Measurement of Competitive Balance in Conference and Divisional Tournament Design # Liam J. A. Lenten * School of Economics and Finance


 Eugene Palmer
 1 years ago
 Views:
Transcription
1 Measuremet of Competitive Balace i Coferece ad Divisioal Touramet Desig # Liam J. A. Lete * School of Ecoomics ad Fiace La Trobe Uiversity Abstract The coferece ad divisioal system has log bee a staple part of touramet desig i the major prosports leagues of North America. This popular but highlyrigid system determies o how may occasios all bilateral pairigs of teams play each other durig the seaso. Despite the virtues of this system, it ecessitates removig the biases it geerates i the set of wiratios from the regular seaso stadigs prior to calculatig withiseaso measures of competitive balace. This paper applies a modified versio of a recet model, a extesio that is geeralizable to ay ubalaced schedule desig i professioal sports leagues worldwide, to correct for this iheret bias for the NFL over the seasos , the results of which suggest the NFL is eve more competitively balaced tha thought previously. JEL Classificatio Number: C4, L83 Keywords: Competitive Balace, Measuremet Methods # Research assistace o this paper was provided by La Nguye. A earlier versio of this paper was preseted at a semiar at Temple Uiversity, Philadelphia, 3 September 200. The author would like to thak the participats of the semiar for their commets ad suggestios, especially Michael Bogao. The author would also like to thak Damie Eldridge, Stefa Késee ad Markus Lag for their iput. * Cotact details: School of Ecoomics ad Fiace, La Trobe Uiversity, Victoria, 3086, AUSTRALIA. Tel: , Fax:
2 . Itroductio The coferece ad divisioal system has log bee cosidered the default touramet desig i the Natioal Basketball Associatio (NBA), Natioal Football League (NFL), Major League Baseball (MLB) ad Natioal Hockey League (NHL), referred to heceforth as the major leagues. The practice is also stadard i other less popular North America leagues, such as Major League Soccer (MLS) ad the Caadia Football League (CFL). Fudametally, the coferece ad divisioal system is a maifestatio of ubalaced schedules, whereby ot all bilateral pairigs of teams play each other a equal umber of times. Other forms of ubalaced schedules exist elsewhere, like where the ubalacedess is largely radomized, for example, the Australia Football League (AFL). Aother alterative is powermatchig like i Scottish Premier League (SPL) soccer, where teams play a fourth time oly teams that fiished i the same half of the stadigs followig three full roudrobis. Other forms are eve more radical such as the Belgia League format from 200, ivolvig may playoff combiatios after two full roudrobis, producig a seaso whereby all teams do ot play the same umber of games (a ueve schedule). Despite these isolated examples, ubalaced schedules are rare i Europea soccer see Cai ad Haddock (2005) for a historical accout of TrasAtlatic sports idustry differeces, icludig touramet desig. There are umerous motivatios i favor of the system. Logistically, it ca be used to coordiate ad miimize travel distaces for teams. Ecoomically, the system fosters local rivalries ad arguably icreases aggregate demad, measured typically by The year refers geerally to the caledar year i which the regular seaso cocludes. 2
3 televisio ratigs or attedaces. Paul (2003) ad Paul, Weibach ad Melvi (2004) demostrate this poit, although the MLB results of Butler (2002) suggest that fas are kee to see their team agaist other teams that they have ot played agaist i past seasos. Notwithstadig these merits, it is far from perfect may fas do ot like seeig their team miss out o a playoff spot to a team i aother divisio with fewer wis. There is also the pereial dilemma of teams havig already cliched a coferece seedig or playoff spot losig games they should otherwise wi due to purposely restig key players i the last few regular seaso games, idirectly ifluecig the playoff destiy of others (see Blodget, 200). More problematically, ubalaced schedules cause biases i fial regular seaso team stadigs, sice they do ot cotrol for the stregth of schedule, rederig ivalid stadard measures that sports ecoomists use to characterize competitive balace i sports leagues (see Lete, 2008 ad 200, for the SPL ad AFL, respectively). This study applies a modified adjustmet procedure to idetify the extet to which stregth of schedule affects competitive balace measures i coferece ad divisioal systems, based o a simple matrix algebra method. This issue is aalogous to the ogoig idustrial orgaizatio debate o the appropriateess of stadard quatitative measures of market cocetratio to accurately reflect true competitive structure. The focus is o NFL data specifically, ad raises a umber of empirical questios that pertai to the uaces of the coferece ad divisioal system as a ubalaced schedule. Firstly, this is a critical ogoig policy issue for major league touramet desig the NHL altered the compositio of its schedule i advace of the 200 seaso, ad as the remaiig leagues cosider expasio from 30 to 32 teams. More geerally, it is difficult to speculate for major league data whether 3
4 adjustig the league stadigs will affect reported competitive balace measures sigificatly. We are also iterested i whether the teamquality equalizatio aspect of the NFL fixture, uique to the major leagues, is successful i makig the touramet more eve. There is also the possibility that some teams (or eve divisios) fare better or worse tha others i the fixture systematically because of the fixed allocatio of frachises to cofereces ad divisios over seasos. The remaider of the paper proceeds accordig to the followig structure: sectio 2 summarizes the relevat literature, while sectio 3 illustrates the stregth of schedule adjustmet methodology. Sectio 4 outlies alterative ubalaced schedule formats used i the various major leagues. The results are preseted ad the fidigs discussed extesively i sectio 5, followed by a brief coclusio i sectio Literature Review The uiversal applicatio of the coferece ad divisioal system across the major leagues has made it the attetio of substatial previous research, though ot quite to the extet imagied. Give the largely North America cotext of the problem, much of this work ceters o team ratigs i ubalaced schedules Faimesser, Fershtma ad Gadal (2009) is a recet example with respect to the highly idiosycratic NCAA College Football. Lete (200) lists various studies utilizig a rage of statistical methods for this purpose, as well as other studies that relate to the more geeral ecoomic implicatios of touramet desig for the professioal sports idustry. Of uderlyig relevace to sports ecoomists ad statisticias is the competitive balace paradigm, which exteds back to Rotteberg s (956) baseball players labor market study. Competitive balace is defied as the degree of parity i sports leagues, ad 4
5 helps describe the ucertaity of outcome hypothesis a corerstoe of sports ecoomics. Sice demad for sport is expected to be greater whe the league is competitively balaced, this represets a rare example i the ecoomics disciplie i which ucertaity is associated with higher utility. Cosequetly, i sports ecoomics there is a large volume of literature comparig leagues (major ad other), such as Quirk ad Fort (992), ad Vrooma (995 ad 2009), to assess relative effectiveess of labor market ad reveuesharig policies used by leagues to maitai/improve competitive balace. Sigleleague timeseries studies, such as Lee ad Fort (2005) ad Fort ad Lee (2007), are also useful for the same reaso. Lee (200) follows a similar directio for the NFL specifically, yieldig the coclusio (recetly topical agai) that the 993 collective bargaiig agreemet (CBA) improved competitive balace. Examiig whether adjustig for ubalaced schedules affects competitive balace measures is motivated largely by its possible implicatios for these studies, sice they ivariably use these commo measures for the purposes of testig. Therefore, there is a questio of how sesitive their coclusios are to ay possible biases i these measures. O these coclusios, umerous ifluetial modelig studies o policy effectiveess, such as Fort ad Quirk (995) ad Vrooma (995), argue that the Coasia ivariace priciple holds i leagues comprised of profitmaximizig teams. That is, free agecy competitive balace is ivariat to the itroductio or removal of labor market ad reveuesharig policies. However, other otable studies preset a differet view Depke (999) fids some (albeit mixed) empirical evidece that the removal of the reserve clause i MLB (freeig up the players labor market), resulted i a 5
