) i i i. Seasonal var iat ion( S) i v. I rr egul arv ari ati on( I) Not e: (1)t hewor dsvar iat ionandmov ementcanbeusedi nter changeabl y,t heyconnot ethe same meani ng ( 2)Cycl ical andseasonal var iat ionsar eof tencal l edshor tter mfl uct uat ionsorper iodi c movements SECULARMOVEMENT( TREND) Secul armov ementi sthesmoot hlong- ter m mov ementort hedi rect ionoft het imeser ies.I t I susual lyil l ust rat edbyat rendcur ve( trendl i ne).Theconst ruct ionoft imepl oti sapr eli minar y step Whi chr eveal sorhel pst odet ectt hepat ter ninher enti tti meser iesdat aort henat ureofat ime Ser ies.Wecant alkoft hel ong- ter m mov ement( trend)ofsal es, product ion, empl oyment ,shar e pr ices, etc 2  EDWARD CARES 3  EDWARD CARES USESOFSECULARMOVEMENTORTREND( T) i. Itr eveal sthegeneral ideaaboutthepatt ernofbehav iorofthephenomenonunder consi derat ion.Thi shelpsi nbusinessfor ecasti ngandplanningoff utur eoperat ions i i. Itf aci li tat est hecompari sonoftr endsoft imeseri esov erdi ff erentper iodsoft ime andtodrawi mport antorvi tal concl usi on( s)aboutthem. i i i. Byisol ati ngthetrendv al uesf rom t hegi venti meser ies, wecanst udyt heshor tter m andthelong- ter mv ar iati onsi ntheti meseri es. SHORTTERM FLUCTUATI ONS( PERI ODI CMOVEMENTS) Theshor t- ter mfl uct uat ionshav ebeencl assi fi edi ntot wocat egor ies, viz: i. Seasonal var iat ion( S) i i. Cy cli cal var iat ion( C) CYCLI CALVARI ATI ON( C) Cycl icalvar iat ionreferstotheriseandf al lofthetimeseri esoveraperiodl ongerthanone year.Ther i seandf al lofthet i meser i esar eoftenref err edtoas“Oscil lations”or“Swings” movement saboutthet r endline.Cycli caldispl aysorshowsatypical busi nesscy cl eswhich comprisesofaper i odofpr osperi ty(economi cboom)foll owedbyaper iodofr ecessi on, depr essionandr ecovery. pr osper it y TYPI CALBUSI NESSCYCLE Recessi on Recov ery Depr essi on USESOFCYCLI CALVARI ATI ON i. I thel psbusi nessexecut ivei nthef ormul ati onofpol i cy( str ategi cpl anni ng) i i. Ithel psgov ernmenttotakeproacti vemeasuresthatwil lprev entdeteri orati onof mildrecessi onorear l yeconomi crecessi oni ntodeeprecessi on(depressi on)and keeptheswi ngsofprosperi tywithi nmeasurabl elimit swi t houtr esulti ngt ostor ming speculat ions( pr opaganda) SEASONALVARI ATI ON( S) Seasonalv ari ati onrefer stothepat ter nofchangeint imeseri eswit hinoneyear.Seasonal var iati onsoccuri nregul arandperi odi cmanneroveraspanofnotl essthanoneyearora 4  EDWARD CARES per iodof12monthsandhav ealmostt hesamepatter nyearaft eryear.Itwi l lint eresty outo not ethatpar tofthetimeseri esrel ati ngtobusi nessandeconomiccyclesareoft