Question

Don’t turn it over How many points do turnovers cost teams in the NFL? The scatterplot shows the relationship between x = number of turnovers and y = number of points scored by teams in the NFL during 2015, along with the least-squares regression line $\stackrel{\wedge }{y}=460.2-4.084x$

a. Interpret the slope of the regression line in context.

b. For this regression line, $\mathrm{s}=57.3$. Interpret this value.

c. Calculate and interpret the residual for the San Francisco $49\mathrm{ers}$, who had $17$turnovers and scored $238$points.

d. How does the point for the $49\mathrm{ers}$affect the least-squares regression line and standard deviation of the residuals? Explain your answer.

Short answer

Part(a) The regression line has a slope of $4.084$

Part(b) $\mathrm{s}=57.3$describes how closely the regression line matches the data.

Part(c) The residual value is $-152.772$

Part(d) It has an impact on the least square regression line and the standard deviation of the residual because it has a negative relationship with the number of points scored in the NFL.

Step-by-step solution

Part(a) Step 1 : Given information

We need to interpret the slope of the regression.

Part(a) Step 2 : Simplify

The scatter plot depicts the association between the number of turnovers (x) and the number of points scored (y) by NFL clubs.
$\stackrel{\wedge }{y}=460.2-4.084x$is the formula.
The regression line has a slope of $4.084$
This means that on average, the amount of points scored by NFL teams drops by $4.084$because of turnovers.

Part(b) Step 1 : Given information

We need to find $\mathrm{s}=57.3$ for this regression line.

Part(b) Step 2 : Simplify

For the regression line, we must interpret the value of $\mathrm{s}=57.3$
Consider that $\stackrel{\wedge }{\mathrm{y}}=460.2-4.084\mathrm{x}$is correct.
It's the least square regression line's standard deviation.
It means that the points in the scatter plot for the number of points scored and the number of turnovers in the NFL are different.
It describes how closely the regression line matches the data.
As a result, the regression line fits around $57\%$of the data.

Part(c) Step 1 : Given information

We need to find and interpret the residual for the San Francisco.

Part(c) Step 2 : Simplify

The scatter plot depicts the association between the number of turnovers $(x=)$and the number of points scored $(y=)$ by NFL clubs.
It is stated as follows: $\stackrel{\wedge }{\mathrm{y}}=460.2-4.084\mathrm{x}$.
We must calculate the residual for a player that had $17$turnovers and $238$ points.
As a result of knowing that $y=238$, we can calculate $\stackrel{\wedge }{y}$.
$$\begin{gathered} \stackrel{\wedge }{y}=460.2-4.084x \\ =460.2-4.084\times 17 \\ =460.2-69.428 \\ =390.772 \end{gathered}$$
There are $17$ turnovers as stated.
As a result, the residual is $y-\stackrel{\wedge }{y}$

$$\begin{gathered} y-\stackrel{\wedge }{y}=238-390.772 \\ =-152.772 \end{gathered}$$

As a result,

This indicates that the genuine value is less than the estimated value, with a difference of $-152.772$. Because the difference is negative, the estimated value is greater than the actual value.

Part(d) Step 1 : Given information

We need to find if the point for the $49\mathrm{ers}$affect the least-squares regression line and standard deviation of the residuals

Part(d) Step 2 : Simplify

The scatter plot depicts the association between the number of turnovers $(x=)$and the number of points scored $(y=)$ by NFL clubs.
It is stated as follows: $\stackrel{\wedge }{\mathrm{y}}=460.2-4.084\mathrm{x}$
It has an impact on the least square regression line and the standard deviation of the residual because it has a negative relationship with the number of points scored in the NFL. This has an impact on the regression line and the standard deviation, indicating that the regression line does not fit the data as well as it should.