6 deterioratio i competitive balace. Meawhile, it is wellkow that these itervetioist policies ted to be effective i leagues of wimaximizig teams. Sice the major leagues do ot ted to be modeled i this way, several factors (other tha team objectives) will collectively determie the (i)effectiveess of such policies. I the case of reveue sharig, these factors iclude the: (i) specificatio of team reveue fuctios (icludig or excludig absolute quality of competitio); (ii) sharig arragemet (atthegate or pooled; equal or performacerelated); ad (iii) equilibrium coditio (fixedsupply Walras or Nash cojectures). O these lies, Késee (2000) cocludes that reveue sharig ca improve competitive balace eve if teams are profit maximizers, while Szymaski ad Késee (2004) demostrate that the cotrary could occur uder very differet coditios. Moreover, Dietl, Lag ad Werer (2009), show that reveue sharig improves competitive balace i certai types of leagues cotaiig (heterogeeous) teams with mixed objectives. 3. Methodology We use a modified versio of the Lete (2008) team ubalaced schedule league model where team i is scheduled to play some other teams k times ad all remaiig teams k + times. This model has the advatage of treatig oly the ubalacedess of the schedule as edogeous, while other factors typically icluded i team ratig models are exogeous. This allows us to idetify purely the effect arisig from the policy uder ivestigatio. However, the model is iadequate to deal with the flexible rage of coferece ad divisioal systems used throughout the various major leagues, a shortcomig we aim to overcome. Therefore, the seaso structure is relaxed such that teams ca play other teams o ay umber of occasios, by allowig m differet values of k (ot just k or k + ). I this settig, a iitial schedule 6
7 (symmetric) matrix for seaso t is specified similarly to Jech (983: p. 247), i which elemet x, deotes the umber of times team i plays team j durig the seaso, as i j x, x,2 L x, = x2, x2,2 L x2, X () M M O M xi, j j = x, x,2 L x, where x { k k k }, i j ; k + = { k Z : k 0}, l{ m}, 2,..., i, j m, Z ; = 0, i l l l,..., x i, i ad xi, j = x j, i, i j. Oe ca similarly defie aother matrix as beig a full set of headtohead expost decimalized wiloss records, W, such that: w, w,2 L w, = w2, w2,2 L w2, W (2) M M O M w, w,2 L w, where kl w i, j 0,,...,,, kl > 0 (3) kl kl ad where wi, j = w j, i, i j ; ad w i, i = 2, i. 2 For k l = 0, we ca assume w i, j = 2 for ow. 3 The iterpretatio of the diagoal elemets is that team i, if hypothetically playig agaist itself (despite beig impossible ituitively), is expected to have a equal probability to wi or lose. The costrait that all teams play a equal seaso legth is imposed to make formal represetatio easier, that is: 2 The set of possible values of w i, j i equatio (3) igores ties uusual i the NFL sice teams have to be level after regulatio time, with o further score i overtime. This occurred oly twice i the sample (Pittsburgh Steelers v Atlata Falcos, Week 0, 2002; ad Ciciati Begals v Philadelphia Eagles, Week, 2008). Assumig equal allocatio of competitio poits for a tie (as i the NFL), the 2kl set of possible values ca istead be geeralized as: w i, j 0,,...,,, kl > 0. 2kl 2kl 3 This is possible oly because we are iterested purely i the diagoal elemets from the product matrix, ad these elemets are quatitatively uaffected by the presece of this assumptio. Nevertheless, this poit is revisited i the latter stages of the appedix. 7
8 j= m xi, j = kl k, i (where l k is the umber of teams that team i plays k l l times), l= though it is ot essetial to the validity of the procedure. This also esures that det [ X] is fixed for all i. I assessig withiseaso competitive balace, the usual startig poit is costructig a colum vector of observed team stregths, to resemble a league table (a cocept familiar to fas). Stregth is measured simply as the wiratio (games wo divided by games played), represetig the actual fial seaso record of orderig of all teams i the league from all games played. For simplicity, the associated vector, V, ca be expressed as [ w w K ] ' V = 2 w (4) where w i = wi, j x j, i j=. These elemets are idetical to the diagoal elemets of the product of the matrices from () ad (2). While valid for ay orderig of teams, it is easiest to visualize if the orderig is idetical to the expost rakig of teams. If it were assumed that this is idetical to the (uobservable) exate orderig of teams, the such a orderig also allows us to defie both extremes agaist which observed competitive balace measures ca be deflated. This could be either: (i) a perfectly ubalaced league (closest outcome possible to a moopoly) as oe i which W is a upper triagular matrix where w i j <, =, i j ; or (ii) a perfectly balaced league, whereby w i, j, = 2, i j. A third possible bechmark is that whereby match outcomes are accordig to the biomial distributio: Pr ( i wis) = Pr( j wis) = 0. 5 which case, for ay k > 0 i (3) ad igorig ties, ca be geeralized as l, i 8
9 where {,2 } z Pr w, z k l z kl z i, j = wi j k = Pr = = l k (5) l z z,...,k l is a idex. The distributioal properties of the dispersio of the resultig set of w i i (4) ca the be calculated (or if impractical, simulated). I order to correct the ubalaced schedule bias i the edofseaso stadigs, a logical way to proceed is calculatig implied wiprobabilities (usig oly actual stadigs) of theoretical uplayed matches required to reset each ( i j) max { } x i, j x i j, equal to. The simple logitstyle rule used by Lete (2008) produced a solutio cotaiig a umber of attractive umerical properties, such as i= w i = w~ t{,..., T} (6), t i= i, t where w ~ i is the adjusted wiratio i seaso t, which is to say that the adjustmet procedure should ot affect the sum of wis across all teams i ay give seaso. However, a more direct way to accout for the wiratio bias is to compare the stregths of team i s actual fullseaso schedule to that from a hypothetical schedule i which every game were agaist a oppoet of average stregth of all teams (excludig itself). The former, A, is simply a colum vector of sum products of the umber of games agaist each oppoet ad the correspodig wiratios, hece A = x i, j XV (7) Meawhile, the latter, Y (also a colum vector), ca similarly be computed as j= xi, j ' j= Y = w w2 K wn (8) 9
10 Usig oly these two pieces of iformatio to determie the adjustmet meas that the distributio of idividual headtohead records do ot matter, which is also cosidered to be a desirable property. 4 I vector form, takig equatio (4) as a base, the adjusted set of wiratios arisig from equatios (7) ad (8) for all teams ca be expressed as follows ~ V ' = [ w~ w~ K w~ ] 2 ' = V + j= x i, j [ A ] ' Y ' (9) where each elemet i V ~, for ay sigle team, i, ca alteratively be calculated accordig to the followig fuctio relatig the adjusted wiratio to the uadjusted wiratio directly, expressed i scalar form as m k l = = = w i w~ 2 i = wi + w k k xi j x l l, i, j (0) l k j l j= for ay orderig of iequalities for k,...,, k2 km. Equatio (0) has the additioal desirable property, oted by Lete (200), that it is ivariat to the assumed umber of full roudrobis, which is crucial for NFL data, sice it cotais zeros ad oes. 4. Touramet Formats Correctio for ubalaced schedules is idetical irrespective of the ature of the touramet desig; however, the applicatio of the proposed procedure is potetially profoudly differet. A quick glace at table demostrates the similarities ad differeces betwee the major league seaso structures. The middle two colums, which reveal how may times team i plays all other teams, are ordered vertically such that k < < k2 <.. < k m km. The curret NFL system was itroduced upo expasio 4 The appedix presets a alterative procedure i which these headtohead records do matter. 0
11 to 32 teams (6 i both cofereces ad 4 i each divisio). This system sees each team playig teams i their ow divisio twice per seaso, automatically allocatig oppoets i 6 out of 6 matches. They also play all teams i oe other divisio withi their ow coferece, plus all teams from oe divisio i the other coferece, allocatig a further 8 matches. The other two matches i the seaso are allocated accordig to a (lagged) powermatchig rule they are agaist teams i the two divisios i the same coferece they would ot play otherwise, that fiished i the same divisioal rakpositio i the previous seaso. To formalize the earlier otatio, this meas collectively that uder the curret system, = 32, 0 k ( ) = k = 8, = 2 k ( ) k 2 = 0, ad 3 = 2 k ( ) k 3 = 3. From 2002, the idea was for a 8year rotatioal system, i which each team would play every other team at least twice oce home, oce away (eve itercoferece teams). This full rotatioal period defies the sample, extedig over the regular seasos from This ivolves the placig ad aggregatio of the pairig of teams ad match results from 2,048 regular seaso matches ito the costructio of 8 separate matrix observatios for both X ad W. A look at the remaider of table reveals the varyig ature of touramet desigs, mostly o the basis of seaso legth, as all other listed examples are based o 30team leagues. I a comparative sese, the NFL touramet desig was similar, except with slightly greater weightig o the itradivisioal ad powerbalacig aspects. Despite a idetical divisioal compositio ( 6 5) ad seaso legth of 82 games, the NHL ad NBA differ markedly i terms of schedule cocetratio, with 5 From 200, a ew rotatioal system applies, similar to the previous system, though altered slightly to reduce the overall travel burde o East Coast teams.
