Empirical result

Empiricalresult

Themodel developed in this research paper is meant to investigate theeffect of team rank, years of experience and age of a player on theincome earned. One or all of these factors may be used incontracting a player in a team. In this case the dependent variable(Y) is the income earned ($). Four independent variables consideredin his paper are team rank (X1),years of experience (X2),age of the player (X3)and points per game (X4). A relationship between the variables is developed by making aregression of the data and using linear models [ CITATION Mat14 l 1033 ].

Hypothesis:Ho

  1. Team ranking determines the income of a player.

  2. Years of experience of a player determine the income of the player

  3. Age of the player determines the income of the player.

  4. Points scored per game determines the income of the player

Thenull hypothesis: Ho

  1. Team ranking does not determine the income of a player.

  2. Years of experience of a player does not determine the income of the player

  3. Age of the player does not determine the income of the player.

  4. Points scored per game do not determine the income of a player.

Theteam rank is given by a number between 1 and 30. Age of the player,and years of experience are given by a whole number. Point score pergame are given by numbers. It is expected that the linearrelationship between Y and X1has a negative slope indicating that a drop in ranking willultimately decrease the income. It is also expected that the linearrelationships between Y and X2,Y and X3,and Y and X4havea positive slope. A more experienced player is expected to get moreincome and consequently an older player will also earn more.

Thetable below shows data collected from different teams involvingdifferent players in different positions and a combination of factorsrequired in our models.

Table1: Data

Season 2008-2009

Player

Team:

Salary($)

Position

Rebound/ game

Asissts/ game

Points/ game

Years of Exp.

