Question

Using 10 years of National Football League (NFL) data, we calculate the following regression line to predict regular season wins (Wins) by number of wins in the 4 pre-season games (PreSeason): \(\widehat{\text { Wins }}=7.5+0.2(\) PreSeason \()\) (a) Which is the explanatory variable, and which is the response variable in this regression line? (b) How many wins does the regression line predict for a team that won 2 games in pre-season? (c) What is the slope of the line? Interpret it in context. (d) What is the intercept of the line? If it is reasonable to do so, interpret it in context. If it is not reasonable, explain why not. (e) How many regular season wins does the regression line predict for a team that wins 100 preseason games? Why is it not appropriate to use the regression line in this case?

Short answer

The explanatory variable is 'PreSeason' and the response variable is 'Wins'. A team that won 2 preseason games is expected to win approximately 8 regular season games. The slope of the line is 0.2, suggesting an increase of 0.2 regular season wins per preseason win. The y-intercept of 7.5 does not provide a reasonable interpretation in this context. Predicting a team winning 100 preseason games would result in 27.5 regular season wins, which is impossible and thus, not a suitable use of this regression model.

Step-by-step solution

Identify the Explanatory and Response Variables

In the regression equation given, the variable 'PreSeason' determines or explains the variable 'Wins'. Therefore, 'PreSeason' is the explanatory variable, while 'Wins' is the response variable.

Predict the Number of Wins Given PreSeason Wins

To determine the number of predicted wins given 2 preseason wins, substitute 'PreSeason' with 2 in the regression equation: \(\widehat{Wins} = 7.5 + 0.2 * 2 = 7.5 + 0.4 = 7.9.\) Therefore, a team that won 2 games in preseason is expected to win around 8 games in the regular season.

Interpret the Slope

The slope in this scenario is 0.2. This suggests that for every additional preseason win, the model predicts an increase of 0.2 wins in the regular season.

Interpret the Y-Intercept

The y-intercept in the context of this problem is 7.5, but there isn't a contextually meaningful explanation for the y-intercept in this case since typically the number of preseason games does not go to 0 nor negative.

Predict Outliers and Discuss Reasonableness

If a team wins 100 preseason games, substituting into the equation will give us \(\widehat{Wins} = 7.5 + 0.2 * 100 = 27.5\). But the maximum number of regular season games is less than this (16), so this is not a reasonable prediction, which exposes one of the limitations of this regression model in predicting extreme cases.

Key concepts

Explanatory Variable

In the context of simple linear regression, the explanatory variable is the one we believe provides some explanation for the variation in another variable. In our NFL example, the variable 'PreSeason' represents the number of wins a team has achieved during the preseason. This is the explanatory variable because we are using it to explain changes or variations in the 'Wins', which is the number of wins in the regular season. Understanding the explanatory variable is essential as it is the foundation upon which predictions are made. One key point to remember is that the explanatory variable is the one manipulated or controlled in an experimental study or observed in a correlational study to determine the effect on the response variable.

The incorporation of explanatory variables into regression models helps to formulate an equation that demonstrates the relationship between two related aspects of the data. In our specific exercise, we are investigating how preseason performance may potentially impact regular season success.

Response Variable

On the flip side of the explanatory variable, we have the response variable, also known as the dependent variable. This is what we seek to predict or understand changes in - in our NFL data scenario, it's the 'Wins' during the regular season. The response variable is essentially the outcome interest. The regression model aims to predict the response variable based on the given values of the explanatory variable.

In the exercise, by analyzing football game data, we assess how well the preseason performance (explanatory variable) can predict the regular season outcomes (response variable). Understanding the dynamic between the explanatory and response variables allows us to create more accurate models for prediction and make thoughtful conclusions based on the regression analysis.

Regression Line Prediction

The regression line prediction is the heart of making forecasts in simple linear regression. It represents the best fit line through the data points on a scatter plot. This line is derived from the relationship between the explanatory and response variables. For our football data, the regression equation \(\widehat{\text{Wins}} = 7.5 + 0.2\text{(PreSeason)}\) predicts the regular season wins based on preseason wins. The process involves substituting different values of 'PreSeason' into the equation to get predicted 'Wins'.

For example, for a team with 2 preseason wins, the regression line predicts approximately 8 regular season wins. It's essential to understand that these predictions are based on aggregated data and are probabilistic. They indicate trends rather than certainties, and using past data to predict future outcomes always comes with a degree of uncertainty.

Slope Interpretation

The slope of the regression line is a critical component in interpreting the relationship between variables. It represents the rate at which the response variable 'Wins' changes for a one-unit change in the explanatory variable 'PreSeason'. In our exercise, the slope is 0.2. What this tells us is that for every additional win in the preseason, the model predicts an increase of 0.2 wins in the regular season. This interpretation provides valuable insight into how influential preseason success is on the outcomes of the regular season according to the data analyzed.

Understanding the slope gives us a sense of the strength and direction of the relationship. As we put this into context, it suggests that preseason performance does have a positive, albeit small, effect on regular season success, according to this model.

Y-Intercept Significance

Finally, we consider the y-intercept in a regression model, which is the value of the response variable when the explanatory variable is zero. The y-intercept in the NFL regression equation is 7.5. This would theoretically represent the predicted wins in the regular season if a team had zero preseason wins. However, interpreting the y-intercept within the context of this scenario isn't necessarily reasonable, because having zero preseason wins is a scenario that doesn't typically occur, and the model is not designed to predict outcomes for such extreme cases.

Nevertheless, the y-intercept is a fundamental part of the regression equation because it adjusts the height of the line and completes the formula needed to make predictions for other values of the explanatory variable. The y-intercept is critical in framing the regression line correctly within the coordinate space of the data points.