12 NHL teams playig a sigificatly higher proportioal umber of games agaist itracoferece teams tha i the NBA. The previous ( ) NHL system was eve more highly cocetrated, with values of k l ragig from 0 to 8. The MLB system is the most difficult to describe, with geeral differeces betwee divisios, depedig o how may teams i both that divisio ad the correspodig coferece ( league i MLB termiology), ad further differeces still o a teamspecific basis. A geeral way to describe the cocetratio of schedules is to use the Herfidahl idex from idustrial orgaizatio. The results are outlied for each of the major leagues i table 2. Iitially, the idex is deflated by the idex from a theoretical idealized schedule, ivolvig the most eve allocatio of games (betwee oppoets) possible holdig both ad seaso legth fixed. O this basis, the MLB schedule is the most cocetrated, due to the log seaso ad there beig little iterleague play. As examples from the 2009 seaso, the idex ratio was 2.5 for the Bosto Red Sox (AL) ad 2.02 for the Philadelphia Phillies (NL). However, if the idex is istead deflated by the bechmark idex from a truly balaced schedule, the the NFL provides the most cocetrated schedule out of the major leagues. While this is somewhat geerated merely by the short seaso, it demostrates that the NFL presets the most iterestig test case for the adjustmet procedure, as it provides the greatest scope for idetificatio of large adjustmets should they be preset. While applicable to ay type of ubalaced schedule, the NFL is easily the most suitable cadidate of the major leagues for this techique. I additio to the cocetrated schedule, it also overcomes the other major leagues for umerous compellig reasos. Firstly, the allocatio of competitio poits is equal for all games 2
13 ulike the NHL, where the total allocatio of poits is greater if the match exteds to overtime, creatig drawbacks for some withiseaso competitive balace measures. Aother problem with the NHL is discotiuity of the sample arisig from the 2005 seaso cacellatio due to the players strike. Secodly, MLB is the least suitable cadidate, owig to the ouiform ature of schedules betwee teams discussed earlier combied with the scarcity of iterleague play (oly about 6% of matches, compared to 25% i the NFL). Fially, the comparative aalysis is more powerful whe the compositio of teams is idetical throughout the sample. This is the case for the NFL; with the same 32 teams sice 2002 (eve the NBA has had two relocatios sice 2005). 5. Results The descriptive statistics of the raw wiratios for all 32 teams over the sample are displayed i table 3. As show, there is cosiderable turover of stadigs over the sample (idicative of betweeseaso competitive balace), with oly the Detroit Lios failig to have a wiig seaso. Iversely, the New Eglad Patriots ad Idiaapolis Colts were the oly teams that did ot have a losig seaso. These three teams, ad the Oaklad Raiders, are also the oly teams to have a mea wiratio over the eight seasos outside the bouds of 3 ad 2 3. The teamspecific stadard error of wiratios over the sample useful as a betweeseaso measure of competitive balace accordig to Humphreys (2002) supports this cotetio. A look at the middle colum of table 4 shows that the mea value of the teamspecific stadard errors is higher tha the aalogous figures for the other major leagues. 3
14 As for withiseaso, oe may use a rage of measures to describe NFL competitive balace. Firstly, we cosider the most widelyused measure the ratio of the actual stadard deviatio to the idealized (biomial distributio of match results as i (5)) stadard deviatio (Noll, 988; Scully, 989), represeted as 2 ( wi 0.5) 2 ASD/ISD = x () i= The righthad colum of table 4 shows that the mea dispersio of NFL wiratios is the lowest (most balaced) of the major leagues, accordig to ASD/ISD. Recallig the sectio 2 debate o labor market ad reveuesharig restrictios ifluecig competitive balace, high NFL balacedess (both withi ad betwee seasos) is due arguably to the cosiderable equalizig effect of the restrictive suite of policies (relative to other major leagues) used by the NFL. However, this has recetly become far less so, with the abolitio of the salary cap i March 200. j= i, j As a meas of robustifyig the results to alterative measures, three others are also reported, as formalized i equatios (2)(4). Respectively, these measures are: (i) the Herfidahl idex of competitive balace (HICB), similar to the baselie Herfidahl idex, except allowig to be timevaryig; (ii) the cocetratio idex of competitive balace, C2ICB, measurig the proportio of total wis by the top twelve teams i that seaso the umber that qualifies for the playoffs; 6 ad (iii) the stadard Gii coefficiet. 7 For (ii) ad (iii), w i are assumed rakordered. 6 The twelve teams assiged by simple rakorder is typically ot the precise set of teams that make the playoffs. This is aother distictive feature of the coferece ad divisioal system. 7 While GINI is variat to chages i ad x j = i, j over time, either chages durig the sample, circumvetig this crucial cocer. Otherwise we would be uable to apply the Utt ad Fort (2002) methodology, as it depeds o the (uobservable but assumed) exate rakorder of teams uder NFL possible combiatios. touramet desig, of which there are 32! ( ) 4
15 2 HICB = 4 w (2) C2ICB = i= 2 i= i w 6 (3) h 2 GINI = 4 wi (4) h= i= Iitially of curiosity are the magitudial chages to the wiratios arisig from the adjustmet. It is show i table 5 that the outliers are quite large, because of the high schedule cocetratio oted previously. Out of 256 adjustmets (for 32 teams over 8 seasos), a total of are greater i absolute magitude tha the value of a wi (0.0625), of which 6 are egative ad 5 positive, ad a further 58 are greater tha the absolute value of halfawi. The largest sigle adjustmets are (Seattle Seahawks i 2007) ad (Clevelad Brows i 2004). The adjustmets are also scatter plotted agaist the origial wiratios i figure. Fortuately, there appears to be little systematic bias, providig techical backig for the methodology. i Normally of acute cocer is the extet to which hypothetical adjustmets create qualitative differeces, i terms of chages to team rakorder. This is ot quite as much of a issue i the coferece ad divisioal system, sice it is recogized that ot ecessarily the best teams will always make the playoffs. Nevertheless, uder the criteria specified for playoff qualificatio, it is still of iterest to see whether differet teams would have qualified otioally uder adjusted wiratios. Specifically, there are a total of oly ie cases where this occurs over the sample, eight of which ivolve AFC teams. Oe case amazigly ivolves three teams (AFC East i 2002), i which New York Jets fall from first to third, promotig New 5
16 Eglad Patriots from secod to the divisio title. The eight remaiig cases ivolve oly two teams swappig positios. Of these, four have o bearig o playoff positios, while a further two (AFC North i 2005 ad NFC East i 2009) ivolve toptwo teams i cases where secod ears a wildcard, affectig oly seedigs. The two remaiig cases would have altered playoff qualificatio, with the Dever Brocos beatig Kasas City Chiefs for a AFC wildcard i 2006 ad New Eglad Patriots toppig Miami Dolphis for the AFC East 2008 divisio title. I additio, there are three further cases (agai all AFC) i which adjusted stadigs would have altered wildcard places, with Dever Brocos replacig Clevelad Brows i 2002, ad Miami Dolphis (2003) ad Baltimore Raves (2004) both replacig the Brocos. The latter case is the oly oe (out of twelve) i which the combied adjustmets of the two teams ivolved overcome the value of a wi. 8 I most cases ivolvig teams tied o wis (4 out of 22), the adjustmet assigs a orderig ot icosistet with that derived from the secodary ad tertiary criteria specified curretly by the NFL to separate teams tied o wis. Aspects of the adjustmets at a aggregate level are also of ituitive appeal to fas. Table 6 provides a profile of possible favoritism i the fixture. Begiig with the righthad colum, it is see that the mea adjustmet for AFC teams is , a sizable figure over 28 observatios. This results from AFC teams wiig the majority of itercoferece games over the sample. Furthermore, the divisioal adjustmets are positive across the board i the AFC, alog with NFC East. By team, Teessee Titas are the oly AFC team with a easier tha average schedule over the 8 seasos, whereas oly 4 NFC teams have positive mea adjustmets, three of 8 I umerous other cases, the adjustmet chages playoff seedigs, however, this is of less cocer. 6