Age of Player

Team Rank

Maurice Williams

Cavaliers

8353000.00

Guard

3.8

4.1

17.8

5

26

1

Lebron James

Cavaliers

14410581.00

Forward

7.6

7.2

28.4

5

24

1

Zydrunas Ilgauskas

Cavaliers

10841615.00

Center

7.5

1

12.9

10

33

1

Kobe Bryant

Lakers

21262500.00

Guard

5.2

4.9

26.8

12

31

2

Pau Gasol

Lakers

15106000.00

Forward

9.6

3.5

18.9

7

28

2

Andrew Bynum

Lakers

2769300.00

Center

8

1.4

14.3

3

21

2

Jameer Nelson

Magic

7600000.00

Guard

3.5

5.4

16.7

4

26

4

Rashard Lewis

Magic

16447871.00

Forward

5.7

2.6

17.7

10

29

4

Dwight Howard

Magic

13041250.00

Center

13.8

1.6

20.6

4

23

4

Chauncey Billups

Nuggets

11050000.00

Guard

3

6.4

17.9

11

32

5

Carmelo Anthony

Nuggets

14410581.00

Forward

6.8

3.4

22.8

5

24

5

Johan Petro

Nuggets

1939893.00

Center

2.3

0.4

2.2

3

23

5

Tracy McGrady

Rockets

19614000.00

Guard

5

4.4

15.6

11

29

9

Ron Artest

Rockets

7400000.00

Forward

5.8

3.3

17.1

9

28

9

Yao Ming

Rockets

15070550.00

Center

9.9

1.8

19.7

6

29

9

Jason Kidd

Mavericks

21372000.00

Guard

6.2

8.7

9

14

35

10

Dirk Nowitzki

Mavericks

18077904.00

Forward

8.4

2.4

25.9

10

30

10

Erick Dampier

Mavericks

11553000.00

Center

7.7

1

5.7

12

33

10

Joe Johnson

Hawks

14232566.00

Guard

4.4

5.8

21.4

7

27

13

Josh Smith

Hawks

10000000.00

Forward

7.4

2.5

16

4

23

13

Zaza Pachulia

Hawks

4000000.00

Center

10.7

1.4

11.8

5

24

13

Allen Iverson

Piston

20840625.00

Guard

3.1

4.9

17.4

12

33

17

Rasheed Wallace

Piston

13680000.00

Forward

7.4

1.4

12

13

34

17

Kwame Brown

Piston

4000000.00

center

5

0.6

4.2

7

26

17

Vince Carter

Nets

14724125.00

Guard

5.1

4.7

20.8

10

32

20

Bobby Simmions

Nets

9920000.00

Forward

3.9

1.3

7.8

6

28

20

Brook Lopez

Nets

2098560.00

Center

8.1

1

13

1

20

20

Jose Calderon

Raptors

7348018.00

Guard

2.9

8.9

12.8

5

27

22

Chris Bosh

Raptors

14410581.00

Forward

10

2.7

22.7

5

24

22

Jermaine O`Neal

Raptors

21372000.00

Center

7

1.6

13.5

12

30

22

Monta Ellis

Warriors

11000000.00

Guard

4.3

3.7

19

3

23

24

Corey Maggette

Warriors

8600000.00

Forward

5.5

1.6

18.6

9

29

24

Andris Biedrins

Warriors

9000000.00

Center

11.6

2

11.9

4

22

24

O.J Mayo

Grizzlies

3875040.00

Guard

3.8

3.2

18.5

1

21

26

Hakim Warrick

Grizzlies

2119102.00

Forward

5

0.8

11.6

5

23

26

Hamed Haddadi

Grizzlies

1572221.00

Center

2.5

0.4

2.5

1

23

26

Baron Davis

Clippers

11250000.00

Guard

3.7

7.7

14.9

9

29

28

Marcus Camby

Clippers

10000000.00

Forward

11

2

10.3

12

34

28

Chris Kaman

Clippers

9500000.00

Center

8

1.5

12

5

26

28

Gilbert Arenas

Wizards

14653466.00

Guard

4.5

10

13

7

27

30

Antawn Jamison

Wizards

9923285.00

Forward

8.5

1.9

22.2

10

32

30

Brendan Haywood

Wizards

5500000.00

Center

7.3

1.3

9.7

7

29

30

Thedata was then analyzed using regressions. Descriptive Statisticalanalysis of the income gives the following information tabled below.

Table2: Descriptive statistics for the dependent variable Y (income $).

Income/ Salary $

Mean

11046181.76

Standard Error

877506.5421

Median

10920807.5

Mode

14410581

Standard Deviation

5686892.36

Sample Variance

3.23407E+13

Kurtosis

-0.681917666

Skewness

0.099271633

Range

19799779

Minimum

1572221

Maximum

21372000

Sum

463939634

Count

42

Confidence Level (95.0%)

1772160.394

Thisshows the average income of a player is $11,046,182.

Aregression analysis done on the variables regressed against incomegives the graphs shown in the figures below.

Regressionon income against team ranking gives the results in figure 1 below.

Figure1: Graph of Y against X1

Regressionon income against years of experience gives the results in figure 2below.

Figure2: Graph of Y against X2

Regressionon income against player’s age gives the results shown in figure 3below.

Figure3: Graph of Y against X3

Regressionon income against points per game gives figure 4 below.

Figure4: graph of Y against (X4)

Thetables below give an analysis of variability for the dependentvariable Y (income $) against the four variables or predictors X1,X2,X3andX4.

Table3: ANOVA test for variable X1

SUMMARY OUTPUT

Regression Statistics-team ranking

Multiple R

0.24139196

R Square

0.05827008

Adjusted R Square

0.03472683

Standard Error

5587276

Observations

42

ANOVA

&nbsp

df

SS

MS

F

Significance F

Regression

1

7.72644E+13

7.72644E+13

2.475023011

0.123545169

Residual

40

1.24871E+15

3.12177E+13

Total

41

1.32597E+15

&nbsp

&nbsp

&nbsp

&nbsp

Coefficients

Standard Error

t Stat

P-value

Lower 95%

Upper 95%

Intercept

13147478.1

1589741.988

8.270196176

3.4139E-10

9934489.731

16360466.48

Team ranking

-139422.51

88622.3506

-1.573220586

0.123545169

-318534.956

39689.94371

Thisresults show that team ranking is not significant at 95% confidencelevel because the p-value &gt 0.05. The intercept is negativeindicating that a drop in team ranking negates the salary of theplayer. The F-statistic is 2.47.

