Question

Check Major League Baseball (MLB) consists of teams that play in the American League and the National League. MLB collects a wide variety of team and player statistics. Some of the often used to evaluate pitching performance are as follows: • ERA: The average number of earned runs given up by the pitcher per nine innings. An earned run is any run that the opponent scores off a particular pitcher except for runs result of errors. SO/IP: The average number of strikeouts per inning pitched. • HR/IP: The average number of home runs per inning pitched. • R/IP: The number of runs given up per inning pitched. The following data show values for these statistics for a random sample of 20 pitchers from the American League for a season. Player Team W L ERA SO/IP HR/IP R/IP 24 5 7 2.39 2.89 2.94 0.10 0.10 13 DET BOS TEX NYY 1.00 0.91 0.92 0.30 0.34 16 7 0.41 19 8 0.97 0.38 0.39 LAA 16 2.99 3.18 3.32 10 0.80 OAK 9 9 0.71 0.42 LAA 11 12 3.39 0.77 0.96 15 9 BOS SEA 3.48 3.48 0.06 0.06 0.07 0.05 0.11 0.10 0.09 0.11 0.12 0.13 0.16 0.42 0.41 0.41 14 14 0.95 13 9 Verlander, ) Beckett, ] Wilson, C Sabathia, C Haren, D McCarthy, B Santana, E Lester, Hernandez, F Buehrle, M Pineda, M Colon, B Tomlin, Pavano, C Danks, J Guthrie, J Lewis, C Scherzer, M Davis, w Porcello, R 0.54 CWS SEA NYY 9 10 1.00 3.60 3.75 4.01 4.24 10 8 12 0.45 0.43 0.53 0.49 0.56 CLE 0.82 0.54 0.46 0.79 7 9 13 4.30 0.10 MIN CWS 8 12 0.11 0.52 4.33 4.33 9 17 0.64 0.12 0.54 0.51 BAL TEX DET 14 10 4.39 0.84 0.16 15 9 4.42 0.89 0.51 0.15 0.12 TB 10 0.57 11 14 4.45 4.76 0.51 0.57 DET 9 0.58 0.10

Answer 1

(a) The regression output is :-

    

SUMMARY OUTPUT Regression Statistics Multiple R 0.751941946 R Square 0.56541669 Adjusted R Square 0.514289242 Standard Error 0.052519766 Observations 20 ANOVA df Significance F 0.000838727 2 Regression Residual Total SS MS F 0.061008461 0.030504 11.05897 0.046891539 0.002758 0.1079 17 19 Intercept SO/IP HR/IP Coefficients 0.55433573 -0.250580596 0.913857324 Standard Error t Stat P-value 0.077464642 7.155984 1.61E-06 0.070872546 -3.53565 0.00254 0.39954821 2.287227 0.035274 Lower 95% 0.390899623 -0.400108595 0.070884295 Upper 95% 0.717771837 -0.101052596 1.756830353 Lower 95.0% 0.390899623 -0.400108595 0.070884295 Upper 95.0% 0.717771837 -0.101052596 1.756830353

Therefore,

R/IP=0.5543)+(-0.2506)SO/IP+(0.9138) HR/IP

$R^2=0.5654$

$R_{a}^2=0.5143$

(b) The data is able to explain 51.43% of the variability .

(c) The output is :-

SUMMARY OUTPUT Regression Statistics Multiple R 0.784805575 R Square 0.615919791 Adjusted R Square 0.570733884 Standard Error 0.432403697 Observations 20 ANOVA df Significance F 0.000293485 2 Regression Residual Total SS MS F 5.097179734 2.54859 13.63079 3.178540266 0.186973 8.27572 17 19 Intercept SO/IP HR/IP Coefficients 3.990968122 -1.843914921 11.17752134 Standard Error t Stat P-value 0.63777888 6.257605 8.67E-06 0.583505089 -3.16007 0.005718 3.289544786 3.397893 0.003423 Lower 95% 2.645372317 -3.075003036 4.237188567 Upper 95% Lower 95.0% 5.336563926 2.645372317 -0.612826807 -3.075003036 18.11785411 4.237188567 Upper 95.0% 5.336563926 -0.612826807 18.11785411

Therefore,

$\textup{ERA=(3.991)+(-1.8439)SO/IP+(11.1775)HR/IP}$

$R^2=0.6159$

$R_{a}^2=0.5707$

The fit is not bad.... ..... the data is able to explain 57.07% of the variability .