17 them i East divisio. The mea adjustmet is a descriptive outlier greater tha two stadard errors usig mea adjustmets from all teams for Oaklad Raiders (positive) ad Seattle Seahawks (egative), with the differece betwee them of greater tha oe full wi per seaso. The Seahawks case demostrates icely the outcomes arisig from the procedure. While their raw wiratio of is slightly above average over the sample, they play a disproportioate umber of games agaist three teams (the other NFC West teams) that have performed weaker tha average the adjusted wiratio (0.4876) takes this ito accout. Returig mometarily to figure ad the lie of best fit, further ispectio also reveals a oticeable egative correlatio ( ρ = ), idicatig that teams achievig a higher wiratio o average received more favorable fixtures, though this factor accouts for a small fractio of the variatio. This result is totally ituitive, sice a team with a fixed level of talet ad performace will (all else equal) achieve a higher wiratio whe allocated a easier schedule. Whe mea adjustmets by divisioal place fiish are computed, as i table 7, it is foud that the meas are icreasig motoically i rakorder, as expected. If, however, the adjustmets are based o rakorder from the previous seaso (oe seaso of observatios lost), the correlatio chages sig ( ρ = 0.02), so do the sigs of all the mea adjustmets i table 7, though all are close to zero. The iterpretatio is that the powerbalacig aspect to the fixture eves up the competitio, though to a lesser degree tha iteded sice chages i stadigs from oe seaso to the ext are difficult to predict. Most crucially, the calculated uadjusted measures of competitive balace are show i the top portio of table 8. It is observed (o the basis of ASD/ISD) that the
18 seaso was comfortably the most competitively balaced of the sample (i fact, most eve sice 995), ad while 2006 was also more eve tha average, all other seasos produced a similar competitive balace levels. This fidig is reiforced by the other various measures reported i table 8. The adjusted competitive balace measures ad adjustmets are also computed ad exhibited i the middle ad bottom portios of table 8, respectively. The iformatio from table 8 is replicated graphically i figure 2. The most strikig result is that these adjustmets are egative for all seasos across all measures, although i some years the magitude of the adjustmets are relatively large (most oticeably i 2003), ad close to zero i other years (2004, 2006 ad 2007). These figures idicate i a quatitative sese that the NFL is cosiderably more competitively balaced whe the ubalaced schedule is accouted for. Naturally, the issue of statistical sigificace of differeces betwee uadjusted ad adjusted arises. While ay basic oparametric sig test or raksum test would defiitively reject the ull of o statistical differece, we seek further support or otherwise to substatiate this. To this ed, the more powerful parametric differece i meas ttest for paired samples is applied. Sice there is o a priori reaso to presume that the adjusted measures should be lower, the twotailed versio is used. The t statistics for ASD/ISD ad HICB of ad respectively are both easily above the 5% critical value of , ad while the values of for C2ICB ad 3.39 for GINI are also sigificat at 5%, they are isigificat at %. Despite this apparetly strog result, cautio is exercised for two reasos: (i) small sample size; ad (ii) the distributio of competitive balace measures may be oormal. Oe possible reaso for our result may be that the level of ability of a team coverges o the level of other teams that they play more ofte. However, eve if true, it is difficult 8
19 to coclude outright that the coferece ad divisioal touramet desig plays ay part i geeratig that result. The same applicatio to data from the CFL a league with oly 8 teams but with a similar (albeit reducedform) coferecestyle ubalaced schedule, produces a differece of meas ttest statistic for the ASD/ISD measure of merely , makig it comfortably isigificat Coclusio This paper has preseted a modified ubalaced schedule model that ca be geeralized to ay league i which teams play other teams ay umber of times. The model was the applied to NFL data over the regular seasos. The major coclusio is that, quatitatively, the adjusted competitive balace metrics imply the league to be more competitively balaced tha idicated by uadjusted measures i every seaso, ad sigificatly so for the sample meas. Qualitatively, the fidigs, while cosistet, are less categorical. The logically followig questio of whether adjusted or uadjusted measures more accurately reflect the ifluece of ucertaity of outcome o demad for America football is oe that is difficult to aswer usig average seaso attedaces, sice virtually all NFL games are sellouts. Regardless, adjusted measures are still of much cosequece to sports ecoomists. Especially so for ecoomists usig microecoomic models to make ifereces about the effectiveess of itervetioist league policies, ad usig withiseaso competitive balace metrics from differet sports leagues as a meas of comparative aalysis to test those ifereces empirically. This is because the use of adjusted measures could alter coclusios about which leagues are more competitively balaced. 9 The CFL is more comparable to the NFL tha the other major leagues. Not oly does it ivolve virtually the same sport ad a almost idetical seaso legth (8 games), but data from seasos ad is used, esurig idetical sample legth. To agai guaratee the same set of teams through the sample, seasos are excluded, whe the illfated Ottawa Reegades also competed. The sample ASD/ISD meas are.404 (uadjusted) ad.4233 (adjusted). 9
20 There was also a appealig story behid the adjustmets geerated from the procedure. At a aggregate level, i terms of both by team ad by rakorder i divisio stadigs, the meas were sufficietly large to raise cocers about the high cocetratio of the schedule. While it is improbable that the league s costituet teams would collectively favor overhaulig the NFL touramet desig radically i the short ru, they would likely be ope to the idea of at least debatig the issue upo beig made aware of this story. The fidigs preseted ad discussed here have crucial implicatios for the various major league commissios, i terms of their ogoig reforms to their labor market ad reveuesharig policies, as well as touramet desig itself. 20
In nite Sequences. Dr. Philippe B. Laval Kennesaw State University. October 9, 2008
I ite Sequeces Dr. Philippe B. Laval Keesaw State Uiversity October 9, 2008 Abstract This had out is a itroductio to i ite sequeces. mai de itios ad presets some elemetary results. It gives the I ite Sequeces
More informationHypothesis testing. Null and alternative hypotheses
Hypothesis testig Aother importat use of samplig distributios is to test hypotheses about populatio parameters, e.g. mea, proportio, regressio coefficiets, etc. For example, it is possible to stipulate
More informationModified Line Search Method for Global Optimization
Modified Lie Search Method for Global Optimizatio Cria Grosa ad Ajith Abraham Ceter of Excellece for Quatifiable Quality of Service Norwegia Uiversity of Sciece ad Techology Trodheim, Norway {cria, ajith}@q2s.tu.o
More informationDepartment of Computer Science, University of Otago
Departmet of Computer Sciece, Uiversity of Otago Techical Report OUCS200609 Permutatios Cotaiig May Patters Authors: M.H. Albert Departmet of Computer Sciece, Uiversity of Otago Micah Colema, Rya Fly
More informationI. Chisquared Distributions
1 M 358K Supplemet to Chapter 23: CHISQUARED DISTRIBUTIONS, TDISTRIBUTIONS, AND DEGREES OF FREEDOM To uderstad tdistributios, we first eed to look at aother family of distributios, the chisquared distributios.