enbeing i nfl uencedbyseasonalf orces.Sal esandpr ofit sinDepar tmentalst ores Seasonal var iat ionshav etwomai ncauses, viz: i. Cl i mat eint hewi destsenseofi t i i. Cust oms CLI MATE Cli mateinf luencest heti mi ngoffar mi ng, useofallsor tsofwoolen(cl othi ng) ,umbr ell a,r ain coats, rai nboots,et c.Al lthisexper ienceseasonal mov ementduetocli mate. CUSTOMS Customssuchashabit s,fashions,andconventi onsofpeopleinthesociet yal socause seasonal vari ati onsi nthedemandsf ort oys,whi chsuddenlyincreasej ustbef oreChr ist mas, the demandsforramssuddenlyincreaseduri ngEld- kabi rcelebrati on. i. Toest abl ishtheseasonalpatter nwhichhelpsi nplanningfut ureoperat ionsand for mul ati ngpol i cydeci sionregardi ngpur chase,producti on,i nventor ycontr ol, personnel r ecrui tment, sell ingandadvert isementprogrammeset cduri ngvari ous partofthey ear. I RREGULARVARI ATI ON( I) I rr egul arv ari ati oni sof tenr efer redt oast heov eral lfl uct uat ionsorst ochast icv ari ati on. Manyanal yst spr efert ocat egor izei rr egul arv ari ati oni ntot wosub- component s,v iz: i. Epi sodi cfl uct uat ion i i. Resi dual fluct uat ion EPI SODI CFLUCTUATI ON Episodi cfl uct uati onsareduetooccasi onali nfl uencesthatmayoccurj ustonceorsev eral ti mes.Theyareusual l yunpredi ctabl ebutcanbeidenti fi ede.gini ti ali mpact(s)ont he 5  EDWARD CARES economyofmaj orst ri kes, floods, fir eout breaks, war s,t sunami ,fami ne, epi demi c,pandemi c, etc Afterepisodicfluct uati onshav ebeenel iminat edorr emoved,t heremai ningv ari ati oni s knownasr esidualvari ati on(chancevar iat ion).Ther esi dualv ari ati onsar eneit her predict ablenori denti fi able. BASI CMATHEMATI CALMODELSOFTI MESERI ES Weassumet het imeser iesdef inedbyt hev aluesyy, 1,2 …y n ofav ari abl eYt( sal esatt ime tt, 1,2 …t n respect ivel y) y t isaf unct ionofobser vat ionsatt imet Sy mbol i cal l yiti swr it tenasy=f (t) Themodel coul dbe Addi ti ve, y=f (t)=T+C+S+I Mul ti pli cat ive, y=f (t)=T*C*S*I ANALYSI SOFTI MESERI ESMODELS Thi sisai medati nvest igat ingt hef act ors, namel y: - Tr end( T) - Cy cli cal var iat ion( C) - Seasonal var iat ion( T) - I rr egul arv ari ati on( I) Theanaly sisofti meseri esisof tenref err edt oasthedecomposi ti oni ntoi tsbasi c const it uentsorcomponentv ari ati onsormov ements. ESTI MATI ONOFTREND( T) Thef oll owi ngmet hodscanbeusedt oest imat etr end: 1.Met hodofmov ingav erage( M.A) 2.Met hodofsemi -av erage( S.A) 3.Leastsquar esmet hod( L.S. ) 4.Fr ee-handmet hod( F.H) METHODOFMOVI NGAVERAGE( M.A) 6  EDWARD CARES M.A.invol vestheusingofanappr opri ateorder (n)sothatCy cli cal ,Seasonal ,andIr regul ar var