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 the 2011 season (NFL web site, February 12, 2012).

Click on the datafile logo to reference the data.

Team	Conference	Yds/Att	Int/Att	Win%
Arizona Cardinals	NFC	6.5	0.042	50.0
Atlanta Falcons	NFC	7.1	0.022	62.5
Carolina Panthers	NFC	7.4	0.033	37.5
Cincinnati Bengals	AFC	6.2	0.026	56.3
Detroit Lions	NFC	7.2	0.024	62.5
Green Bay Packers	NFC	8.9	0.014	93.8
Houstan Texans	AFC	7.5	0.019	62.5
Indianapolis Colts	AFC	5.6	0.026	12.5
Jacksonville Jaguars	AFC	4.6	0.032	31.3
Minnesota Vikings	NFC	5.8	0.033	18.8
New England Patriots	AFC	8.3	0.020	81.3
New Orleans Saints	NFC	8.1	0.021	81.3
Oakland Raiders	AFC	7.6	0.044	50.0
San Francisco 49ers	NFC	6.5	0.011	81.3
Tennessee Titans	AFC	6.7	0.024	56.3
Washington Redskins	NFC	6.4	0.041	31.3

Let x₁ represent Yds/Att.
Let x₂ represent Int/Att.

(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. If required, round your answer to three decimal digits. For subtractive or negative numbers use a minus sign even if there is a + sign before the blank. (Example: -300)
	ŷ =   +  x1
	What proportion of variation in the sample values of proportion of games won does this model explain? If required, round your answer to one decimal digit.
	%
(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. If required, round your answer to three decimal digits. For subtractive or negative numbers use a minus sign even if there is a + sign before the blank. (Example: -300)
	ŷ =   +  x2
	What proportion of variation in the sample values of proportion of games won does this model explain? If required, round your answer to one decimal digit.
	%
(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. If required, round your answer to three decimal digits. For subtractive or negative numbers use a minus sign even if there is a + sign before the blank. (Example: -300)
	ŷ =   +  x1 +  x2
	What proportion of variation in the sample values of proportion of games won does this model explain? If required, round your answer to one decimal digit.
	%
(d)	The average number of passing yards per attempt for the Buffalo Bills during the 2011 season was 6.7, and the team’s number of interceptions thrown per attempt was 0.043. Use the estimated regression equation developed in part (c) to predict the percentage of games won by the Buffalo Bills during the 2011 season. (Note: For the 2011 the 2011 season, the Buffalo Bills' record was 7 wins and 9 loses.)
	If required, round your answer to one decimal digit. Do not round intermediate calculations.
	%
	Compare your prediction to the actual percentage of games won by the Buffalo Bills. If required, round your answer to one decimal digit.
	The Buffalo Bills performed - Select your answer -better worse Item 12  than what we predicted by  %.
(e)	Did the estimated regression equation that uses only the average number of passing yards per attempt as the independent variable to predict the percentage of games won provide a good fit?

Answer 1

In order to solve this question I used R software.

R codes and output:

> d=read.table('football.csv',header=TRUE, sep=',')
> head(d)
Yds.Att Int.Att Win
1 6.5 0.042 50.0
2 7.1 0.022 62.5
3 7.4 0.033 37.5
4 6.2 0.026 56.3
5 7.2 0.024 62.5
6 8.9 0.014 93.8
> attach(d)
The following objects are masked from d (pos = 3):

Int.Att, Win, Yds.Att

> fit_1=lm(Win~Yds.Att)
> summary(fit_1)

Call:
lm(formula = Win ~ Yds.Att)

Residuals:
Min 1Q Median 3Q Max
-25.020 -15.072 3.643 6.847 33.531

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -58.77 26.18 -2.245 0.041423 *
Yds.Att 16.39 3.75 4.371 0.000639 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 15.87 on 14 degrees of freedom
Multiple R-squared: 0.5771, Adjusted R-squared: 0.5469
F-statistic: 19.11 on 1 and 14 DF, p-value: 0.0006393

> fit_2=lm(Win~Int.Att)
> summary(fit_2)

Call:
lm(formula = Win ~ Int.Att)

Residuals:
Min 1Q Median 3Q Max
-43.425 -5.277 0.274 16.172 22.883

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 97.54 13.86 7.036 5.9e-06 ***
Int.Att -1600.49 484.63 -3.303 0.00524 **
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 18.3 on 14 degrees of freedom
Multiple R-squared: 0.4379, Adjusted R-squared: 0.3977
F-statistic: 10.91 on 1 and 14 DF, p-value: 0.005236

> fit_3=lm(Win~Yds.Att+Int.Att)
> summary(fit_3)

Call:
lm(formula = Win ~ Yds.Att + Int.Att)

Residuals:
Min 1Q Median 3Q Max
-26.075 -3.099 1.149 6.265 17.112

Coefficients:
Estimate Std. Error t value Pr(>|t|)   
(Intercept) -5.763 27.147 -0.212 0.83517   
Yds.Att 12.949 3.186 4.065 0.00134 **
Int.Att -1083.788 357.117 -3.035 0.00958 **
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 12.6 on 13 degrees of freedom
Multiple R-squared: 0.7525, Adjusted R-squared: 0.7144
F-statistic: 19.76 on 2 and 13 DF, p-value: 0.0001144

Que.a

Y= -58.77 + 16.39 X1

Proportion of variation = 57.7%

Que.b

Y = 97.538 - 1600.491 X2

Proportion of variation= 43.8%

Que.c

Y = -5.763 + 12.949 X1 - 1083.788 X2

Proportion of variation = 75.3%

Que.d

> data=data.frame(Yds.Att=6.7,Int.Att=0.043 )

> predict(fit_3,newdata=data)

1 34.39452

Predicted value = 34.4%

Actual win % = 7/(7+9) * 100 = 43.8%

The Buffalo Bills performed worse than what we predicted by  9.4% (43.8 - 34.4)

Que.e

Goodness of fit of any model is checked by F test. For this model,

F-statistic: 19.11 on 1 and 14 DF, p-value: 0.0006393

Since p-value is less than 0.0006393, which is less than 0.05, hence we reject null hypothesis and conclude that this model provide good fit.