More informationINVESTMENT PERFORMANCE COUNCIL (IPC)
INVESTMENT PEFOMANCE COUNCIL (IPC) INVITATION TO COMMENT: Global Ivestmet Performace Stadards (GIPS ) Guidace Statemet o Calculatio Methodology The Associatio for Ivestmet Maagemet ad esearch (AIM) seeks
More informationAnalyzing Longitudinal Data from Complex Surveys Using SUDAAN
Aalyzig Logitudial Data from Complex Surveys Usig SUDAAN Darryl Creel Statistics ad Epidemiology, RTI Iteratioal, 312 Trotter Farm Drive, Rockville, MD, 20850 Abstract SUDAAN: Software for the Statistical
More informationSECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES
SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES Read Sectio 1.5 (pages 5 9) Overview I Sectio 1.5 we lear to work with summatio otatio ad formulas. We will also itroduce a brief overview of sequeces,
More informationTaking DCOP to the Real World: Efficient Complete Solutions for Distributed MultiEvent Scheduling
Taig DCOP to the Real World: Efficiet Complete Solutios for Distributed MultiEvet Schedulig Rajiv T. Maheswara, Milid Tambe, Emma Bowrig, Joatha P. Pearce, ad Pradeep araatham Uiversity of Souther Califoria
More informationLesson 17 Pearson s Correlation Coefficient
Outlie Measures of Relatioships Pearso s Correlatio Coefficiet (r) types of data scatter plots measure of directio measure of stregth Computatio covariatio of X ad Y uique variatio i X ad Y measurig
More informationProblem Set 1 Oligopoly, market shares and concentration indexes
Advaced Idustrial Ecoomics Sprig 2016 Joha Steek 29 April 2016 Problem Set 1 Oligopoly, market shares ad cocetratio idexes 1 1 Price Competitio... 3 1.1 Courot Oligopoly with Homogeous Goods ad Differet
More informationCase Study. Normal and t Distributions. Density Plot. Normal Distributions
Case Study Normal ad t Distributios Bret Halo ad Bret Larget Departmet of Statistics Uiversity of Wiscosi Madiso October 11 13, 2011 Case Study Body temperature varies withi idividuals over time (it ca
More informationOutput Analysis (2, Chapters 10 &11 Law)
B. Maddah ENMG 6 Simulatio 05/0/07 Output Aalysis (, Chapters 10 &11 Law) Comparig alterative system cofiguratio Sice the output of a simulatio is radom, the comparig differet systems via simulatio should
More informationHow to read A Mutual Fund shareholder report
Ivestor BulletI How to read A Mutual Fud shareholder report The SEC s Office of Ivestor Educatio ad Advocacy is issuig this Ivestor Bulleti to educate idividual ivestors about mutual fud shareholder reports.
More informationCenter, Spread, and Shape in Inference: Claims, Caveats, and Insights
Ceter, Spread, ad Shape i Iferece: Claims, Caveats, ad Isights Dr. Nacy Pfeig (Uiversity of Pittsburgh) AMATYC November 2008 Prelimiary Activities 1. I would like to produce a iterval estimate for the
More informationCovariance and correlation
Covariace ad correlatio The mea ad sd help us summarize a buch of umbers which are measuremets of just oe thig. A fudametal ad totally differet questio is how oe thig relates to aother. Stat 0: Quatitative
More informationCHAPTER 3 DIGITAL CODING OF SIGNALS
CHAPTER 3 DIGITAL CODING OF SIGNALS Computers are ofte used to automate the recordig of measuremets. The trasducers ad sigal coditioig circuits produce a voltage sigal that is proportioal to a quatity
More informationINVESTMENT PERFORMANCE COUNCIL (IPC) Guidance Statement on Calculation Methodology
Adoptio Date: 4 March 2004 Effective Date: 1 Jue 2004 Retroactive Applicatio: No Public Commet Period: Aug Nov 2002 INVESTMENT PERFORMANCE COUNCIL (IPC) Preface Guidace Statemet o Calculatio Methodology
More informationThe analysis of the Cournot oligopoly model considering the subjective motive in the strategy selection
The aalysis of the Courot oligopoly model cosiderig the subjective motive i the strategy selectio Shigehito Furuyama Teruhisa Nakai Departmet of Systems Maagemet Egieerig Faculty of Egieerig Kasai Uiversity
More informationHere are a couple of warnings to my students who may be here to get a copy of what happened on a day that you missed.
This documet was writte ad copyrighted by Paul Dawkis. Use of this documet ad its olie versio is govered by the Terms ad Coditios of Use located at http://tutorial.math.lamar.edu/terms.asp. The olie versio
More informationTrading the randomness  Designing an optimal trading strategy under a drifted random walk price model
Tradig the radomess  Desigig a optimal tradig strategy uder a drifted radom walk price model Yuao Wu Math 20 Project Paper Professor Zachary Hamaker Abstract: I this paper the author iteds to explore
More informationCHAPTER 3 THE TIME VALUE OF MONEY
CHAPTER 3 THE TIME VALUE OF MONEY OVERVIEW A dollar i the had today is worth more tha a dollar to be received i the future because, if you had it ow, you could ivest that dollar ad ear iterest. Of all
More informationUnit 20 Hypotheses Testing
Uit 2 Hypotheses Testig Objectives: To uderstad how to formulate a ull hypothesis ad a alterative hypothesis about a populatio proportio, ad how to choose a sigificace level To uderstad how to collect
More informationChapter 10. Hypothesis Tests Regarding a Parameter. 10.1 The Language of Hypothesis Testing
Chapter 10 Hypothesis Tests Regardig a Parameter A secod type of statistical iferece is hypothesis testig. Here, rather tha use either a poit (or iterval) estimate from a simple radom sample to approximate
More information9.8: THE POWER OF A TEST
9.8: The Power of a Test CD91 9.8: THE POWER OF A TEST I the iitial discussio of statistical hypothesis testig, the two types of risks that are take whe decisios are made about populatio parameters based
More information.04. This means $1000 is multiplied by 1.02 five times, once for each of the remaining sixmonth
Questio 1: What is a ordiary auity? Let s look at a ordiary auity that is certai ad simple. By this, we mea a auity over a fixed term whose paymet period matches the iterest coversio period. Additioally,
More informationTHE ROLE OF EXPORTS IN ECONOMIC GROWTH WITH REFERENCE TO ETHIOPIAN COUNTRY
 THE ROLE OF EXPORTS IN ECONOMIC GROWTH WITH REFERENCE TO ETHIOPIAN COUNTRY BY: FAYE ENSERMU CHEMEDA EthioItalia Cooperatio ArsiBale Rural developmet Project Paper Prepared for the Coferece o Aual Meetig
More information5: Introduction to Estimation
5: Itroductio to Estimatio Cotets Acroyms ad symbols... 1 Statistical iferece... Estimatig µ with cofidece... 3 Samplig distributio of the mea... 3 Cofidece Iterval for μ whe σ is kow before had... 4 Sample
More information0.7 0.6 0.2 0 0 96 96.5 97 97.5 98 98.5 99 99.5 100 100.5 96.5 97 97.5 98 98.5 99 99.5 100 100.5
Sectio 13 KolmogorovSmirov test. Suppose that we have a i.i.d. sample X 1,..., X with some ukow distributio P ad we would like to test the hypothesis that P is equal to a particular distributio P 0, i.e.
