Question

The National Football League (NFL) records a variety of performance data for individuals and teams. To investigate the importance of passing on the percentage of games won by a team, the following data show the conference (Conf), average number of passing yards per attempt (Yds/Att), the number of interceptions thrown per attempt (Int/Att), and the percentage of games won (Win%) for a random sample of 16 NFL teams for a season

Team	Conf	Yds/Att	Int/Att	Win%
Arizona Cardinals	NFC	6.7	0.042	50.1
Atlanta Falcons	NFC	7.3	0.024	62.7
Carolina Panthers	NFC	7.5	0.034	37.5
Cincinnati Bengals	AFC	6.0	0.026	56.1
Detroit Lions	NFC	7.0	0.025	62.3
Green Bay Packers	NFC	8.9	0.014	93.9
Houstan Texans	AFC	7.6	0.019	62.6
Indianapolis Colts	AFC	5.7	0.027	12.2
Jacksonville Jaguars	AFC	4.7	0.031	31.3
Minnesota Vikings	NFC	5.9	0.035	18.5
New England Patriots	AFC	8.1	0.020	81.1
New Orleans Saints	NFC	8.1	0.021	81.2
Oakland Raiders	AFC	7.8	0.043	50.2
San Francisco 49ers	NFC	6.6	0.010	81.3
Tennessee Titans	AFC	6.6	0.025	56.2
Washington Redskins	NFC	6.2	0.043	31.1

a. Develop the estimated regression equation that could be used to predict the percentage of games won given the average number of passing yards per attempt (to 1 decimal). Enter negative value as negative number.

    

b. Develop the estimated regression equation that could be used to predict the percentage of games won given the number of interceptions thrown per attempt (to 1 decimal). Enter negative value as negative number.

    

c. Develop the estimated regression equation that could be used to predict the percentage of games won given the average number of passing yards per attempt and the number of interceptions thrown per attempt (to 1 decimal). Enter negative value as negative number.

      

d. The average number of passing yards per attempt for the Kansas City Chiefs was 6.2 and the number of interceptions thrown per attempt was 0.036 . Use the estimated regression equation developed in part (c) to predict the percentage of games won by the Kansas City Chiefs. (Note: For a season the Kansas City Chiefs' record was 7 wins and 9 losses.) Compare your prediction to the actual percentage of games won by the Kansas City Chiefs (to whole number).

Predicted percentage		Actual percentage
	- Select your answer -<>=Item 9

Answer 1

Consider the variables

Y : Percentage of games won

X1 : Average number of passing yards per attempt

X2 : Number of interaction thrown per attempt.

a) We have to obtain regression equation that could be used to predict the percentage of games won given average number of passing yards per attempt.

i.e. We have to obtain simple linear equation Y on X1

By using R

> y=c(50.1,62.7,37.5,56.1,62.3,93.9,62.6,12.2,31.3,18.5,81.1,81.2,50.2,81.3,56.2,31.2)
> x1=c(6.7,7.3,7.5,6,7,8.9,7.6,5.7,4.7,5.9,8.1,8.1,7.8,6.6,6.6,6.2)
> a=lm(y~x1)
> a

Call:
lm(formula = y ~ x1)

Coefficients:
(Intercept) x1
-57.78 16.20

From R-output

The regression equation Y on X1 is

Y = -57.78 + 16.20 * X1.

b) We have to obtain regression equation that could be used to predict the percentage of games won given number of interaction thrown per attempt.

i.e. We have to obtain simple linear equation Y on X2

by using R

> y=c(50.1,62.7,37.5,56.1,62.3,93.9,62.6,12.2,31.3,18.5,81.1,81.2,50.2,81.3,56.2,31.2)
> x2=c(0.042,0.024,0.034,0.026,0.025,0.014,0.019,0.027,0.031,0.035,0.02,0.021,0.043,0.01,0.025,0.043)
> b=lm(y~x2)
> b

Call:
lm(formula = y ~ x2)

Coefficients:
(Intercept) x2
99.08 -1633.15
From R -output

The line of regression Y on X2 is

Y = 99.08 - 1633.15 * X2

c) We have to obtain regression equation that could be used to predict the percentage of games won given average number of passing yards per attempt and number of interaction thrown per attempt.

i.e. We have to obtain multiple linear regression Y on X1 and X2.

Y is dependent variable and X1 and X2 are independent variables.

By using R

> y=c(50.1,62.7,37.5,56.1,62.3,93.9,62.6,12.2,31.3,18.5,81.1,81.2,50.2,81.3,56.2,31.2)
> x1=c(6.7,7.3,7.5,6,7,8.9,7.6,5.7,4.7,5.9,8.1,8.1,7.8,6.6,6.6,6.2)
> x2=c(0.042,0.024,0.034,0.026,0.025,0.014,0.019,0.027,0.031,0.035,0.02,0.021,0.043,0.01,0.025,0.043)
> d=data.frame(y,x1,x2)
> d
y x1 x2
1 50.1 6.7 0.042
2 62.7 7.3 0.024
3 37.5 7.5 0.034
4 56.1 6.0 0.026
5 62.3 7.0 0.025
6 93.9 8.9 0.014
7 62.6 7.6 0.019
8 12.2 5.7 0.027
9 31.3 4.7 0.031
10 18.5 5.9 0.035
11 81.1 8.1 0.020
12 81.2 8.1 0.021
13 50.2 7.8 0.043
14 81.3 6.6 0.010
15 56.2 6.6 0.025
16 31.2 6.2 0.043

> c=glm(data=d,y~x1+x2)
> c

Call: glm(formula = y ~ x1 + x2, data = d)

Coefficients:
(Intercept) x1 x2
-0.2792 12.5391 -1173.6054

Degrees of Freedom: 15 Total (i.e. Null); 13 Residual
Null Deviance: 8383
Residual Deviance: 1976 AIC: 130.5

From R-output

Y = -0.2792 + 12.5391 * X1 - 1173*6054 * X2

d) Given

X1 : 6.2 and X2 = 0.036

Predicted percentage of won = -0.2792 + 12.5391 * (6.2) - 1173.6054*0.036 = 35.24

Predicted Percentage = 35.24

Since Total number of games = 9+ 7= 16

number of wins = 7

Actual percentage of win = 7 / 16 * 100 = 43.75

The difference between actual and predicted percentage (error) = 43.75 - 35.24 = 8.51.