Question

The number of passes EX1242 completed and the total number of passing yards for Tom Brady, quarterback for the New England Patriots, were recorded for the 16 regular games in the 2006 football season. \({ }^{8}\) Week 6 was a bye and no data was reported. $$ \begin{array}{ccc} \text { Week } & \text { Completions } & \text { Total Yards } \\ \hline 1 & 11 & 163 \\ 2 & 15 & 220 \\ 3 & 31 & 320 \\ 4 & 15 & 188 \\ 5 & 16 & 140 \\ 6 & * & * \\ 7 & 18 & 195 \\ 8 & 29 & 372 \\ 9 & 20 & 201 \\ 10 & 24 & 253 \\ 11 & 20 & 244 \\ 12 & 22 & 267 \\ 13 & 27 & 305 \\ 14 & 12 & 78 \\ 15 & 16 & 109 \\ 16 & 28 & 249 \\ 17 & 15 & 225 \end{array} $$ a. What is the least-squares line relating the total passing yards to the number of pass completions for Tom Brady? b. What proportion of the total variation is explained by the regression of total passing yards \((y)\) on the number of pass completions \((x) ?\) c. If they are available, examine the diagnostic plots to check the validity of the regression assumptions.

Short answer

Question: Based on the provided step-by-step solution, calculate the least-squares line and R-squared value for the correlation between pass completions and total passing yards for Tom Brady in the 2006 football season (excluding Week 6). Answer: To calculate the least-squares line and the R-squared value for the given problem, follow the steps provided in the step-by-step solution. First, calculate the means of pass completions and total passing yards. Then, calculate SSX, SCPD, and the slope of the regression line. Next, calculate the y-intercept of the regression line and form the least-squares line equation. Finally, to find the R-squared value, calculate the squared sum of the total passing yards (SSY), the Sum of Squared Errors (SSE), and divide the result by SSY.

Step-by-step solution

1. Calculate the means of pass completions and total passing yards

First, excluding Week 6, calculate the sum of pass completions and total passing yards for all available weeks. Then, divide each sum by the number of available weeks (15) to find the means for both variables.

2. Calculate SSX and SCPD

Using the means calculated in step 1, find the difference between each individual value and their corresponding mean, then square these differences for the pass completions (x) variable. The sum of these squared differences is the SSX. To calculate the SCPD, multiply the differences of pass completions and differences of total passing yards for each week, then sum these products.

3. Calculate the slope of the regression line

Divide SCPD by SSX to get the slope of the regression line (b_1).

4. Calculate the y-intercept of the regression line

Using the slope (b_1) calculated in step 3 and the means of pass completions and total passing yards (calculated in step 1), use the following formula to find the y-intercept of the regression line (b_0): \(b_0 = \bar{y} - b_1 * \bar{x}\)

5. Form the least-squares line

Using the b_0 and b_1 from steps 4 and 3, form the least-squares line equation, \(y = b_0 + b_1x\) Now, we'll calculate the R-squared value:

6. Calculate the squared sum of the total passing yards (SSY)

Using the mean of total passing yards calculated in step 1, find the difference between each individual value and their corresponding mean, then square these differences. The sum of these squared differences is the SSY.

7. Calculate the Sum of Squared Errors (SSE)

Using the least-squares line equation found in step 5, find the predicted y values for each x value. Then, subtract each predicted y-value from the actual y-value, square the differences, and sum them to obtain the SSE.

8. Calculate R-squared

Subtract the SSE from SSY and divide the result by SSY to obtain R-squared, which is the proportion of the total variation explained by the regression. By following these steps, we can calculate the least-squares line and the R-squared value for the given problem. The diagnostic plots are not available, so we cannot check the validity of regression assumptions in part c.

Key concepts

Regression Line

In linear regression, a regression line is a straight line that best fits the data points on a scatter plot. It represents the relationship between the dependent variable (\(y\)) and the independent variable (\(x\)). In our exercise, we find a regression line to relate Tom Brady's passing completions to the total passing yards.
The equation of the regression line is expressed as \(y = b_0 + b_1x\), where \(b_0\) is the y-intercept and \(b_1\) is the slope. The y-intercept \(b_0\) is where the line crosses the y-axis, representing the estimated value of \(y\) when \(x = 0\). The slope \(b_1\) indicates how much \(y\) changes for a one-unit change in \(x\).
When organizing data and calculating these components, it’s essential to use accurate measurements to ensure a reliable model is constructed.

Least Squares Method

The least squares method is a statistical technique used to determine the best-fitting line by minimizing the sum of the squares of the vertical distances (errors) between the observed values and the values predicted by the line.
The steps involve calculating the means of the two datasets, the sum of the squared differences (SSX) for the independent variable, and the sum of the cross-products (SCPD) for both datasets. The slope \(b_1\) is then computed as \(b_1 = \frac{SCPD}{SSX}\).
Once the slope is determined, the y-intercept \(b_0\) is calculated using the formula \(b_0 = \bar{y} - b_1 \bar{x}\), where \(\bar{y}\) and \(\bar{x}\) are the means of the y and x-values, respectively. The least squares line then represents the equation \(y = b_0 + b_1x\).
This method ensures that the total error is minimized, providing a robust line that represents the trend of the data.

Regression Analysis

Regression analysis is a powerful statistical method that allows us to explore the relationship between two or more variables. It helps predict one variable based on the knowledge of others, identify trends, and uncover causal relationships.
In the context of the exercise, regression analysis was used to discern how Tom Brady's pass completions impacted the total passing yards. By plotting its data points and establishing the regression line, we can see how well it predicts passing yards based on completions.
This analysis provides insights into:

• The strength and direction of the relationship (quantified by the slope)
• How well the regression line predicts values (described by R-squared)
• The statistical significance of the model

This helps in making informed decisions or predictions based on the established model.

R-squared

R-squared, represented as \(R^2\), is a statistical measure that represents the proportion of the variance for the dependent variable that's explained by the independent variable(s) in the regression model.
It is calculated as the ratio of explained variation (SSY - SSE) to the total variation (SSY). The value of \(R^2\) ranges between 0 and 1.

• \(R^2 = 1\) indicates that the regression line perfectly fits the data with all observed values explained by the model.
• \(R^2 = 0\) means that the model explains none of the variability of the response data around its mean.

A higher \(R^2\) indicates a better fit for the model. For Tom Brady's completion vs passing yards analysis, \(R^2\) demonstrates how much of the total passing yard variation is accounted for by the number of completions.