Overview of Numerical Methods: Applications of Analytical Methods in Sports

Abstract

In this chapter, we discuss and highlight research from interesting and relevant publications that feature applications of these analytical methodologies to many economics, education, strategy development and assessment, health issues, and biomechanics.

Appendices

Exercises

• 1. In a study of the relationship between wind assistance and jump distance in the long jump, the researchers collected information on the long jump distances (in m) and the corresponding wind assistances (in m/s) for 90 recent attempts in NCAA competitions. These data are provided in the following table.

  

  Create an appropriate graphical display of the relationship between long jump distance and wind assistance. What relationship does your graph suggest?

• 2. A researcher interested in metatarsalgia in runners has collected data from 1021 randomly selected runners who have been competitive for at least 5 years. Each runner was asked if she or he had been diagnosed with metatarsalgia in the past 5 years. In addition, each runner was classified as a rearfoot runner (strikes the ground with the heel first), midfoot runner (strikes the ground with the foot flat), or a forefoot runner (runs on the balls of the feet). The following table summarizes the data this researcher has collected:

  

• (a) Create a bar chart of the absolute frequencies of runners diagnosed with metatarsalgia in the past 5 years for the three runner classifications (rearfoot, midfoot, and forefoot). What does this graph suggest?

• (b) Create a bar chart of the absolute frequencies of the three runner classifications (rearfoot, midfoot, and forefoot). What does this graph suggest?

• (c) Create a bar chart of the relative frequencies of runners diagnosed with metatarsalgia in the past 5 years for the three runner classifications (rearfoot, midfoot, and forefoot). What does this graph suggest?

• (d) Review the results of parts (a), (b), and (c). Do you think the bar chart in part (a) or the bar chart in part (c) provides a more meaningful description of the incidence of metatarsalgia in the past 5 years for the three runner classifications (rearfoot, midfoot, and forefoot)?

• 3. Collect data on the mean number of goals per game scored by each National Hockey League team for each of the past 25 seasons. Find the minimum, each quintile, and the maximum for each of these seasons and use the results to create a change point plot. Does your graph suggest any systematic changes in scoring in the National Hockey League over the past 25 seasons?

• 4. The following data on Major League Baseball franchise value (in $ million) and age of the stadium in which the franchise plays its home games for the 2017 season have been collected from Forbes.com and BallparkDigest.com, respectively.

  

  Perform a regression analysis similar to Miller (2009) on these data to estimate the relationship between the value of the Major League Baseball franchise and the age of the team’s playing facility.

• (a) What is the estimated change in Major League Baseball franchise value (in $ million) associated with a 1 year increase in the age of stadium in which the team plays?

• (b) Are your results similar to those reported by Miller? If not, what could explain the discrepancy between Miller’s findings and your results?

• 5. The following table contains the annual total gross domestic product (GDP) in millions of dollars for the Oklahoma City metropolitan statistical area (MSA) from 2001 through 2016 as reported by the Federal Reserve Bank of St. Louis.

  

  After the 2008 season, the National Basketball Association’s Seattle SuperSonics relocated to Oklahoma City and were renamed the Oklahoma City Thunder; the Thunder played its first season in Oklahoma City in 2009. Use the data in this table and regression analysis to assess the impact of the Thunder on the Oklahoma City MSA’s economy.

• (a) Create a new variable called LagGDP that is equal to the GDP of the previous year (i.e., in 2002 LagGDP = 37,158, in 2003 LagGDP = 38,574, etc.). Perform a regression analysis on the data for the years prior to the relocation of the Seattle SuperSonics to Oklahoma City (2001–2008) using GDP as the dependent variable and LagGDP as the independent variable. What does your regression model suggest about the Oklahoma City MSA’s economy during this period? Note that you will not be able to use the year 2001 in estimating your regression analysis because you do not have data on the GDP for the year 2000 (and so do not have a value for LagGDP for 2001).

• (b) Use the regression model for the Oklahoma City MSA’s GDP that you estimated in part (a) to estimate the Oklahoma City MSA’s GDP from 2009 to 2016 and calculate the error terms for these years. What does this suggest about the contribution the Oklahoma Thunder has made to the Oklahoma City MSAs GDP?

Solutions to Exercises

• 1. The appropriate graphical display of the relationship between long jump distance and wind assistance is a scatter plot. Such a scatter plot follows.

  

As one would expect, this graph suggests that wind assistance and long jump distance have a positive relationship.

• 2.

  • (a) A bar chart of the absolute frequencies of runners diagnosed with metatarsalgia in the past 5 years for the three runner classifications (rearfoot, midfoot, and forefoot) follows.

  

• (b) This graph suggests that far more rearfoot runners than midfoot or forefoot runners were diagnosed with metatarsalgia over the past 5 years.

• (c) A bar chart of the absolute frequencies of rearfoot, midfoot, and forefoot runners follows.

  

• (d) This graph suggests that the rearfoot running technique is used far more frequently than the midfoot or forefoot running techniques.

• (e) A bar chart of the relative frequencies of rearfoot, midfoot, and forefoot runners follows.

• (f) This graph suggests that rearfoot runners were diagnosed with metatarsalgia far more frequently than midfoot or forefoot runners over the past 5 years.

• (g) The graph in (a) is misleading because, as the graph in (b) shows, the rearfoot running technique is used far more frequently than either the midfoot or forefoot running technique. Thus, the relative frequency graph in (c) more meaningful and revealing.