iati onsmaybeel i minatedf rom t heti meser i es.Thus,l eav ingbehi ndonlytheSecular mov ement( trendorl ong-t erm movement). Exampl esonmov ingAv erage( M.A) 1.Letyy…y bet 1,2, ,n i meser iesobser vat ions( readi ngs). Obt ain( i)M. A(3)( ii )M. A(2)( ii i)M. A(4) ( i)TheM. A.ofor der3i sgi venas y+y+y y+y+y y+y+y M. A( = 1 2 3,2 3 4,3 4 5, 3) et c 3 3 3 ( ii ) TheM. A.ofor der2i sgi venas y t M. T(2 C. M.T( 2) C. M.A( 2 ) ) Y 1 Y1+ Y2 Y Y1+2Y2+y 3 A/ 4 2 =A Y2+ Y3 Y Y2+2Y3+y 4 B/ 4 3 =B Y3+ Y4 Y Y3+2Y4+y 5 C/ 4 4 =C Y4+ Y5 Y Y4+2Y5+y 6 D/ 4 5 =D 7  EDWARD CARES Y5+ Y6 Y Y5+2Y6+y 7 E/ 4 6 =E Y6+ Y7 Y 7 Not ethat: M.Ti sMov ingTot alandC.M.Ai stheCent eredMov ingAv erage y t M. T(4) C. M.T( 4) C. M.A( 4) Y1 Y2 Y1+Y2+y 3+y4 Y3 Y1+2Y2+2y 3+2y 4+y5=E E/ 8 Y2+Y3+y 4+y5 Y4 Y2+2Y3+2y 4+2y 5+y6=F F/ 8 Y3+Y4+y 5+y6 Y5 Y3+2Y4+2y 5+2y 6+y7=G G/ 8 Y4+Y5+y 6+y7 Y6 Y4+2Y5+2y 6+2y 7+y8=H H/ 8 Y5+Y6+y 7+y8 Y7 Y8 EXERCI SE1:Gi vent het imeser iesdat aonpr oduct ionofr ice( tons)i ntheTabl e1bel ow Year Pr oduct ion( tons) M. T(3) M. A(3) 2009 40 2010 36. 5 119. 8 39. 9 8  EDWARD CARES 2011 43. 3 124. 3 41. 43 2012 44. 5 126. 7 42. 23 2013 38. 9 121. 5 40. 5 2014 38. 1 109. 6 36. 53 2015 32. 6 109. 4 36. 47 2016 38. 7 113. 0 37. 67 2017 41. 7 121. 5 40. 5 2018 41. 1 116. 6 38. 87 2019 33. 8 EXERCI SE2:Gi vent het imeser iesdat aonpr oduct ionofr ice( tons)i ntheTabl e2bel ow Year Pr oduct ion( tons) M. T(2) C. M.T( 2) C. M.A( 2) M. T(4) C. M.T( 4) C. M.A( 4) 2009 40 2010 36. 5 76. 5 156. 3 39. 075 2011 43. 3 79. 8 167. 6 41. 9 164. 3 327. 5 40. 9375 2012 44. 5 87. 8 171. 2 42. 80 163. 2 328. 0 41. 0000 2013 38. 9 83. 4 160. 4 40. 10 164. 8 318. 9 39. 8625 2014 38. 1 77. 0 147. 7 36. 925 154. 1 302. 4 37. 8000 2015 32. 6 70. 7 142. 0 35. 50 148. 3 299. 4 37. 4250 2016 38. 7 71. 3 151. 7 37. 925 151. 1 305. 2 38. 1500 2017 41. 7 80. 4 163. 2 40. 80 154. 1 309. 4 38. 6750 2018 41. 1 82. 8 157. 7 39. 425 155. 3 2019 33. 8 74. 9 EXERCI SE3:Gi vent het imeser iesdat aonpr oduct ionofr ice( tons)i ntheTabl e3bel ow 9  EDWARD CARES Year Pr oduct ion( tons) M. T(5) M. A(5) 2009 40 2010 36. 5 2011 43. 3 203. 2 40. 64 2012 44. 5 201. 3 40. 26 2013 38. 9 197. 4 39. 48 2014 38. 1 192. 8 38. 56 2015 32. 6 159. 3 31. 86 2016 38. 7 192. 2 38. 44 2017 41. 7 187. 9 37. 58 2018 41. 1 2019 33. 8 WEI GHTEDMOVI NGAVERAGE( W.M. A) Gi vent hesequenceofobserv ations2, 6,1, 5,3, 7,2i fthewei ght s1, 4and1ar eused, obt aint he wei ght edMovingAverageoforder3 Tabl e4 y t W.M. T(3) W. M.A( 3) y 1=2 y 2=6 ( y1* w1+y 2*w2+y 3*w3)/( w1+w2+w3) = 27/ 6 4. 5 y 3=1 ( y2* w1+y 3*w2+y 4*w3)/( w1+w2+w3) =15/ 6 2. 5 y 4=5 ( y3* w1+y 4*w2+y 5*w3)/( w1+w2+w3) =24/ 6 4. 