91Ó°ÊÓ

In least-squares regression, the slope \( m \) of the line is a critical component. It quantifies the rate of change in the dependent variable \( y \) for every one-unit increase in the independent variable \( x \). This gives us insight into the relationship between \( x \) (number of fouls over the opposing team) and \( y \) (percentage of wins). The formula to calculate the slope is as follows:

\[m = \frac{n(\Sigma xy) - (\Sigma x)(\Sigma y)}{n(\Sigma x^2) - (\Sigma x)^2}\]

Where:

• \( n \) is the number of data points
• \( \Sigma xy \) is the sum of the product of each pair of values \( x \) and \( y \)
• \( \Sigma x \) is the sum of all \( x \) values
• \( \Sigma y \) is the sum of all \( y \) values
• \( \Sigma x^2 \) is the sum of the squares of \( x \) values

Given data such as \( n = 4\), \( \Sigma xy = 411 \), \( \Sigma x = 13 \), \( \Sigma y = 154 \), and \( \Sigma x^2 = 65 \), we can plug these into the formula. After simplification, we find \( m \approx -3.934 \). This slope means there is a negative relationship between \( x \) and \( y \); as the number of excess fouls increases, the winning percentage tends to decrease.

The y-intercept, \( b \), is where the regression line crosses the y-axis. This value represents the expected value of \( y \) when \( x \) is zero. In context, it tells us the predicted percentage of wins when there is no difference in fouls between the teams.

The formula for calculating \( b \) is:

\[b = \frac{\Sigma y - m(\Sigma x)}{n}\]

Here, using the calculated slope \( m \approx -3.934 \), along with known values \( \Sigma y = 154 \), \( \Sigma x = 13 \), and \( n = 4 \), we substitute to find:

\[b = \frac{154 - (-3.934)(13)}{4} = \frac{154 + 51.142}{4} \approx 51.285\]

This indicates that, theoretically, if both teams committed an equal number of fouls, the team with more fouls still manages about a 51.285% win rate. The y-intercept offers a baseline prediction when there aren't any disparities in fouling.

Statistical forecasting using the least-squares regression line enables us to predict future outcomes based on current data. The regression line equation \( y = mx + b \) encapsulates both the slope and y-intercept. We use this equation to forecast the success rate \( y \) based on the number of fouls \( x \).

For example, if a team has made four more fouls than the opposing team (\( x = 4 \)), we substitute \( x \) into the regression line to compute \( y \):

\[y = -3.934(4) + 51.285\]

By doing the calculation:

• \( y = -15.736 + 51.285 \)
• \( y = 35.549 \)

This forecast suggests that with four additional fouls, the expected win percentage decreases to approximately 35.549%. Statistical forecasting provides a valuable tool for anticipating future performance and making informed decisions.