Table4: ANOVA test for variable X2

SUMMARY OUTPUT

Regression Statistics- years of experience

Multiple R

0.678616509

R Square

0.460520366

Adjusted R Square

0.447033375

Standard Error

4228871.296

Observations

42

ANOVA

&nbsp

df

SS

MS

F

Significance F

Regression

1

6.1064E+14

6.10636E+14

34.14552377

7.84644E-07

Residual

40

7.1533E+14

1.78834E+13

Total

41

1.326E+15

&nbsp

&nbsp

&nbsp

&nbsp

Coefficients

Standard Error

t Stat

P-value

Lower 95%

Upper 95%

Lower 95.0%

Intercept

3308479.811

1476221.97

2.241180445

0.030631907

324923.948

6292035.673

Years of experience

1079679.342

184768.487

5.843417131

7.84644E-07

706248.3039

1453110.38

&nbsp

Table5: ANOVA test for variable X3

SUMMARY OUTPUT

Regression Statistics- age of player

Multiple R

0.586076186

R Square

0.343485296

Adjusted R Square

0.327072428

Standard Error

4665080.793

Observations

42

ANOVA

&nbsp

df

SS

MS

F

Significance F

Regression

1

4.55451E+14

4.55451E+14

20.9278052

4.54101E-05

Residual

40

8.70519E+14

2.1763E+13

Total

41

1.32597E+15

&nbsp

&nbsp

&nbsp

&nbsp

Coefficients

Standard Error

t Stat

P-value

Lower 95%

Upper 95%

Lower 95.0%

Intercept

-11331119.05

4944224.624

-2.291788887

0.027259563

-21323769.7

-1338468.439

Age of player

817257.9427

178647.6501

4.574691815

4.54101E-05

456197.5771

1178318.308

&nbsp

Table6:ANOVAtest for variable X3

Regression Statistics-points per game

Multiple R

0.44335508

R Square

0.19656373

Adjusted R Square

0.15427761

Standard Error

5140681.94

Observations

21

ANOVA

&nbsp

df

SS

MS

F

Regression

1

1.22842E+14

1.22842E+14

4.648422055

Residual

19

5.02106E+14

2.64266E+13

Total

20

6.24948E+14

&nbsp

&nbsp

&nbsp

Coefficients

Standard Error

t Stat

P-value

Intercept

3499077.08

3121868.534

1.120827813

0.276328577

Points per game

457373.108

212137.6968

2.156019957

0.044110432

Theresults obtained from the figures above show that the experience of aplayer and the age are significant at 95% confidence levels indetermining the income of a player because their p-values are lessthan 0.05. Team ranking and points per game do not significantlydetermine income. A player with no experience is expected to have anincome of $3,308,480. Figures 1, 2, 3 and 4 confirm the expectationsand predictions made in this paper about how the variables arerelated. The expectations were correct and they are furthersupported by the ANOVA results in tables 3, 4 and 5. As for thehypotheses, the following can be concluded:

  1. Hypothesis 1: accept the null hypothesis. Team ranking is not significant in determining the income of the player.

  2. Hypothesis 2: reject null hypothesis. Experience of the player is significant in determining the income of a player.

  3. Hypothesis 3: reject null hypothesis. Age of the player is significant in determining the income of the player.

  4. Hypothesis 4: accept null hypothesis. Points per game do no determine the income of a player significantly.

Theprediction lines or models that confirm the expectations are:-

  1. For relationship between team ranking and income Y = 13147478 – 139423X1. This is a negative relationship since the slope of the model line is negative.

  2. For the years of experience and income Y = 3308479 + 1079679X2. This is a positive relationship. It is also a model with the highest positive slope indicating that this is the strongest factor that affects the income of the player.

  3. For the age and income Y = 11331119 + 817258X3. This is a positive relationship. The slope is lower than that of years of experience.

  4. For points per game Y = 3499077 + 457373X4. This is a positive relationship and therefore an increase in points per game will result in an increase in income although the increase is not significant.

Fromthe R-square value, the effect of ranking on income is only 5.8%. The t-value is negative hence showing a negative correlation betweenthe two variables and therefore team ranking can not affect incomesignificantly.

Yearsof experience has an R-value of 0.4605. This implies that 46% ofvariability in income of a player is determined by the years ofexperience. The t-value is 5.84 showing that there is a strongcorrelation between the income of a player and years of experience.The F-statistic is 34.

