Question

Player Salary PC TD Age

1 25.5566 65.2 28 27
2 22.0441 60.5 27 26
3 20.5000 62.3 27 28
4 19.0047 66.1 26 38
5 17.0000 67.9 29 28
6 15.0052 55 16 27
7 14.0057 68.8 33 33
8 12.9895 70.6 34 30
9 12.5073 60.3 22 33
10 12.0000 68.4 33 40
11 11.2554 48.8 3 24
12 9.5000 60.5 21 29
13 8.6018 63.1 21 24
14 8.6000 64.7 30 26
15 8.5000 60.9 15 31
16 8.0073 65.7 28 32
17 7.9073 58.3 22 24
18 7.7516 66.6 26 27
19 6.5073 56.7 5 32
20 6.3250 55.5 8 34
21 6.2562 60 17 34
22 4.0073 60.5 18 25
23 3.1000 53.3 13 21
24 2.9956 55.9 9 27
25 2.8648 64.5 20 27
26 2.5450 53.8 12 23
27 2.1673 58.7 10 26
28 2.0079 53.1 8 25
29 1.3800 54.5 10 21
30 1.0950 62.1 21 27
31 0.9503 60.8 12 24
32 0.6260 63.1 26 29

American football is the highest-paying sport on a per-game basis. Given that the quarterback is considered the most important player on the team, he is typically well-compensated. A sports statistician examines the factors that influence a quarterback’s salary. He believes that a quarterback’s pass completion rate is the most important variable affecting salary. He also wonders how total touchdowns scored and a quarterback’s age might impact salary. The statistician collects data on salary (Salary in $ millions), pass completion rate (PC in %), total touchdowns scored (TD), and age for 32 quarterbacks during a recent season. A portion of the data is shown in the accompanying table.

Player Salary PC TD Age
1 25.5566 65.2 28 27
2 22.0441 60.5 27 26
⋮ ⋮ ⋮ ⋮ ⋮
32 0.6260 63.1 26 29

a. Estimate the model defined as Salary = β0 + β1PC + β2TD + β3Age + ε

Answer 1

To estimate the model relating a quarterback's salary to pass completion rate, total touchdowns, and age, a multiple regression analysis is conducted to determine the relevant coefficients that predict salary based on these variables.

Step-by-step explanation:

To estimate the model Salary = β0 + β1PC + β2TD + β3Age + ε, a sports statistician would typically use statistical analysis software to perform a multiple regression analysis. This analysis will assess how much of the variance in a quarterback's salary can be explained by the quarterback's pass completion rate (PC), total touchdowns scored (TD), and age. It involves calculating coefficients β0 (intercept), β1 (slope for PC), β2 (slope for TD), and β3 (slope for Age) that will minimize the sum of squared residuals (ε).

In the context provided, the slope for endorsements (β1) of 1.99 as an example could be interpreted in the football model to imply that for each percentage point increase in the pass completion rate, a player's salary would increase on average by 1.99 million dollars, if all other factors remain constant. However, to determine the actual slopes and the intercept for the NFL quarterback salary model, we would need to see the regression output which would give us the coefficients (β0, β1, β2, β3) and the significance of each predictor. Note that this is just an illustrative explanation, actual model results could significantly vary.

The model aims to determine the relationship between a quarterback's salary and their pass completion rate, total touchdowns scored, and age.

The final model can be written as Salary = β0 + β1PC + β2TD + β3Age + ε, where the coefficients β1, β2, and β3 represent the impact of PC, TD, and Age on the quarterback's salary.

Factors such as global demand for 'stars' and the related notion of a winner-take-all labor market are contextual considerations that emphasize the disparities in salaries that might be seen even after accounting for statistical models.