More information1 Computing the Standard Deviation of Sample Means
Computig the Stadard Deviatio of Sample Meas Quality cotrol charts are based o sample meas ot o idividual values withi a sample. A sample is a group of items, which are cosidered all together for our aalysis.
More informationPSYCHOLOGICAL STATISTICS
UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION B Sc. Cousellig Psychology (0 Adm.) IV SEMESTER COMPLEMENTARY COURSE PSYCHOLOGICAL STATISTICS QUESTION BANK. Iferetial statistics is the brach of statistics
More informationExample Consider the following set of data, showing the number of times a sample of 5 students check their per day:
Sectio 82: Measures of cetral tedecy Whe thikig about questios such as: how may calories do I eat per day? or how much time do I sped talkig per day?, we quickly realize that the aswer will vary from day
More informationPresent Value Tax Expenditure Estimate of Tax Assistance for Retirement Saving
Preset Value Tax Expediture Estimate of Tax Assistace for Retiremet Savig Tax Policy Brach Departmet of Fiace Jue 30, 1998 2 Preset Value Tax Expediture Estimate of Tax Assistace for Retiremet Savig This
More informationDetermining the sample size
Determiig the sample size Oe of the most commo questios ay statisticia gets asked is How large a sample size do I eed? Researchers are ofte surprised to fid out that the aswer depeds o a umber of factors
More informationThe following example will help us understand The Sampling Distribution of the Mean. C1 C2 C3 C4 C5 50 miles 84 miles 38 miles 120 miles 48 miles
The followig eample will help us uderstad The Samplig Distributio of the Mea Review: The populatio is the etire collectio of all idividuals or objects of iterest The sample is the portio of the populatio
More informationDiscrete Mathematics and Probability Theory Spring 2014 Anant Sahai Note 13
EECS 70 Discrete Mathematics ad Probability Theory Sprig 2014 Aat Sahai Note 13 Itroductio At this poit, we have see eough examples that it is worth just takig stock of our model of probability ad may
More informationInference on Proportion. Chapter 8 Tests of Statistical Hypotheses. Sampling Distribution of Sample Proportion. Confidence Interval
Chapter 8 Tests of Statistical Hypotheses 8. Tests about Proportios HT  Iferece o Proportio Parameter: Populatio Proportio p (or π) (Percetage of people has o health isurace) x Statistic: Sample Proportio
More informationKey Ideas Section 81: Overview hypothesis testing Hypothesis Hypothesis Test Section 82: Basics of Hypothesis Testing Null Hypothesis
Chapter 8 Key Ideas Hypothesis (Null ad Alterative), Hypothesis Test, Test Statistic, Pvalue Type I Error, Type II Error, Sigificace Level, Power Sectio 81: Overview Cofidece Itervals (Chapter 7) are
More informationChapter 7 Methods of Finding Estimators
Chapter 7 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 011 Chapter 7 Methods of Fidig Estimators Sectio 7.1 Itroductio Defiitio 7.1.1 A poit estimator is ay fuctio W( X) W( X1, X,, X ) of
More informationhp calculators HP 12C Statistics  average and standard deviation Average and standard deviation concepts HP12C average and standard deviation
HP 1C Statistics  average ad stadard deviatio Average ad stadard deviatio cocepts HP1C average ad stadard deviatio Practice calculatig averages ad stadard deviatios with oe or two variables HP 1C Statistics
More informationTHE ARITHMETIC OF INTEGERS.  multiplication, exponentiation, division, addition, and subtraction
THE ARITHMETIC OF INTEGERS  multiplicatio, expoetiatio, divisio, additio, ad subtractio What to do ad what ot to do. THE INTEGERS Recall that a iteger is oe of the whole umbers, which may be either positive,
More informationSoving Recurrence Relations
Sovig Recurrece Relatios Part 1. Homogeeous liear 2d degree relatios with costat coefficiets. Cosider the recurrece relatio ( ) T () + at ( 1) + bt ( 2) = 0 This is called a homogeeous liear 2d degree
More informationInvesting in Stocks WHAT ARE THE DIFFERENT CLASSIFICATIONS OF STOCKS? WHY INVEST IN STOCKS? CAN YOU LOSE MONEY?
Ivestig i Stocks Ivestig i Stocks Busiesses sell shares of stock to ivestors as a way to raise moey to fiace expasio, pay off debt ad provide operatig capital. Ecoomic coditios: Employmet, iflatio, ivetory
More informationHOSPITAL NURSE STAFFING SURVEY
2012 Ceter for Nursig Workforce St udies HOSPITAL NURSE STAFFING SURVEY Vacacy ad Turover Itroductio The Hospital Nurse Staffig Survey (HNSS) assesses the size ad effects of the ursig shortage i hospitals,
More informationMeasures of Spread and Boxplots Discrete Math, Section 9.4
Measures of Spread ad Boxplots Discrete Math, Sectio 9.4 We start with a example: Example 1: Comparig Mea ad Media Compute the mea ad media of each data set: S 1 = {4, 6, 8, 10, 1, 14, 16} S = {4, 7, 9,
More information3. Covariance and Correlation
Virtual Laboratories > 3. Expected Value > 1 2 3 4 5 6 3. Covariace ad Correlatio Recall that by takig the expected value of various trasformatios of a radom variable, we ca measure may iterestig characteristics
More informationBiology 171L Environment and Ecology Lab Lab 2: Descriptive Statistics, Presenting Data and Graphing Relationships
Biology 171L Eviromet ad Ecology Lab Lab : Descriptive Statistics, Presetig Data ad Graphig Relatioships Itroductio Log lists of data are ofte ot very useful for idetifyig geeral treds i the data or the
More information1 Correlation and Regression Analysis
1 Correlatio ad Regressio Aalysis I this sectio we will be ivestigatig the relatioship betwee two cotiuous variable, such as height ad weight, the cocetratio of a ijected drug ad heart rate, or the cosumptio
More informationEkkehart Schlicht: Economic Surplus and Derived Demand
Ekkehart Schlicht: Ecoomic Surplus ad Derived Demad Muich Discussio Paper No. 200617 Departmet of Ecoomics Uiversity of Muich Volkswirtschaftliche Fakultät LudwigMaximiliasUiversität Müche Olie at http://epub.ub.uimueche.de/940/
More informationThe second difference is the sequence of differences of the first difference sequence, 2
Differece Equatios I differetial equatios, you look for a fuctio that satisfies ad equatio ivolvig derivatives. I differece equatios, istead of a fuctio of a cotiuous variable (such as time), we look for
More informationA GUIDE TO LEVEL 3 VALUE ADDED IN 2013 SCHOOL AND COLLEGE PERFORMANCE TABLES
A GUIDE TO LEVEL 3 VALUE ADDED IN 2013 SCHOOL AND COLLEGE PERFORMANCE TABLES Cotets Page No. Summary Iterpretig School ad College Value Added Scores 2 What is Value Added? 3 The Learer Achievemet Tracker
More informationExample 2 Find the square root of 0. The only square root of 0 is 0 (since 0 is not positive or negative, so those choices don t exist here).