0 y 5=3 ( y4* w1+y 5*w2+y 6*w3)/( w1+w2+w3) =24/ 6 4. 0 y 6=7 ( y5* w1+y 6*w2+y 7*w3)/( w1+w2+w3) =33/ 6 5. 5 y 7=2 DEMERI TSOFMOVI NGAVERAGE 10  EDWARD CARES - Dat aar elostatt hebegi nni ngandendoft imeser ies. f oroddor derofM.A, (n- 1)/ 2dat aar elostbot hatt hebegi nni ngandend f orev enor derofM. A,( n-1)dat aar elostbot hatt hebegi nni ngandend - M.A.maygener atenewcy clesormov ement snotpr esenti ntheor igi naldat a - M.Aar eser iousl yaf fect edbyt hepr esenceofout li ersorext remev alues - I tisnotusual lysui tabl eforf orecast ingorpr edi cti on - Thesel ect ionofappr opr iat eor derf orM. Aisusual lybyt her uleoft humb METHODOFSEMI -AVERAGE Pr ocedur es i. Separ ateordi vi det het imeser iesdat aint otwoequalpar ts. i i. Cal cul atet heav erageofeachpar tsepar atel y,t husobt aini ngt wopoi ntsont hegr aphoft he t imeser ies. EXAMPLE Consi dert het imeser iesdat aont her icepr oduct ion( tons) Obt aint heSemi -av erage, hencedet ermi net her ateofi ncr easeordecr easeoft het rendv alues Sol uti on( Tabl e5) Year 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 pr oduct ion 40 36. 5 43. 3 44. 5 38. 9 38. 1 32. 6 38. 7 41. 7 41. 1 33. 8 Par t1, Tot al=203. 2 Omi tt ed Par t2, Tot al=187. 9 Av erage1=40. 64 Av erage2=37. 58 40. 64 dy 40. 64- 37. 58 M=Sl ope( Rat eofdownwar dtr end) = =0. 51 dx 2017- 2011 37. 58 2011 2017 11  EDWARD CARES Fi gur e2:Gr aphshowi ngt her ateofdownwar dtr end MERI TOFSEMI -AVERAGEMETHOD - I tissi mpl etoappl y(anal yze) DEMERI TSOFSEMI -AVERAGEMETHOD - I tleadst opoorr esul tswhenusedi ndi scr imi nat ely - I tisappl icabl etoonl yli neart rendorappr oxi mat elyl ineart rend. FREEHANDMETHOD I nt hi smet hodtheorigi naldataareplot tedi nagraph,t henasmoot hedl inei sdrawnsubject ivel y,whi chi n theopini onoft heStati sti ciandescr ibesont hegraphthegrowt hfactori nvol ved.Thef oll owingsali ent point sarer el evantandmustbeensur edwhenusingf reehandmethod: i. t henumberofpoi ntsabov eandbel owt hesmoot hed/ trendl i near eequal i i. thet otalofthevert ical devi ati onsofthedat apoint sabov ethet rendl i nei sequal tot het otal of t hevert icaldevi ati onsofthedatapoi ntsbel owthetr endl ine i i i t hesum ofsquar esoft hev ert ical dev iat ionsoft heobser vat ions( dat apoi nts)f rom t het rendl i ne i sa mini mum Adv ant agesoff reehandmet hod i. I tist hesi mpl est ,qui ckestandeasi estmet hodofest imat ingt het rend i i. I tisf lexi blet hant her igi dmat hemat ical funct ions/ model s,hencef it sthecur vemor ecl osel yto t hedata Di sadv ant agesoft hef reehandmet hod i. I tishi ghlysubj ect ive; sincenot woper sonsar eli