Question

The scatterplot shows the relationship between the number of yards allowed by teams in the National Football League and the number of wins for that team in a recent season, along with the least-squares regression line. Computer output is also provided.

a. State the equation of the least-squares regression line. Define any variables you use.

b. Calculate and interpret the residual for the Seattle Seahawks, who allowed 4668 yards and won 10 games.

c. The Carolina Panthers allowed 5167 yards and won 15 games. What effect does the point representing the Panthers have on the equation of the least-squares regression line? Explain.

Short answer

1. The required result is $\hat{y}=b_0+b_{1^x}=25.66-0.003131x$.
2. Residual $=-1.044492$.
3. The y intercept increases as the slope decreases.

Step-by-step solution

Part (a) Step 1: Given information

The graph is

Part (a) Step 2: Calculation

In this case, the number of yards allotted is the independent variable $\left(x\right)$, and the number of wins for that team in the most recent season is the dependent variable$\left(y\right)$

The general regression line of equation is as follows:

$\hat{Y}=a+b\hat{X}$.

Let $\hat{Y}$represent the predicted number of vins and $\hat{Y}$represent the predicted number of yards allowed.

The coefficient of a constant term is calculated from the output.$a=25.66$.

The output yields a constant term's coefficient as $b=-0.003131$.

The regression line using least squares is$\hat{Y}=25.66-0.003131\hat{X}$.

Part (b) Step 1: Given information

Given:

x= number of yards allowed is $4668$.

y number of wins is 10

Part (b) Step 2: Calculation

When team SS allows 4668 yards, the predicted number of wins is,

$$\begin{gathered} \hat{Y}=25.66-0.003131\left(4668\right) \\ \hat{Y}=11.0445 \end{gathered}$$

The actual number of games won is 10 .

The residual error is,

$$\begin{gathered} Y-\hat{Y}=10-11.0445 \\ =-1.0445 \end{gathered}$$

The residual error is $-1.0445$

This means that if a team gave up $4668$yards, the actual number of Seattle Seahawks victories was 1.0445 less than predicted by the fitted regression equation.

Part (c)  Step 1: Given information

The graph below is

Part (c) Step 2: Explanation

When team CP allows 5167 yards, the predicted number of wins is,

$$\begin{gathered} \hat{Y}=25.66-0.003131\left(5167\right) \\ \hat{Y}=9.48212 \end{gathered}$$

The total amount of games won is $15$

The residual error is as follows:

$$\begin{gathered} Y-\hat{Y}=13-9.48212 \\ =5.51788 \end{gathered}$$

The residual error is $5.51788$.

The scatter plot clearly shows that the association between the variable line is negative, implying that the slope of the line is negative; in other words, as the number of yards allowed increases, so does the number of vins.

In the case of team CP, despite having more yards allowed than team SS, the team CP has more wins.

The point of $\mathrm{CP}$ is to increase the slope of the line, which means it increases the negative slope, making the regression line steeper.

As the line steepens, the $y-$-intercept increases, and the point $CP$ increases the $y$ intercept.