Question

Runs and Wins in Baseball In Exercise 2.150 on page \(104,\) we looked at the relationship between total hits by team in the 2014 season and division (NL or AL) in baseball. Two other variables in the BaseballHits dataset are the number of wins and the number of runs scored during the season. The dataset consists of values for each variable from all 30 MLB teams. From these data we calculate the regression line: \(\widehat{\text { Wins }}=34.85+0.070(\) Runs \()\) (a) Which is the explanatory and which is the response variable in this regression line? (b) Interpret the intercept and slope in context. (c) The San Francisco Giants won 88 games while scoring 665 runs in 2014. Predict the number of games won by San Francisco using the regression line. Calculate the residual. Were the Giants efficient at winning games with 665 runs?

Short answer

The explanatory variable is 'Runs' and the response variable is 'Wins'. The intercept has no practical interpretation in this context and the slope indicate for every additional run scored, there is an increase of \(0.070\) games won on average. The Giants were predicted to win \(83\) games and the residual was \(5.05\), indicating they were more efficient at winning games than the typical team in 2014.

Step-by-step solution

Identify the Explanatory and Response Variables

In the context of the regression analysis in this exercise, the explanatory variable, also known as the independent variable, is 'Runs', which possibly explains or predicts the 'wins'. The response variable, also known as the dependent variable, is 'Wins', which is the outcome we are interested in predicting or explaining.

Interpretation of Intercept and Slope

The intercept \( 34.85 \) is the estimated number of games a team would win if they scored no runs, which is not physically possible, so in this context, the intercept has no practical interpretation. The slope \( 0.070 \) means that for each additional run scored, we predict the team will win an additional \(0.070\) games on average.

Predict Number of Games won by Giants and Calculate the Residual

To predict the number of games won by the San Francisco Giants, we substitute the number of runs \(665\) into our regression equation: \( \widehat{\text { Wins }} = 34.85 + 0.070 \times 665 = 82.95 \). This means the Giants would have expected to win about \(83\) games based on the number of runs they scored. The residual is the difference between the observed and predicted responses: \( \text { residual } = \text { observed } - \text { predicted } = 88 - 82.95 = 5.05 \). A positive residual of \(5.05\) suggests that the Giants were more efficient at converting runs into wins than the typical team in 2014.

Key concepts

Explanatory and Response Variables in Regression Analysis

Understanding the relationship between two variables is fundamental to regression analysis. When we perform a regression, we have an explanatory variable (or independent variable) and a response variable (or dependent variable). The explanatory variable is used to explain, or predict, the values of the response variable.

For instance, in the context of baseball, if we are looking at how the number of runs a team scores can predict the number of games they win, 'Runs' would be our explanatory variable. It's the factor we think influences changes in the 'Wins', which is our response variable. By plotting these variables on a scatter plot, with 'Runs' on the x-axis and 'Wins' on the y-axis, we establish a visual representation of the relationship we want to investigate. A regression line then maps out the predicted 'Wins' for each value of 'Runs', giving us a clearer picture of our data's trend.

Interpretation of Intercept and Slope

In regression, a line is described by the equation \(y = mx + c\), where \(m\) represents the slope and \(c\) is the y-intercept. Translating this into the context of our baseball example, we have the regression line \(\hat{\text{Wins}} = 34.85 + 0.070\times\text{Runs}\).

• The intercept (\(34.85\)) theoretically defines the number of wins a team would have if they didn't score any runs. It's a starting value, but in real-life applications like baseball, it often lacks a practical interpretation since a team can't win games without scoring runs.
• The slope (\(0.070\)) is much more insightful as it represents the average increase in the number of wins as a result of one additional run being scored. Put simply, a slope of \(0.070\) suggests that for each extra run, a team's wins are predicted to increase by \(0.070\) games on average.

These components of the regression line are vital for interpreting the data and making predictions.

Prediction and Residual Calculation

Regression analysis allows us to make predictions about one variable based on the value of another. Using the specified regression equation, we can estimate outcomes, which in our case is the number of wins (\(\hat{\text{Wins}}\)) a baseball team is predicted to have given a certain number of runs.

To calculate the residual, we measure the difference between the observed outcome (actual number of wins) and the predicted outcome (the value given by the regression line). The formula for this is: \(\text{residual} = \text{observed} - \text{predicted}\). A positive residual shows that the actual wins were higher than predicted, which could be interpreted as a team being more efficient at converting runs into wins than average. Conversely, a negative residual would indicate less efficiency. Calculating and analyzing residuals helps us understand the accuracy of our regression model and the performance of teams relative to expectations.