Overview of Numerical Methods: Introduction to Analytical Methods in Sports

Abstract

In this chapter we discuss the history of applications of analytical methods to problems in sports and provide an overview of some analytical methods (graphs, probability, regression analysis, and mathematical programming) that are commonly applied to various problems in sports. We consider the long and complicated relationship between statistics and sports and the struggle to gain acceptance for statistical analyses in professional sports. We also highlight some analytical methods commonly applied to sports problems, including graphs, probability and probability models, regression analysis, and mathematical programming. Finally, we briefly discuss ways in which instructors use sports examples to teach analytics methods.

Solutions to Exercises

• 1.

  • (a) The following line plot shows the number of points scored per game by the University of Alabama men’s basketball team during the 2016–2017 regular season.

    

    We can see that the team generally scored between 50 and 80 points per game.

  • (b) The following line plot shows the number of points allowed per game by the University of Alabama men’s basketball team during the 2016–2017 regular season.

    

    We can see that the team generally allowed between 55 and 80 points per game.

  • (c) The following overlay of line plots shows the number of points scored and points allowed per game by the University of Alabama men’s basketball team during the 2016–2017 regular season.

    

    The graph shows that the University of Alabama men’s basketball team won slightly more games than it lost (its regular season record was 17 wins and 13 losses).

• 2.

  • (a) The following histograms show the number of home runs hit at each age by Willie McCovey, Jimmie Foxx, Mike Schmidt, Mel Ott, Ken Griffey, Jr., and Harmon Killebrew.

    

    

    

    

    

    

    We can see that Jimmie Foxx, Mel Ott, and Ken Griffey, Jr. were productive at younger ages and Willie McCovey, Mike Schmidt, and Harmon Killebrew did not become major league home run threats until they were a bit older.

  • (c) The following side-by-side bar chart simultaneously shows the number of home runs hit by Willie McCovey, Jimmie Foxx, Mike Schmidt, Mel Ott, Ken Griffey, Jr., and Harmon Killebrew at each age.

    

    We can see that home run productivity generally increases as a player ages, peaks around ages 27–30, and then decreases as the player ages.

• 3. Data for the 2017 Major League season are provided in the table that follows.

  

  Let x = Spring Training W/L% and y = Regular Season W/L%. We have that:

  $\sum \limits_{i=1}^n{x}_i{y}_i=75492,\sum \limits_{i=1}^n{x}_i=1500,\sum \limits_{i=1}^n{y}_i=1500,\sum \limits_{i=1}^n{x}_i^2=77468,\mathrm{and}\ n=30$

  so

  ${b}_1=\frac{75492-\frac{(1500)(1500)}{30}}{77468-\frac{1500^2}{30}}=0.2063$

  and

  ${b}_0=\frac{1500}{30}-(0.2063)\left(\frac{1500}{30}\right)=39.6906$

  Thus, the estimated Regular Season W/L% ($\hat{y}$) for some Spring Training W/L% (x) is:

  $\hat{y}=39.6906+0.2063x$

  The results suggest that a 1% increase in a team’s Spring Training W/L% (x) coincides with a 0.2063% increase in the team’s Regular Season W/L% (y). A change in a team’s Spring Training W/L% does not appear to coincide with much change in its Regular Season W/L%.

• 4. The number of free throws the WNBA player makes (x) in n = 6 attempts is binomially distributed with p = 0.75, so

  (a) The probability she will make exactly three of these free throw attempts~is:

  $\Pr (3)=\left(\begin{array}{c}6\\ {}3\end{array}\right){0.75}^3{\left(1-0.75\right)}^{6-3}=0.131835938$

  (b) The probability she will make all six of these free throw attempts is:

  $\Pr (6)=\left(\begin{array}{c}6\\ {}6\end{array}\right){0.75}^6{\left(1-0.75\right)}^{6-6}=0.177978516$

  (c) The probability she will make none of these free throw attempts is:

  $\Pr (0)=\left(\begin{array}{c}6\\ {}0\end{array}\right){0.75}^0{\left(1-0.75\right)}^{6-0}=0.000244141$

• 5.

  • (a) IAE’s goal is to maximize its profit, and the profit IAE will earn is a function of the number of standard-size tennis racquets (X ₁) and the number of oversized tennis racquets (X ₂) it produces during the next week. The linear programming model that will determine the number of racquets of each type IAE should manufacture over the next week to maximize the total profit is:

  Maximize	30X 1 + 45X 2
  Subject to	4.4X 1 + 14X 2 ≤ 2850 (ounces of titanium alloy)
  	20X 1 + 24X 2 ≤ 9600 (manufacturing time)a
  	0.8X 1 − 0.2X 2 ≥ 0 (ratio of standard rackets to total racquets produced)b
  	X 1 ≥ 0
  	X 2 ≥ 0

  1. ^aNote that four manufacturing employees can each work 40 h during the week, so total manufacturing time available in minutes is 4 ^∗ 40 ^∗ 60 = 9600.
  2. ^bNote that the constraint for the ratio of standard racquets to total racquets produced can be written X ₁ ≥ 0.2(X ₁ + X ₂), and rearranging the terms in this constrain gives us 0.8X ₁ − X ₂ ≥ 0.

• (b) The number of standard-size tennis racquets (X ₁) = 378.440367, the number of oversized tennis racquets (X ₂) = 84.63302752, and the maximum total profit is $15,161.69725.

• (c) The optimal number of standard-size tennis racquets and oversized tennis racquets to produce during the week are not integer. We could address this concern by adding the following two constraints to the formulation:

X ₁ ∈ 

X ₂ ∈ 

and resolving the problem as an integer programming problem. For this integer programming problem, the number of standard-size tennis racquets (X ₁) = 379, the number of oversized tennis racquets (X ₂) = 84, and the maximum total profit is $15,150.00 (note that simply rounding the optimal values of the decision variables from a linear programming problem does not generally yield the optimal integer solution).