kel ytodr awt hesamet rendl i ne/ smoot hedl i ne f orthesamet i meseriesdat a. i i. Theuser smusthaveat hor oughunder standi ngoft heeconomi cbackgr oundoft hepar ti cul ar ti meseri esv ari able LEASTSQUARESMETHOD Thiscanbeusedt oobtai ntheequati onofanappropri atetr endline/ curvefrom whi chwecancomput e thetr endli nevalues/pr edict edvaluesandmakeforecast.Theleastsquaresmethodmini mi zestheerror sum ofsquar esi.eitmini mizesthesum oft hedeviat ionsbetweentheplott edpoint s(ori ginaldata)and thecomputedtr endv al ues Weassumeal i neart rendoft hef orm 12  EDWARD CARES Wher e i sthet imeser iesobser vat ioni nti met a, ist hei nter ceptony -axi s b,i sthesl ope/ regr essi oncoef fi cientoft imet i ,st her andom er rort erm Est imat orsofaandb , usual l yt=x, ist hei ndependentv ari abl e r espect ivel y TheTabl e6bel owi sdat aonpr oduct ionofcoal (1948- 1958)f itt heLSt rendequat ion Year T 1948 40 1 1 40 1949 36. 5 2 4 73 1950 43. 3 3 9 129. 9 1951 44. 5 4 16 178 1952 38. 9 5 25 194. 5 1953 38. 1 6 36 228. 6 1954 32. 6 7 49 228. 2 1955 38. 7 8 64 309. 6 1956 41. 7 9 81 375. 3 1957 41. 1 10 100 411 1958 33. 8 11 121 371. 8 Tot al ∑ y t =429. 2 ∑ t =66 ∑2 t=506 ∑t y=2539. 9 , Tabl e7 Year T 1948 40 1 40. 6227 1949 36. 5 2 40. 3018 1950 43. 3 3 39. 9809 13  EDWARD CARES 1951 44. 5 4 39. 66 1952 38. 9 5 39. 3391 1953 38. 1 6 39. 0182 1954 32. 6 7 38. 6973 1955 38. 7 8 38. 3764 1956 41. 7 9 38. 0555 1957 41. 1 10 37. 7346 1958 33. 8 11 37. 4137 1959 - 12 37. 0928 1960 - 13 36.7719 Note: thef orecastar einr edink whil ethepr edi ctedar einbl acki nk Al ter nat ivel y,wecanobt aint het rendusi ngt hef ormul abel ow , sucht hatweassume, ( val i dassumpt ionwhenusi ngt his f ormul a) Tabl e8 Year t =x x2 Xy 1948 40 - 5 25 - 200 14  EDWARD CARES 1949 36. 5 - 4 16 - 146 1950 43. 3 - 3 9 - 129. 9 1951 44. 5 - 2 4 - 89 1952 38. 9 - 1 1 - 38. 9 1953 38. 1 0 0 0 1954 32. 6 1 1 32. 6 1955 38. 7 2 4 77. 4 1956 41. 7 3 9 125. 1 1957 41. 1 4 16 164. 4 1958 33. 8 5 25 169 Tot al 429. 2 0 110 - 35. 3 Mer it sofLeastsquar esmet hod - Iti samat hemat ical met hodmeasur ingt rendf reef rom subj ect iveopi nion/ out li ers - Itgivesli neofbestf itt otheti meseri esdatai.ei tmi nimi zest hesum ofsquar eddev iat ionsbet weent he ori ginaldataandt heesti matedtr endval ues -Nodatapoi nti slosthencewecancomput ethet rendv aluesf oral lthegi vent imeper iodi nthet ime ser ies - Itcanbeusedf orf utur epr edi cti on( for ecast)atanyper iodt - Iti stheonl ytechni quewhi chenabl esust oobt aint her ateofgr owt hperannum f ory ear lydat aincaseof l i neartr end. Li mit ati onsofLeastsquar esmet hod - Ifasi ngl eobser