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?
	The input in the box below will not be graded, but may be reviewed and considered by your instructor.

Answer 1

using excel>data>data analysis>Regression

we have

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.759693
R Square	0.577133
Adjusted R Square	0.546928
Standard Error	15.87319
Observations	16
ANOVA
	df	SS	MS	F	Significance F
Regression	1	4814.254	4814.254	19.10735	0.000639
Residual	14	3527.416	251.9583
Total	15	8341.67
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	-58.7703	26.17541	-2.24525	0.041423	-114.911	-2.62964
Yds/Att	16.39063	3.749689	4.371195	0.000639	8.348341	24.43291

(a)	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 = -58.770 + 16.391 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.
	57.7%

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.661739
R Square	0.437898
Adjusted R Square	0.397748
Standard Error	18.3008
Observations	16
ANOVA
	df	SS	MS	F	Significance F
Regression	1	3652.8	3652.8	10.90651	0.005236
Residual	14	4688.87	334.9193
Total	15	8341.67
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	97.53825	13.86182	7.03647	5.9E-06	67.80762	127.2689
Int/Att	-1600.49	484.63	-3.3025	0.005236	-2639.92	-561.063

(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 = 97.538+ -600.49 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. 43.8%

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.867461
R Square	0.752489
Adjusted R Square	0.71441
Standard Error	12.60237
Observations	16
ANOVA
	df	SS	MS	F	Significance F
Regression	2	6277.014	3138.507	19.76145	0.000114
Residual	13	2064.656	158.8197
Total	15	8341.67
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	-5.76328	27.1468	-0.2123	0.835165	-64.4104	52.88381
Yds/Att	12.94936	3.18567	4.064877	0.001338	6.067136	19.83158
Int/Att	-1083.79	357.1165	-3.03483	0.009575	-1855.29	-312.285

(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 = -5.763 + 12.949 x1 + -1083.79 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. 75.3%

(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.)
	ŷ = -5.763 + 12.949*6.7 + -1083.79*0.043
	=34.4 %
(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?
	yes