Question

The National Football League (NFL) records a variety of performance data for individuals and teams. To...

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

 Yds/Att Int/Att Win% Team Conf 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).

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.

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