BEGINNING ALGEBRA Roots ad Radicals (revised summer, 00 Olso) Packet to Supplemet the Curret Textbook  Part Review of Square Roots & Irratioals (This portio ca be ay time before Part ad should mostly
More informationConfidence Intervals for One Mean
Chapter 420 Cofidece Itervals for Oe Mea Itroductio This routie calculates the sample size ecessary to achieve a specified distace from the mea to the cofidece limit(s) at a stated cofidece level for a
More informationVolatility of rates of return on the example of wheat futures. Sławomir Juszczyk. Rafał Balina
Overcomig the Crisis: Ecoomic ad Fiacial Developmets i Asia ad Europe Edited by Štefa Bojec, Josef C. Brada, ad Masaaki Kuboiwa http://www.hippocampus.si/isbn/9789616832328/cotets.pdf Volatility of
More informationEstimating Probability Distributions by Observing Betting Practices
5th Iteratioal Symposium o Imprecise Probability: Theories ad Applicatios, Prague, Czech Republic, 007 Estimatig Probability Distributios by Observig Bettig Practices Dr C Lych Natioal Uiversity of Irelad,
More information7. Sample Covariance and Correlation
1 of 8 7/16/2009 6:06 AM Virtual Laboratories > 6. Radom Samples > 1 2 3 4 5 6 7 7. Sample Covariace ad Correlatio The Bivariate Model Suppose agai that we have a basic radom experimet, ad that X ad Y
More information3.1 Measures of Central Tendency. Introduction 5/28/2013. Data Description. Outline. Objectives. Objectives. Traditional Statistics Average
5/8/013 C H 3A P T E R Outlie 3 1 Measures of Cetral Tedecy 3 Measures of Variatio 3 3 3 Measuresof Positio 3 4 Exploratory Data Aalysis Copyright 013 The McGraw Hill Compaies, Ic. C H 3A P T E R Objectives
More information1. C. The formula for the confidence interval for a population mean is: x t, which was
s 1. C. The formula for the cofidece iterval for a populatio mea is: x t, which was based o the sample Mea. So, x is guarateed to be i the iterval you form.. D. Use the rule : pvalue
More informationIncremental calculation of weighted mean and variance
Icremetal calculatio of weighted mea ad variace Toy Fich faf@cam.ac.uk dot@dotat.at Uiversity of Cambridge Computig Service February 009 Abstract I these otes I eplai how to derive formulae for umerically
More informationA Combined Continuous/Binary Genetic Algorithm for Microstrip Antenna Design
A Combied Cotiuous/Biary Geetic Algorithm for Microstrip Atea Desig Rady L. Haupt The Pesylvaia State Uiversity Applied Research Laboratory P. O. Box 30 State College, PA 168040030 haupt@ieee.org Abstract:
More information5.4 Amortization. Question 1: How do you find the present value of an annuity? Question 2: How is a loan amortized?
5.4 Amortizatio Questio 1: How do you fid the preset value of a auity? Questio 2: How is a loa amortized? Questio 3: How do you make a amortizatio table? Oe of the most commo fiacial istrumets a perso
More informationLesson 15 ANOVA (analysis of variance)
Outlie Variability betwee group variability withi group variability total variability Fratio Computatio sums of squares (betwee/withi/total degrees of freedom (betwee/withi/total mea square (betwee/withi
More informationDesigning Incentives for Online Question and Answer Forums
Desigig Icetives for Olie Questio ad Aswer Forums Shaili Jai School of Egieerig ad Applied Scieces Harvard Uiversity Cambridge, MA 0238 USA shailij@eecs.harvard.edu Yilig Che School of Egieerig ad Applied
More informationMARTINGALES AND A BASIC APPLICATION
MARTINGALES AND A BASIC APPLICATION TURNER SMITH Abstract. This paper will develop the measuretheoretic approach to probability i order to preset the defiitio of martigales. From there we will apply this
More information5 Boolean Decision Trees (February 11)
5 Boolea Decisio Trees (February 11) 5.1 Graph Coectivity Suppose we are give a udirected graph G, represeted as a boolea adjacecy matrix = (a ij ), where a ij = 1 if ad oly if vertices i ad j are coected
More informationNonlife insurance mathematics. Nils F. Haavardsson, University of Oslo and DNB Skadeforsikring
Nolife isurace mathematics Nils F. Haavardsso, Uiversity of Oslo ad DNB Skadeforsikrig Mai issues so far Why does isurace work? How is risk premium defied ad why is it importat? How ca claim frequecy
More informationCOMPARISON OF THE EFFICIENCY OF SCONTROL CHART AND EWMAS 2 CONTROL CHART FOR THE CHANGES IN A PROCESS
COMPARISON OF THE EFFICIENCY OF SCONTROL CHART AND EWMAS CONTROL CHART FOR THE CHANGES IN A PROCESS Supraee Lisawadi Departmet of Mathematics ad Statistics, Faculty of Sciece ad Techoology, Thammasat
More informationMEI Structured Mathematics. Module Summary Sheets. Statistics 2 (Version B: reference to new book)
MEI Mathematics i Educatio ad Idustry MEI Structured Mathematics Module Summary Sheets Statistics (Versio B: referece to ew book) Topic : The Poisso Distributio Topic : The Normal Distributio Topic 3:
More informationSubject CT5 Contingencies Core Technical Syllabus
Subject CT5 Cotigecies Core Techical Syllabus for the 2015 exams 1 Jue 2014 Aim The aim of the Cotigecies subject is to provide a groudig i the mathematical techiques which ca be used to model ad value
More informationChapter 7: Confidence Interval and Sample Size
Chapter 7: Cofidece Iterval ad Sample Size Learig Objectives Upo successful completio of Chapter 7, you will be able to: Fid the cofidece iterval for the mea, proportio, ad variace. Determie the miimum
More informationWeek 3 Conditional probabilities, Bayes formula, WEEK 3 page 1 Expected value of a random variable
Week 3 Coditioal probabilities, Bayes formula, WEEK 3 page 1 Expected value of a radom variable We recall our discussio of 5 card poker hads. Example 13 : a) What is the probability of evet A that a 5
More informationMathematical goals. Starting points. Materials required. Time needed
Level A1 of challege: C A1 Mathematical goals Startig poits Materials required Time eeded Iterpretig algebraic expressios To help learers to: traslate betwee words, symbols, tables, ad area represetatios
More informationwhere: T = number of years of cash flow in investment's life n = the year in which the cash flow X n i = IRR = the internal rate of return
EVALUATING ALTERNATIVE CAPITAL INVESTMENT PROGRAMS By Ke D. Duft, Extesio Ecoomist I the March 98 issue of this publicatio we reviewed the procedure by which a capital ivestmet project was assessed. The
More informationTHE REGRESSION MODEL IN MATRIX FORM. For simple linear regression, meaning one predictor, the model is. for i = 1, 2, 3,, n
We will cosider the liear regressio model i matrix form. For simple liear regressio, meaig oe predictor, the model is i = + x i + ε i for i =,,,, This model icludes the assumptio that the ε i s are a sample
More informationRISK TRANSFER FOR DESIGNBUILD TEAMS
WILLIS CONSTRUCTION PRACTICE IBEAM Jauary 2010 www.willis.com RISK TRANSFER FOR DESIGNBUILD TEAMS DesigbuilD work is icreasig each quarter. cosequetly, we are fieldig more iquiries from cliets regardig
More informationVladimir N. Burkov, Dmitri A. Novikov MODELS AND METHODS OF MULTIPROJECTS MANAGEMENT
Keywords: project maagemet, resource allocatio, etwork plaig Vladimir N Burkov, Dmitri A Novikov MODELS AND METHODS OF MULTIPROJECTS MANAGEMENT The paper deals with the problems of resource allocatio betwee
More informationYour organization has a Class B IP address of 166.144.0.0 Before you implement subnetting, the Network ID and Host ID are divided as follows:
Subettig Subettig is used to subdivide a sigle class of etwork i to multiple smaller etworks. Example: Your orgaizatio has a Class B IP address of 166.144.0.0 Before you implemet subettig, the Network
More informationSequences and Series
CHAPTER 9 Sequeces ad Series 9.. Covergece: Defiitio ad Examples Sequeces The purpose of this chapter is to itroduce a particular way of geeratig algorithms for fidig the values of fuctios defied by their
More informationLecture 4: Cauchy sequences, BolzanoWeierstrass, and the Squeeze theorem
Lecture 4: Cauchy sequeces, BolzaoWeierstrass, ad the Squeeze theorem The purpose of this lecture is more modest tha the previous oes. It is to state certai coditios uder which we are guarateed that limits
More informationDEPARTMENT OF TREASURY FORECASTING ACCURACY OF THE ACT BUDGET ESTIMATES
DEPARTMENT OF TREASURY FORECASTING ACCURACY OF THE ACT BUDGET ESTIMATES May 008 . Executive Summary Every public or private orgaisatio that produces projectios or forecasts also evaluates their performace.
