Question

You want to determine whether there is a relationship between an athlete's weight \(x\) (in pounds) and the athlete's maximum bench-press weight \(y\) (in pounds). An equation that models the data you obtained is \(y=1.4 x-39\) (a) Estimate the values of \(x\) that predict a maximum bench-press weight of at least 200 pounds. (b) Do you think an athlete's weight is a good indicator of the athlete's maximum bench-press weight? What other factors might influence an individual's bench-press weight?

Short answer

Part A: An athlete needs to weigh at least approximately 171 pounds to have a maximum bench-press weight of 200 pounds. Part B: An athlete’s weight might be one of many factors influencing their bench press capabilities, and it might not be the only or most significant one. Other factors that might influence this could include muscle mass, physical conditioning, genetics, and body fat percentage.

Step-by-step solution

Set up the equation to solve for \(x\)

The first step is to set up the equation \(y=1.4x-39\) to solve for \(x\). In this case, since we want \(y\) to be at least 200, we set up the following inequality: \(1.4x-39\geq200\)

Solve the inequality for \(x\)

We begin by adding 39 to both sides of the inequality, getting \(1.4x\geq239\). Then, we divide by 1.4 on both sides to find \(x\geq170.71\). This indicates that an athlete needs to weigh at least approximately 171 pounds to have a maximum bench-press weight of 200 pounds.

Evaluate the relationship between athlete's weight and bench-press weight

While an athlete’s weight can be indicative of their potential bench pressing capacity, other factors might also influence an individual's bench press weight such as muscle mass, physical conditioning, body fat percentage, or genetics. Data analysis would be needed to confirm or refute this hypothesis. In general, without further data, it might be unwise to say definitively that an athlete's weight is a good, sole indicator of their bench press capabilities.

Key concepts

Inequality Solving

In the world of mathematics, solving inequalities is an essential skill, particularly when dealing with real-world scenarios like the one presented in the exercise. Here, the goal is to estimate the values of an athlete's weight that would predict a maximum bench-press weight of at least 200 pounds. This requires setting up and solving an inequality based on a given linear equation. To tackle this, we start by formulating the inequality from the equation: \(1.4x - 39 \geq 200\). The aim is to isolate \(x\) and find its least possible value. By first adding 39 to both sides, we simplify the inequality to \(1.4x \geq 239\). Dividing every term by 1.4, we determine that \(x \geq 170.71\). Thus, an athlete must weigh at least approximately 171 pounds to bench-press at least 200 pounds. This process of solving inequalities is crucial in many aspects of data analysis and predictive modeling, as it allows us to determine the thresholds or limits within varying scenarios.

Data Analysis

Data analysis involves systematically applying statistical or logical techniques to describe and evaluate data. In the context of this exercise, data analysis is used to understand the relationship between an athlete's weight and their bench-press capacity. Although weight is a factor, relying solely on it as a predictor might be misleading. To gain a comprehensive understanding, it's important to consider other contributing variables:

• Muscle mass, which directly contributes to strength capabilities.
• Physical conditioning and overall health, impacting performance.
• Body fat percentage, affecting strength in various ways.
• Genetics, which can predispose individuals to greater strength.

Utilizing data analysis effectively means considering all these factors to draw meaningful and accurate conclusions. Therefore, it's always important to have a complete dataset for reliable insights.

Predictive Modeling

Predictive modeling is a statistical technique employed to predict future outcomes based on historical data. In this exercise, the linear equation \(y = 1.4x - 39\) provides a predictive model estimating the maximum bench-press weight based on the athlete's body weight.This model assumes a linear relationship between the two variables, suggesting that as the athlete's body weight increases, so does their potential bench-press weight. It's important to acknowledge, however, that predictive models should be used with caution. They usually work best when complemented with a robust analysis of data, accounting for other factors such as those mentioned in data analysis.Furthermore, predictive models should be tested and refined with real-world data for accuracy. By continuously adjusting them based on new insights and data, you can improve their predictive power, making them invaluable tools in sports science and beyond.