Problem 14 - Calculus and its applications

Question

It has been established that most world records in track and field can be modeled by a linear function. The table below shows world high-jump records for various years. $$ \begin{array}{|cc|} \hline \begin{array}{c} \text { NUMBER OF YEARS, } x, \\ \text { SINCE } 1912 \end{array} & \begin{array}{c} \text { WORLD RECORD IN HIGH } \\ \text { JUMP, } y \text { (in inches) } \end{array} \\ \hline 0 \text { (George Horme) } & 78.0 \\ 44 \text { (Charles Dumas) } & 84.5 \\ 61 \text { (Dwight Stones) } & 90.5 \\ 77 \text { (Javier Sotomayer) } & 96.0 \\ \text { 81 (Javier Sotomayer) } & 96.5 \end{array} $$ a) Find the regression line, \(y=m x+b\). b) Use the regression line to predict the world record in the high jump in 2020 and in \(2050 .\) c) Does your answer in part (b) for 2050 seem realistic? Explain why extrapolating so far into the future could be a problem.

Step-by-Step Solution

Answer

Find the regression line equation, predict values for 2020 and 2050, and analyze the reasonableness of long-term projections using that equation.

Step 1: Understanding the Data

We are given a set of data points representing world high-jump records over various years since 1912. Our goal is to find a linear regression line (a line of best fit) that models these data points.

Step 2: Setting Up the Regression Formula

A linear regression line is defined by the equation \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. Our task is to calculate these values using the given data points.

Step 3: Calculating the Slope \( m \)

The slope \( m \) of the regression line is calculated using the formula \( m = \frac{N(\sum xy) - (\sum x)(\sum y)}{N(\sum x^2) - (\sum x)^2} \), where \( N \) is the number of data points. Plug the data into the formula to find \( m \).

Step 4: Calculating the Y-intercept \( b \)

Once the slope \( m \) is determined, calculate the y-intercept \( b \) using the formula \( b = \frac{(\sum y) - m(\sum x)}{N} \). Substitute the values from the data set into the formula to find \( b \).

Step 5: Forming the Regression Line

With \( m \) and \( b \) calculated, form the linear regression equation \( y = mx + b \), which models the high-jump records as a linear function over time since 1912.

Step 6: Predicting Future Records

Substitute \( x = 108 \) and \( x = 138 \) into the regression equation to predict the high-jump records for the years 2020 and 2050, respectively.

Step 7: Evaluating the Predictive Realism of 2050

Consider the limitations of using a linear model to predict far future events, accounting for factors like biological, technical limits, and changes in athletic training, which may not continue linearly.

Key Concepts

Understanding Slope Calculation

Calculating the slope of a linear regression line is a crucial step in data modeling. The slope, represented by the symbol \( m \), indicates how much the dependent variable (in this case, the world high-jump record in inches) changes with respect to changes in the independent variable (years since 1912). A positive slope signifies an upward trend, meaning as years progress, the high-jump records increase. To determine \( m \), we use a specific formula: - \( m = \frac{N(\sum xy) - (\sum x)(\sum y)}{N(\sum x^2) - (\sum x)^2} \) - Here, \( N \) is the number of data points, \( \sum xy \) is the sum of the product of each pair of \( x \) and \( y \), \( \sum x^2 \) is the sum of the squares of \( x \)-values, \( \sum x \) is the sum of the \( x \) values, and \( \sum y \) is the sum of the \( y \) values.
By substituting the actual data into this formula, we can find the slope \( m \) of the regression line. This figure tells us how rapidly the high-jump records change over time. It's essential for later determined aspects of the linear function, like the y-intercept.

Calculating the Y-intercept

The y-intercept, denoted as \( b \), is another essential component of a linear regression line equation. It represents the point where the line crosses the y-axis. In practical terms, this value indicates the estimated high-jump record at the initial year (x = 0), which is 1912 in our dataset. After calculating the slope \( m \), the y-intercept can be determined using the formula: - \( b = \frac{(\sum y) - m(\sum x)}{N} \) - Here, \( \sum y \) is the sum of all \( y \) values, \( m \) is the slope we previously calculated, \( \sum x \) is the sum of all \( x \) values, and \( N \) is the number of data points.
This calculation helps in completing the equation \( y = mx + b \), which provides a full model to predict world high-jump records based on any given year since 1912. The y-intercept helps us understand what initial state or starting point does our data begin from, extrapolating the line back to when no records might exist.

The Role of Data Modeling

Data modeling, particularly through linear regression, is key in predicting trends and making informed guesses about future outcomes from past data. In the context of world high-jump records, linear regression creates a model that simplifies and summarizes the historical data into a comprehensible trend line. This aids in forecasting future world record heights. Key benefits and considerations include: - **Prediction**: You can estimate future high-jump records by extending the regression line beyond the known data. This involves plugging future \( x \) values into the model equation \( y = mx + b \). - **Simplicity**: A linear model reduces complex data into clear trends, as long as the relationship between the variables is mostly linear over the observed range. - **Extrapolation Challenges**: While linear models are great for short-term predictions, their accuracy can falter when used to extrapolate too far into the future, due to unforeseen changes in conditions, technology, or biology.
Thus, while data modeling through linear regression is immensely useful, always consider the realistic limitations and avoid assuming a linear trend indefinitely into the future.