Agehas a positive correlation since the t-value is 4.57 also showingthat age determines the income of a player. With an R-value of0.3434, it means that 34.34% of a players’ income is determined bythe age of the player. The F-statistic is 21.

Pointsper game will determine 19.66% of variability in income as indicatedby the R-square value. The t-value of 2.15 indicates a positivecorrelation and therefore points per game have a positive correlationwith income. The F-statistic is 4.6.

Amodel that would combine the linear relations to fit the multipleregressions would be Y = B1– B2X1+B3X2+B4X3+B5X4in which the constants represented by B will take certain values orproportions of income.

Themultiple regression model considered in this research paper will havesatisfactorily addressed the expectations of the researcher, andthere was enough evidence shown to conclude that the income of aplayer is strongly influenced and determined by the years ofexperience of the player, and the age of the player. However, a dropin team ranking results in a drop in the income of the player. Butan increase in points per game is likely to give an increase inincome although not significant.

Conclusion

Thisresearch paper reveals that the income of a player will depend on anumber of variable factors or predictors. The variables that wereconsidered in this case were team ranking, years of experience of theplayer, age of the player and points per game. In order to makeproper judgment, statistical tools were used in analyzing datacollected from several teams. The results clearly indicated that theincome of a player is determined by these factors in differentstrengths. Experience of the player was the strongest determiningfactor compared to team ranking, age of the player and points pergame. This implies that the contract of a player must be subjectedto a multiple regression model developed by management in order todecide on the salary or income of the player. The model willconsider all the necessary factors to arrive at the income of theplayer at a given time. This will remove complains that may arisefrom players about differing incomes, and it will also present anobjective way of paying the players of the team.

However,for the results to have a standard model that can be applied overtime, sampling of the teams should be wide, covering a large numberof teams which can give a proper representation of the population. In this paper, 14 teams were considered in the data collection andworking out the correlations. A more standard model can be developedif 30 or more teams are considered for statistical analyses. It isalso assumed that the data collected is correct and honest asconcerns variables like age of the player and years of experience. If the data is biased, the model developed will also be biased andthis will ultimately be a limitation to the application of the modelas a tool for determining the income of a player. Nevertheless,determining the income of a player requires the management to giveweight to a number of variables as indicated in this paper withoutsubjectivity, before deciding on the salary of a player.

Futureresearchers can also test whether the total number of games a playerhas undertaken can also be a significant factor in determining theincome of the player. But contemporary managers cannot afford to paya player without looking at several influencing factors such as theones considered in this paper.

References

Matlab. &quotMatlab and simulink.&quot Matlab. 10 November 2014 &lthttp://www.mathworks.com/help/stats/f-statistic-and-t-statistic&gt.

Summary.

Playersincome based on Team ranking, Years of experience, Age of player, andpoints per game.

Variabilityof income for many players is based on a number of factors whichinclude team ranking, years of experience age of the player andpoints per game. Questions often arise as to why some players arepaid less as compared to others either in the same team or adifferent one, but with the same roles. Construction of a model toanalyze data can help in getting some answers to such questions andhence help management to make decisions that are unbiased.

Inthis research paper, data was collected from 14 teams during the 2008and 2009 season and recorded as in table 1 of this research paper. ANOVA tests and regressions were carried out multiple linear modelsto address several expectations of the researchers. From the resultsof the models, conclusions were made to address the question ofvariability in income of players.

Thetests were done at 95% confidence levels and found out to agree withthe expectations of the research team. It was established that theexperience of the player, the age and points per game have a positiveeffect on variability in income. This implies that a contract signedbetween a player and a given team will definitely look into these andother factors modeled by the teams in order to determine the salaryof a given player. However, team ranking and points per game, thoughpositive in linear relation did not account for significantvariability in the income of a player.

Fromthese tests and results, a multiple regression model was developed todetermine the income of the player by considering the variables orpredictors. It is clear from this research paper that paying aplayer without making considerations of the factors one is subjectedto may result in under payment or over payment hence resulting incomplains. To avoid this problem, it is necessary to develop astandard model that would be used in determining the income ofplayers.