Chapter 7, Problem AP2.23CPT

(a)

To determine

To find: the equation of the least-squares regression line.

Answer to Problem AP2.23CPT

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

Explanation of Solution

Calculation:

The least- squares regression line equation:

  $\widehat{y}=b_0+b_{1x}$

The constant $b_0$ is given in the row “Constant”

  $b_0=25.66$

  $b_1=-0.003131$

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

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

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

(b)

To determine

To Calculate: and interpret the residual for the researcher, who allowed 4668 yards and won 10 games.

Answer to Problem AP2.23CPT

Residual = -1.044492

Explanation of Solution

Given:

  $\begin{array}{l}x=\text{number}\text{of}\text{yards}\text{allowed}=4668\\ y=\text{number}\text{of}\text{wins}=10\end{array}$

Calculation:

The least- squares regression line equation

  $\widehat{y}=b_0+b_{1^x}$

  $b_0=25.66$

  $b_1=-0.003131$

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

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

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

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

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

  $\begin{array}{c}\text{Residual=}y-\widehat{y}\\ =10-11.04492\\ =-1.04492\end{array}$

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

(c)

To determine

To find: the effect does the point representing the panthers have on the equation of the least-squares regression line.

Answer to Problem AP2.23CPT

The slope decreases and the y intercept increases

Explanation of Solution

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.