Question

eBook 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 Seattle Seahawks during the 2011 season was 6.8, and the team’s number of interceptions thrown per attempt was 0.028. Use the estimated regression equation developed in part (c) to predict the percentage of games won by the Seattle Seahawks during the 2011 season. (Note: For the 2011 the 2011 season, the Seattle Seahawks' 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 Seattle Seahawks. If required, round your answer to one decimal digit. The Seattle Seahawks performed - Select your answer -betterworseItem 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

 > # reding the data > df = read.csv(file.choose(),header = T) > str(df) 'data.frame': 16 obs. of 5 variables: $ Team : Factor w/ 16 levels "Arizona Cardinals",..: 1 2 3 4 5 6 7 8 9 10 ... $ Conference: Factor w/ 2 levels "AFC","NFC": 2 2 2 1 2 2 1 1 1 2 ... $ Yds.Att : num 6.5 7.1 7.4 6.2 7.2 8.9 7.5 5.6 4.6 5.8 ... $ Int.Att : num 0.042 0.022 0.033 0.026 0.024 0.014 0.019 0.026 0.032 0.033 ... $ Win. : num 50 62.5 37.5 56.3 62.5 93.8 62.5 12.5 31.3 18.8 ... > head(df)  Team Conference Yds.Att Int.Att Win. 1 Arizona Cardinals NFC 6.5 0.042 50.0 2 Atlanta Falcons NFC 7.1 0.022 62.5 3 Carolina Panthers NFC 7.4 0.033 37.5 4 Cincinnati Bengals AFC 6.2 0.026 56.3 5 Detroit Lions NFC 7.2 0.024 62.5 6 Green Bay Packers NFC 8.9 0.014 93.8  > mod = lm(Win.~ Yds.Att, data = df) > summary(mod)  Call: lm(formula = Win. ~ Yds.Att, data = df) 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 

# the fitted model is ŷ = -58.7 + 16.39*x1

Since R-squared: 0.5771, so the model explain 57.7 % of total variation

#b)

> mod = lm(Win.~ Int.Att, data = df) > summary(mod) Call: lm(formula = Win. ~ Int.Att, data = df) 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

# the fitted model is ŷ = 97.54 -1600.49*x2 Since R-squared: 0.4379, so the model explain 43.79 % of total variation

#c)

 > mod = lm(Win.~ Int.Att + Yds.Att, data = df) > summary(mod)  Call: lm(formula = Win. ~ Int.Att + Yds.Att, data = df) 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 Int.Att -1083.788 357.117 -3.035 0.00958 ** Yds.Att 12.949 3.186 4.065 0.00134 ** --- 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 

the fitted model is ŷ = -5.763 + 12.949*x1 -1083.788*x2

Since R-squared: 0.7525, so the model explain 75.25 % of total variation

D) it is given that The average number of passing yards per attempt for the Seattle Seahawks during the 2011 season was 6.8, and the team’s number of interceptions thrown per attempt was 0.028.

ie. x1 = 6.8 and x2 = 0.028

therefore,the percentage of games won by the Seattle Seahawks during the 2011 season. is

ŷ = -5.763 + 12.949*6.8 -1083.788*0.028

e) T he estimated regression equation that uses only the average number of passing yards per attempt as the independent variable has explain only 57.7% of variation ,so it is not good fit.

ŷ = 51.949

For the 2011 season, the Seattle Seahawks' record was 7 wins and 9 loses.

Therefore, the winning percentage is 7/16 = 0.43

the actual winnig percentage is 43% while our model predict 51.94% , this means the modelmis not very good.