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 passing yards per attempt (Yds/Att), the number of interceptions thrown per attempt (Int/Att), and the percentage of games won following data show the conference (Conf), average number (Win%) for a random sample of 16 NFL teams for a season Team Conf Yds/Att Int/Att Win% Arizona Cardinals NFC 6.7 0.043 49.9 Atlanta Falcons NFC 0.022 62,4 7.1 Carolina Panthers NFC 7,2 0.033 37.2 Cincinnati Bengals AFC 6,0 0,027 56.2 Detroit Lions NFC 7.3 0.023 62.4 Green Bay Packers NFC 8.7 0.016 93.7 AFC Houstan Texans 7.7 0.020 62.7 Indianapolis Colts AFC 5.4 0.025 12.5 Jacksonville Jaguars AFC 4,7 0,032 31,4 Minnesota Vikings NFC 0.034 18.8 5.6 0.022 81.4 New England Patriots AFC 8.4 New Orleans Saints NFC 8.3 0.020 81.1 Oakland Raiders AFC 7.6 0.043 49.7 San Francisco 49ers 6.5 NFC 0.010 81.5 Tennessee Titans AFC 6.9 0.026 56,4 NFC 0.043 Washington Redskins 6.2 31.0 Washington Redskins 31.0 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 Yds/Att Win% 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. Win% Int/Att 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. Win% Yds/Att Int/Att + 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). Actual percentage Predicted percentage -Select your answer-

Answer 1

(a)

Here X is YDS/ATT

Y is Win %

(X-Mx)2 (X-Mx)(Y- My) X-Mx Y-My 0.0375 -0.1938 -4.3688 0.8464 0.2062 0.0425 8.1313 1.6771 0.0938 0.3062 -17.0688 -5,2273 -0.8938 0.4062 1.9313 0.7988 -1.7261 0.165 8.1313 3.3033 3.2625 1.8063 39.4313 71.2227 0.8062 0.65 8.4313 6.7977 2.2313 -1.4938 -41.7688 62.3921 4.8125 -2.1938 -22.8688 50.1683 -1.2938 1.6738 45.8877 -35.4688 2. 2688 1.5063 27.1313 40,8664 1.9775 1.4063 26.8313 37.7314 0.7062 0.4988 -3.2267 -4.5687 0.155 -0.3938 27.2313 -10.7223 0.0062 2.1313 0.0133 0.4813 -0.6938 -23.2688 16.1427 SS: 19.1494 SP: 316.1469

Sum of X = 110.3
Sum of Y = 868.3
Mean X = 6.8938
Mean Y = 54.2688
Sum of squares (SS_X) = 19.1494
Sum of products (SP) = 316.1469

Regression Equation = ŷ = bX + a

b = SP/SS_X = 316.15/19.15 = 16.50951

a = M_Y - bM_X = 54.27 - (16.51*6.89) = -59.54371

ŷ = 16.50951X - 59.54371

Win %= - 59.5 + (16.5)YDS/ATT

(b)

Here X is INT/ATT

Y is Win %

(X- Mx)2 (X-M)(Y- My) X-Mx Y-My 0.0002 -0.068 0.0156 -4.3688 -0.0054 8.1313 -0.0442 0.0056 -17.0688 1.9313 -0.0949 -0.0008 -0.0004 -0.0044 8.1313 -0.0361 0.0001 -0.0114 39.4313 -0.451 0.0001 -0.0074 8.4313 -0.0627 -0.0024 0.1018 -41.7688 -0.1043 0.0046 -22.8688 0.0066 -35.4688 -0.2328 -0.0054 27.1313 -0.1475 0.0001 -0.0074 26.8313 -0.1996 0.0002 0.0156 -4.5687 -0.0711 0.0003 -0.4748 -0.0174 27.2313 2.1313 0 -0.0014 -0.0031 0.0002 0.0156 -23.2688 -0.3621 ss: 0.0015 SP: -2.2513

Sum of X = 0.439
Sum of Y = 868.3
Mean X = 0.0274
Mean Y = 54.2688
Sum of squares (SS_X) = 0.0015
Sum of products (SP) = -2.2513

Regression Equation = ŷ = bX + a

b = SP/SS_X = -2.25/0 = -1548.40304

a = M_Y - bM_X = 54.27 - (-1548.4*0.03) = 96.75306

ŷ = -1548.40304X + 96.75306

Win% = 96.8 + (-1548.4) int/att

(c)

Here Y is Win%

X1 is yds/att

X2 is int/att

(X2-M2)2 SPxly (X1-Mx1)2 SP 2y SPx1x2 X2-M2 X1-Mx1 Y-My -0.068 -0.194 0.016 0.038 0.846 -0.003 -4.369 1.677 -0.044 -0.001 0.206 -0.005 8.131 0.043 -5.227 -0.095 0.002 0.306 0.006 -17.069 0.094 0.799 -1.726 -0.001 -0.894 1.931 8.131 3.303 -0.036 -0.002 0.406 -0.004 0.165 71.223 -0.451 -0.021 1.806 -0.011 39.431 3.263 6.798 0.806 -0.063 -0.006 -0.007 8.431 0.65 62.392 0.102 0.004 -0.002 -1.494 -41.769 2.231 -0.01 0.005 50.168 -0.104 -2.194 -22.869 4.813 45.888 -0.233 -0.008 -1.294 0.007 -35.469 1.674 1.506 1.406 -0.148 40.866 -0.008 -0.005 27.131 2.269 37.731 -0.2 -0.01 -0.007 26.831 1.978 -3.227 -0.071 0.011 0.706 0.016 -4.569 0.499 -10.722 -0.475 0.007 -0.394 -0.017 27.231 0.155 -0.003 0.013 0.006 -0.001 2.131 -0.011 16.143 -0.362 -0.694 0.016 -23.269 0.481 SPX2Y SPX1Y SPX1X2: SSX1: SSX2: 19.149 0.001 316.147 -2.251 -0.057

Sum of X₁ = 110.3
Sum of X₂ = 0.439
Sum of Y = 868.3
Mean X₁ = 6.8938
Mean X₂ = 0.0274
Mean Y = 54.2688
Sum of squares (SS_X1) = 19.1494
Sum of squares (SS_X2) = 0.0015
Sum of products (SP_X1Y) = 316.1469
Sum of products (SP_X2Y) = -2.2513
Sum of products (SP_X1X2) = -0.057

Regression Equation = ŷ = b₁X₁ + b₂X₂ + a

b₁ = ((SP_X1Y)*(SS_X2)-(SP_X1X2)*(SP_X2Y)) / ((SS_X1)*(SS_X2)-(SP_X1X2)*(SP_X1X2)) = 0.33/0.02 = 13.474

b₂ = ((SP_X2Y)*(SS_X1)-(SP_X1X2)*(SP_X1Y)) / ((SS_X1)*(SS_X2)-(SP_X1X2)*(SP_X1X2)) = -25.1/0.02 = -1020.5752

a = M_Y - b₁M_X1 - b₂M_X2 = 54.27 - (13.47*6.89) - (-1020.58*0.03) = -10.61563

win% = - 10.6 + 13.5 yds/att + (-1020.6)int/att

(d)

win% = - 10.6 + 13.5 yds/att + (-1020.6)int/att

=  - 10.6 + (13.5)(6.2) + (-1020.6)(0.036)

= -10.6 + 83.7 - 36.7416

= 36.4 %

Actual Percentage:-

Number of wins = 7

Total Matches = 7+9 = 16

So Actual Percentage = 7 / 16 = 0.4375

So 43.75%