• 3. The number of goals scored by each National Hockey League team for each of the past 25 seasons is provided in the table below:

  

  

  The minimum, each quintile, and the maximum for each of these seasons are provided in the flowing table:

  

  

  Using these results to create a change point plot yields the following graph:

  

  First note the abrupt decreases in goals scored during the 1994–1995 and 2012–2013 seasons. These are primarily due to the 1994–1995 NHL lockout and the 2012–2013 NHL lockout, which each shortened the NHL regular season to 48 games.

  Next note the gap in the graph during the 2004–2005 season. This is due to the 2004–2005 NHL lockout, which resulted in cancellation of the entire 2004–2005 NHL season.

  The NHL’s labor troubles make it somewhat difficult to assess whether any changes occurred in the distribution of goals scored by teams across these seasons, but across the remaining years there appear to be either no dramatic shift or possibly a very slight downward trend.

  Note that your graph and conclusions may differ if you collect data over a different set of 25 NHL seasons.

• 4. (a) Let x = Age of Stadium in Which the Team Plays and y = Major League Baseball Franchise Value. We have that:

  $\sum \limits_{i=1}^n{x}_i{y}_i\,{=}\,1400675,\!\sum \limits_{i=1}^n{x}_i\,{=}\,791,\!\sum \limits_{i=1}^n{y}_i\,{=}\,46105,\!\sum \limits_{i=1}^n{x}_i^2\,{=}\,39155,\mathrm{and}\ n\,{=}\,30\vspace*{9pt}$ so

  $$ {b}_1=\frac{1400675-\frac{(791)(46105)}{30}}{39155-\frac{791^2}{30}}=10.1120 $$

  and

  $$ {b}_0=\frac{46105}{30}-(10.1120)\left(\frac{791}{30}\right)=1270.2126 $$

  Thus, the estimated Major League Baseball Franchise Value ($\hat{y}$) for some Age of Stadium in Which the Team Plays (x) is

  $$ \hat{y}=1270.2126+10.1120x\vspace*{-6pt} $$

  These results suggest that a 1 year increase in Age of Stadium in Which the Team Plays (x) coincides with a $10.112 million increase in the estimated Major League Baseball Franchise Value (y).

  • (b) The results in (a) are not similar to those reported by Miller. This is likely because three of the four most valuable Major League Baseball franchises during this season (Los Angeles Dodgers, Boston Red Sox, and Chicago Cubs) have the three oldest stadiums (these franchises are represented by the three red points in the following scatter plot).

  

  Consider the impact on the model of eliminating the Los Angeles Dodgers, Boston Red Sox, and Chicago Cubs from the data. If we do so and then re-estimate the regression model, we have

  $\sum \limits_{i=1}^n{x}_i{y}_i\,{=}\,690400, \sum \limits_{i=1}^n{x}_i\,{=}\,528, \sum \limits_{i=1}^n{y}_i\,{=}\,37980, \sum \limits_{i=1}^n{x}_i^2\,{=}\,39155,\mathrm{and}\ n\,{=}\,27\vspace*{9pt}$ so

  $$ {b}_1=\frac{690400-\frac{(528)(37980)}{27}}{39155-\frac{528^2}{27}}=\hbox{-} 12.5448 $$

  and

  $$ {b}_0=\frac{37980}{27}-\left(\hbox{-} 12.5448\right)\left(\frac{528}{27}\right)=1651.9864 $$

  The estimated Major League Baseball Franchise Value ($\hat{y}$) for some Age of Stadium in Which the Team Plays (x) is now

  $$ \hat{y}=1651.9864\hbox{-} 12.5448x $$

  These results suggest that a 1 year increase in Age of Stadium in Which the Team Plays (x) coincides with a $12.5448 million decrease in the estimated Major League Baseball Franchise Value (y). This is consistent with Miller’s findings.

  A superior approach would be to include a variable that represents the size of the market (perhaps population) as a second explanatory in a multiple regression.

• 5. (a) Let x = LagGDP and y = GDP. We have that

  $\sum \limits_{i=1}^n{x}_i{y}_i=15295183317,\sum \limits_{i=1}^n{x}_i=314988,\sum \limits_{i=1}^n{y}_i=334749,\sum \limits_{i=1}^n{x}_i^2=14396756506,\mathrm{and}\ n=7\vspace*{6pt}$

  so

  $$ {b}_1=\frac{15295183317-\frac{(314988)(334749)}{7}}{14396756506-\frac{314988^2}{7}}=1.0414 $$

  and

  $$ {\displaystyle \begin{array}{l}{b}_0=\frac{334749}{7}-(1.0414)\left(\frac{314988}{7}\right)=962.0385\end{array}} $$

  Thus, the estimated GDP ($\hat{y}$) for some LagGDP (x) is

  $$ \hat{y}=962.0385+1.0414x $$

  These results suggest that a $1 million increase in LagGDP (x) coincides with a $1.0414 million increase in the estimated GDP (y).

  • (b) The regression model estimates of the Oklahoma City MSA’s GDP from 2009 to 2016 and corresponding error terms are provided in the following table.

  

  The errors are negative and decreasing in magnitude through 2012, and they become positive in 2013, indicating the impact the Oklahoma City Thunder may have had on the MSA’s GDP quickly dissipated after the team arrived. However, the errors were negative and increasing in magnitude from 2014 to 2016, indicating the MSA’s GDP grew faster than predicted by the model over these years. Further investigation would be required to assess if this is in any way attributable to the Oklahoma Thunder’s continued presence in the Oklahoma City MSA.