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.

$\hat{y}=b_0+b_{1^x}=25.66-0.003131x$

The least- squares regression line equation:

$\hat{y}=b_0+b_{1x}$

The constant $b_0$is given in the row "Constant"

$$\begin{gathered} b_0=25.66 \\ {b}_1=-0.003131 \end{gathered}$$

Replacing $b_0$by $25.66$and $b_1$by $-0.003131$in the least- squares regression line equation:

$\hat{y}=b_0+b_{1^x}=25.66-0.003131x$

Here x represents the number of yards allowed, y represents the number of wins.

$x=$number of yards allowed $=4668$

$y=$number of wins $=10$

The least- squares regression line equation

$$\begin{gathered} \hat{y}=b_0+b_{1^x} \\ {b}_0=25.66 \\ {b}_1=-0.003131 \end{gathered}$$

Replacing $b_0$by $25.66$and $b_1$by $-0.003131$in the least-squares regression line

$\hat{y}=b_0+b_{1x}=25.66-0.003131x$

Where $x$ is representing the number of yards allowed and y represents the number of wins.

Replacing x in the regression line by 4668 and calculate

$\hat{y}=25.66-0.003131\left(4668\right)=25.66-14.615508=11.04492$

This residual is the difference between the observed. y-value and the expected value

$$\begin{gathered} \text{Residual}=y-\hat{y} \\ =10-11.04492 \\ =-1.04492 \end{gathered}$$

This then means that it is overestimated the number of wins for the researcher by role="math" localid="1654564363060" $1.04492$wins, when making a prediction using the regression line.

It is observed that the point representing with 5167 yards allowed and 15 games is the point that lies highest in the scatter plot The point pull the least-squares regression line up slightly, which therefore implies that the regression line would lie higher to the left in the scatter plot when the point is included in the calculation of the regression line.

Although, this then implies that the least-squares regression line becomes more decreasing and therefore the slope will decrease.

It is also observed that the regression line would be intersect the y axis at a higher point and therefore the y intercept will be at higher when the points is included.

Therefore the slope decreases and the y-intercept increases.