vat ioni saddedt othet imeser ies, thenaf reshcal cul ati onbecomesnecessar y - Comput ati on/ cal cul ati oni stedi ousandt imeconsumi ng - Fut urepr edi cti ons/ for ecastbasedonl eastsquar esmet hodi gnor est hecy cli cal ,seasonal andi rr egul ar v ari ati ons - LeastSquar esmet hodi snotappl i cabl etononl i neart imeser ies SEASONALVARI ATI ON Def ini ti onofseasonal indexnumber s Seasonal indexnumber sar enumber sshowi ngr elat ivev aluesofav ari abl edur ingmont hs/quar ter sint he year. Note:Theaver ageseasonali ndexf orthewhol eyearshoul dbe100%.Thi simpl i est hatt hesum oft he seasonali ndexnumbersshoul dbe1200% Met hodsofEst imat ingSeasonal Index( Seasonal Var iat ion)ofat imeser ies, Viz: 15  EDWARD CARES i. av erageper cent agemet hod i i r ati otomov ingav eragemet hod i ii Rat iot otr endmet hod i v l inkr elat ivemet hod Weshal ll imi tour sel vest o(i -i ii ) Est imat ionofSeasonalVar iat ionUsi ngAv eragePer cent ageMet hod LEARNI NGBYEXAMPLES Pr ocedur es - expr esst hedat aforeachmont hasper cent agef ort hey ear - Obt aint heav erage( mean)oft heper cent ageoft hecor respondi ngmont hfordi ff erenty ear s Ther esul ti ngper cent agesgi vet heseasonal indi ces Note:t hatwhentheaverage( /mean)i snotequal to100%,i tmeanst hatt hesum oftheper cent agesi s notequalt o1200%.Thisimpl iest hatt her eisneedtoadj ustforseasonal i tybymulti plyi ngt hemonthl y meansbyasui tabl efact orof wher e i sthet otal oft hemont hlyav erageper cent ages Whi chgi vest headj ust edseasonal indi ces(i ncaseofmont hlydat aov ery ear s) 2.Whent imeser iesi nvol vesquar ter lydat a,t hesum oft hequar ter lyav eragesshoul dbeequal to4 Ot her wisey oumul ti plybyasui tabl efact orof wher e i sthet otal oft hequar ter lyav erages Example1.Obt aintheSeasonalI ndexf ort hedat aonsal es( #M)ofaf ir m 2015- 2018usi ngav erage percent agemethod.SeeTable9bel ow year Jan Feb March Apri l May June Jul y Aug Sept Oct Nov Dec Total 2015 92 90 92 94 96 98 98 108 106 112 102 88 1176 2016 90 92 92 96 98 102 104 108 104 110 106 90 1192 2017 92 90 96 94 102 104 110 116 120 108 106 92 1230 2018 94 96 100 96 104 100 9 108 110 114 98 86 1204 Tot al 368 368 380 380 400 404 410 440 440 444 412 356 4802 Monthl y Av. 92 92 95 95 100 101 102. 5 110 110 111 103 89 1200. 5 S. I 91. 96 91. 96 94. 96 94. 96 99. 96 100. 96 102. 46 109. 96 109. 96 110. 96 102. 96 88. 96 1200. 02 , , , , , Exampl e2.Cal cul atet heSeasonalI ndexf ort hedat aont hequar ter lysal esofToy sint ernat ional ($M) 16  EDWARD CARES Usi ngRat iot omov ingav eragemet hod Year Wi nter Spr ing Summer Ful l 2013 6. 7 4. 6 10. 0 12. 7 2014 6. 5 4. 6 9. 8 13. 