Question

Rushing for points What is the relationship between rushing yards and points scored in the National Football League? The table gives the number of rushing yards and the number of points scored for each of the 16 games played by the Jacksonville Jaguars in a recent season.

a. Make a scatterplot with rushing yards as the explanatory variable. Describe what you see.

b. The number of rushing yards in Game 16 is an outlier in the x-direction. What effect do you think this game has on the correlation? On the equation of the least-squares regression line? Calculate the correlation and equation of the least-squares regression line with and without this game to confirm your answers.

c. The number of points scored in Game 13 is an outlier in the y-direction. What effect do you think this game has on the correlation? On the equation of the least-squares regression line? Calculate the correlation and equation of the least-squares regression line with and without this game to confirm your answers.

Short answer

Part (a) The scatterplot confirms a weak relationship because the points seem to lie far apart.

Part (b) The game $16$makes the correlation increase.

Part (c) The game $13$ makes the correlation decrease.

Step-by-step solution

Part (a) Step 1: Given information

Part (a) Step 2: Explanation

The scatterplot with rushing yards as the explanatory variable is as:

Because the scatterplot slopes upwards, we can conclude that the scatterplot confirms a positive linear connection. Because the points appear to be widely apart, the scatterplot indicates a weak association.

Part (b) Step 1: Explanation

For the case with outlier:

Using a calculator, press on STAT and then select 1 : Edit. and then enter the data of sugar in the list L1 and enter the data of calories in the list L2

Next, press on STAT select CALC, and then select Linreg(a + bx) . Next, we need to finish the command by entering L1L2

Linreg(a + bx)L1L2

Finally, pressing on ENTER then gives us the following result:

y = a + bx

a = 11.3891

b = 0.0308

r = 0.1

This then implies the regression line as:

Y= a + bx

⇒Y = 11.3891 + 0.0308x

For the case without outlier:

Using a calculator, press on STAT and then select 1 : Edit. and then enter the data of sugar in the list L1 and enter the data of calories in the list L2

Next, press on STAT select CALC, and then select Linreg(a + bx) . Next, we need to finish the command by entering L1L2

Linreg(a + bx)L1L2

Finally, pressing on ENTER then gives us the following result:

y = a + bx

a = 14.0336

b = 0.0076

r = 0.0180

This then implies the regression line as:

Y= a + bx

⇒Y = 14.0336 + 0.0076x

Thus, we note that the correlation coefficient with the outlier is more than the correlation coefficient without the outlier. We then note that the outlier increases the correlation due to the fact that game 16 as the correlation is much higher. When the outlier is removed the regression line becomes much more horizontal and thus game 16 makes the regression line less horizontal.

Part (c) Step 1: Explanation

For the case with outlier:

Using calculator, press on STAT and then select 1 : Edit . and then enter the data of sugar in the list L1 and enter the data of calories in the list L2

Next, press on STAT select CALC and then select Linreg(a + bx) . Next we need to finish thecommand by entering L1L2

Linreg(a + bx)L1L2

Finally, pressing on ENTER then gives us the following result:

y = a + bx

a = 11.3891

b = 0.0308

r = 0.1

This then implies the regression line as:

Y= a + bx

⇒ Y= 11.3891 + 0.0308x

For the case without outlier:

Using calculator, press on STAT and then select 1: Edit . and then enter the data of sugar in the list L1 and enter the data of calories in the list L2

Next, press on STAT select CALC and then select Linreg(a + bx) . Next we need to finish thecommand by entering L1L2

Linreg(a + bx)L1L2

Finally, pressing on ENTER then gives us the following result:

y = a + bx

a = 7.2776

b = 0.0505

r = 0.3182

This then implies the regression line as:

Y= a + bx

⇒Y = 7.2776 + 0.0505x

Thus, we note that the correlation coeff icient with the outlier is more than the correlationcoeff icient without outlier. Then the game 13 makes the correlation decrease as the correlationcoeff icient is much lower. When the outlier is removed the regression line becomes much steeperand thus the game 13 makes the regression line steep.