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Chapter 15: Problem 58

One measurement of the quality of a quarterback in the National Football League is known as the quarterback passer rating. The rating formula is \(R(c, t, i, y)=\frac{50+20 c+80 t-100 i+100 y}{24},\) where \(c \%\) of a quarterback's passes were completed, \(t \%\) of his passes were thrown for touchdowns, \(i \%\) of his passes were intercepted, and an average of \(y\) yards were gained per attempted pass. a. In the \(2016 / 17\) NFL playoffs, Atlanta Falcons quarterback Matt Ryan completed \(71.43 \%\) of his passes, \(9.18 \%\) of his passes were thrown for touchdowns, none of his passes were intercepted, and he gained an average of 10.35 yards per passing attempt. What was his passer rating in the 2016 playoffs? b. In the 2016 regular season, New England Patriots quarterback Tom Brady completed \(67.36 \%\) of his passes, \(6.48 \%\) of his passes were thrown for touchdowns, \(0.46 \%\) of his passes were intercepted, and he gained an average of 8.23 yards per passing attempt. What was his passer rating in the 2016 regular season? c. If \(c, t,\) and \(y\) remain fixed, what happens to the quarterback passer rating as \(i\) increases? Explain your answer with and without mathematics.

Short Answer

Expert verified
Answer: As the number of intercepted passes increases, the quarterback's passer rating will decrease, negatively affecting their overall performance rating.

Step by step solution

01

Calculate Matt Ryan's passer rating

We are given Matt Ryan's performance statistics: \(c = 71.43\%\), \(t = 9.18\%\), \(i = 0\%\), and \(y = 10.35\). Plug these values into the rating formula \(R(c, t, i, y)=\frac{50+20c+80t-100i+100y}{24}\): \(R(71.43, 9.18, 0, 10.35) = \frac{50 + 20(71.43) + 80(9.18) - 100(0) + 100(10.35)}{24}\). Now, let's calculate the passer rating.
02

Calculate Tom Brady's passer rating

We are given Tom Brady's performance statistics: \(c = 67.36\%\), \(t = 6.48\%\), \(i = 0.46\%\), and \(y = 8.23\). Plug these values into the rating formula \(R(c, t, i, y)=\frac{50+20c+80t-100i+100y}{24}\): \(R(67.36, 6.48, 0.46, 8.23) = \frac{50 + 20(67.36) + 80(6.48) - 100(0.46) + 100(8.23)}{24}\). Now, let's calculate the passer rating.
03

Analyze the effect of increasing intercepted passes

We will analyze the change in quarterback passer rating as \(i\) increases, keeping other variables \(c\), \(t\), and \(y\) constant. When \(i\) increases, the term \(-100i\) in the numerator will decrease. As a result, the numerator of the formula will decrease, while the denominator remains the same. This means that the overall passer rating will decrease as \(i\) increases. In conclusion, if the number of intercepted passes increases, the quarterback's passer rating will be negatively affected.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sports Statistics in Calculus
In the realm of sports analytics, calculus offers a sophisticated methodology to measure and evaluate performance. By using derivatives and integrals, we can better understand the dynamic aspects of sports statistics such as velocity, acceleration, and momentum. In the specific case of determining a quarterback's passer rating in football, calculus does not come into play directly within the formula provided. The passer rating is computed using a formula that is primarily algebraic, where each component of a quarterback's performance—completion percentage, touchdown percentage, interception percentage, and average yards gained per attempt—is plugged directly into an evaluation equation.

However, understanding the underlying patterns and trends of these statistics over time could benefit from calculus. For instance, if we're interested in finding out the rate at which a quarterback's performance is improving or declining over successive games, the derivative of their performance function with respect to time can provide that insight. This is where calculus intertwines gracefully with sports statistics, turning raw data into powerful information about a player's trajectory.
Application of Calculus in Sports Analytics
The application of calculus in sports analytics lies in the ability to model and predict trajectories, rates of change, and optimal decision-making scenarios. Building on the concepts of differentiation and integration, analysts can forecast a player's future performance based on past statistics. For a quarterback, besides the basic assessment using the passer rating, calculus enables analysts to create more sophisticated models that could predict how changes in their training or strategy might influence their future statistics.

For example, using integral calculus to predict total yards gained over a season based on per-game averages, or differential calculus to analyze the sensitivity of the passer rating to each input variable, can provide deeper insights. These models help in decision-making processes such as drafting players or adjusting training programs. The mathematical principles of calculus thus become indispensable tools as they help quantify the seemingly qualitative aspects of sports performance.
Interpreting Sports Performance Data
Interpreting sports performance data is all about transforming numbers into meaningful insights. While the quarterback passer rating formula may seem simple, it is a precisely structured equation to evaluate performance effectively. Each variable within this formula corresponds to an aspect of the quarterback's game which when taken together, yield a composite picture of overall efficiency.

The interpretation process involves examining the weights assigned to each aspect of performance, as indicated by coefficients in the formula. For a quarterback, a higher completion percentage and average yards per attempt are both positively correlated with a good performance and thus have positive coefficients. Conversely, interceptions are harmful, reflected by the negative coefficient in the formula. An increase in interception percentage would therefore lead to a decrease in the overall rating, illustrating how a single variable can impact the composite assessment of a player.

Being able to interpret this data allows coaches, commentators, and fans to have a more nuanced appreciation of a quarterback's contributions to the game. When these insights are shared with the public, they enhance the spectator experience by providing a deeper understanding of the intricacies of the sport.

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