More informationODBC. Getting Started With Sage Timberline Office ODBC
ODBC Gettig Started With Sage Timberlie Office ODBC NOTICE This documet ad the Sage Timberlie Office software may be used oly i accordace with the accompayig Sage Timberlie Office Ed User Licese Agreemet.
More informationRecursion and Recurrences
Chapter 5 Recursio ad Recurreces 5.1 Growth Rates of Solutios to Recurreces Divide ad Coquer Algorithms Oe of the most basic ad powerful algorithmic techiques is divide ad coquer. Cosider, for example,
More informationCS103A Handout 23 Winter 2002 February 22, 2002 Solving Recurrence Relations
CS3A Hadout 3 Witer 00 February, 00 Solvig Recurrece Relatios Itroductio A wide variety of recurrece problems occur i models. Some of these recurrece relatios ca be solved usig iteratio or some other ad
More informationLECTURE 13: Crossvalidation
LECTURE 3: Crossvalidatio Resampli methods Cross Validatio Bootstrap Bias ad variace estimatio with the Bootstrap Threeway data partitioi Itroductio to Patter Aalysis Ricardo GutierrezOsua Texas A&M
More informationEntropy of bicapacities
Etropy of bicapacities Iva Kojadiovic LINA CNRS FRE 2729 Site école polytechique de l uiv. de Nates Rue Christia Pauc 44306 Nates, Frace iva.kojadiovic@uivates.fr JeaLuc Marichal Applied Mathematics
More informationNr. 2. Interpolation of Discount Factors. Heinz Cremers Willi Schwarz. Mai 1996
Nr 2 Iterpolatio of Discout Factors Heiz Cremers Willi Schwarz Mai 1996 Autore: Herausgeber: Prof Dr Heiz Cremers Quatitative Methode ud Spezielle Bakbetriebslehre Hochschule für Bakwirtschaft Dr Willi
More informationAP Calculus BC 2003 Scoring Guidelines Form B
AP Calculus BC Scorig Guidelies Form B The materials icluded i these files are iteded for use by AP teachers for course ad exam preparatio; permissio for ay other use must be sought from the Advaced Placemet
More informationDescriptive Statistics
Descriptive Statistics We leared to describe data sets graphically. We ca also describe a data set umerically. Measures of Locatio Defiitio The sample mea is the arithmetic average of values. We deote
More informationSequences II. Chapter 3. 3.1 Convergent Sequences
Chapter 3 Sequeces II 3. Coverget Sequeces Plot a graph of the sequece a ) = 2, 3 2, 4 3, 5 + 4,...,,... To what limit do you thik this sequece teds? What ca you say about the sequece a )? For ǫ = 0.,
More informationCS103X: Discrete Structures Homework 4 Solutions
CS103X: Discrete Structures Homewor 4 Solutios Due February 22, 2008 Exercise 1 10 poits. Silico Valley questios: a How may possible sixfigure salaries i whole dollar amouts are there that cotai at least
More informationFOUNDATIONS OF MATHEMATICS AND PRECALCULUS GRADE 10
FOUNDATIONS OF MATHEMATICS AND PRECALCULUS GRADE 10 [C] Commuicatio Measuremet A1. Solve problems that ivolve liear measuremet, usig: SI ad imperial uits of measure estimatio strategies measuremet strategies.
More informationA Fuzzy Model of Software Project Effort Estimation
TJFS: Turkish Joural of Fuzzy Systems (eissn: 309 90) A Official Joural of Turkish Fuzzy Systems Associatio Vol.4, No.2, pp. 6876, 203 A Fuzzy Model of Software Project Effort Estimatio Oumout Chouseioglou
More informationChapter 5 O A Cojecture Of Erdíos Proceedigs NCUR VIII è1994è, Vol II, pp 794í798 Jeærey F Gold Departmet of Mathematics, Departmet of Physics Uiversity of Utah Do H Tucker Departmet of Mathematics Uiversity
More informationGregory Carey, 1998 Linear Transformations & Composites  1. Linear Transformations and Linear Composites
Gregory Carey, 1998 Liear Trasformatios & Composites  1 Liear Trasformatios ad Liear Composites I Liear Trasformatios of Variables Meas ad Stadard Deviatios of Liear Trasformatios A liear trasformatio
More informationChapter 5 Unit 1. IET 350 Engineering Economics. Learning Objectives Chapter 5. Learning Objectives Unit 1. Annual Amount and Gradient Functions
Chapter 5 Uit Aual Amout ad Gradiet Fuctios IET 350 Egieerig Ecoomics Learig Objectives Chapter 5 Upo completio of this chapter you should uderstad: Calculatig future values from aual amouts. Calculatig
More informationNormal Distribution.
Normal Distributio www.icrf.l Normal distributio I probability theory, the ormal or Gaussia distributio, is a cotiuous probability distributio that is ofte used as a first approimatio to describe realvalued
More informationNATIONAL SENIOR CERTIFICATE GRADE 12
NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS P EXEMPLAR 04 MARKS: 50 TIME: 3 hours This questio paper cosists of 8 pages ad iformatio sheet. Please tur over Mathematics/P DBE/04 NSC Grade Eemplar INSTRUCTIONS
More informationStatement of cash flows
6 Statemet of cash flows this chapter covers... I this chapter we study the statemet of cash flows, which liks profit from the statemet of profit or loss ad other comprehesive icome with chages i assets
More informationCREATIVE MARKETING PROJECT 2016
CREATIVE MARKETING PROJECT 2016 The Creative Marketig Project is a chapter project that develops i chapter members a aalytical ad creative approach to the marketig process, actively egages chapter members
More informationPrescribing costs in primary care
Prescribig costs i primary care LONDON: The Statioery Office 13.50 Ordered by the House of Commos to be prited o 14 May 2007 REPORT BY THE COMPTROLLER AND AUDITOR GENERAL HC 454 Sessio 20062007 18 May
More information