6 2015 6. 9 5. 0 10. 4 14. 1 2016 7. 0 5. 5 10. 8 15. 0 2017 7. 1 5. 7 11. 1 14. 5 2018 8. 0 6. 2 11. 4 14. 9 Pr ocedur es - obtainthe4y earmovingt ot al - Obt ai nt he2-y earcenteredmov i ngt ot al - Obt ai nt he4-y earcenteredmov i ngAv erage - CalculatetheSeasonalIndex(S. I) Note3:Whent hesum oft hequart erlyaveragesisequalt o4,theresult ingv aluesaretheS.I Otherwiseyouadjustforseasonal itybymul ti ply ingwithasui tabl ef act ormentionedear li er Sol uti onTabl e10 C( 1) C( 2) C( 3) C( 4) C( 5) C( 6) C( 7) 2- yr-C- Year Quar ter Sal es( $M) 4- yrMT MT 4- yr.C.M. A S. I=C( 3)/ C(( 6) 2013 Wi nter 6. 7 Spr ing 4. 6 34 Summer 10. 0 33. 8 67. 8 8. 475 1. 1799 Ful l 12. 7 33. 8 67. 6 8. 45 1. 503 2014 Wi nter 6. 5 33. 6 67. 4 8. 425 0. 7715 Spr ing 4. 6 34. 5 68. 1 8. 5125 0. 5404 Summer 9. 8 34. 9 69. 4 8. 675 1. 1297 Ful l 13. 6 35. 3 70. 2 8. 775 1. 5499 2015 Wi nter 6. 9 35. 9 71. 2 8. 9 0. 7753 Spr ing 5. 0 36. 4 72. 3 9. 0375 0. 5533 Summer 10. 4 36. 5 72. 9 9. 1125 1. 1413 Ful l 14. 1 37 73. 5 9. 1875 1. 5347 2016 Wi nter 7. 0 37. 4 74. 4 9. 3 0. 7527 Spr ing 5. 5 38. 3 75. 7 9. 4625 0. 5812 Summer 10. 8 38. 4 76. 7 9. 5875 1. 1265 Ful l 15. 0 38. 6 77 9. 625 1. 5584 2017 Wi nter 7. 1 38. 9 77. 5 9. 6875 0. 7329 Spr ing 5. 7 38. 4 77. 3 9. 6625 0. 5899 Summer 11. 1 39. 3 77. 7 9. 7125 1. 1429 Ful l 14. 5 39. 8 79. 1 9. 8875 1. 4665 17  EDWARD CARES 2018 Wi nter 8. 0 40. 1 79. 9 9. 9875 0. 801 Spr ing 6. 2 40. 5 80. 6 10. 075 0. 6154 Summer 11. 4 Ful l 14. 9 Tabl e11: Toascer tai ntoadj ustf orseasonal i ty Year Wi nter Spr ing Summer Ful l 2013 - - 1. 1799 1. 5030 2014 0. 7715 0. 5404 1. 1297 1. 5499 2015 0. 7753 0. 5533 1. 1413 1. 5347 2016 0. 7527 0. 5812 1. 1265 1. 5584 2017 0. 7329 0. 5899 1. 1429 1. 4665 2018 0. 8010 0. 6154 - - S. f=0. 9977 Tot al 3. 8334 2. 8802 5. 7203 7. 6125 Mean 0. 7667 0. 5760 1. 1441 1. 5225 4. 0093 Adj ust ed 0. 7649 0. 5747 1. 1415 1. 5190 4. 0001 S. I Rat iot otr endmet hodt hepr ocedur esar e: - Obtainthetrendusingtheleastsquaresmethod - Determinether ateofi ncrement/decrement - Basedonst ep2abov egenerat ethetrendvalues - Expressquar t erlyval uesaspercentageofthetrend - Veri fyiftheresulti ngvaluesareS.I Example3 Cal cul atet heS. Iusi ngr ati otot rendmet hodf ordat aonpr oduct ioni n(000)t onesofcoal sbyaf ir m Year I I I I II I V Tot al Av erage 2014 30 40 36 34 140 35 2015 34 52 50 40 180 45 2016 40 58 54 48 200 50 2017 54 76 68 62 260 65 18  EDWARD CARES 2018 80 92 86 82 340 85 SOLUTI ON Fi tthet rendl i neequat ion, , wher e and r esp, sinceweassumet hat , Thef it tedt rendbecomes X Year Year lyt otal Av.Q.v alue=Y t =X XY Fi tt edt rend 201