Question

Major League Baseball (MLB) consists of teams that play in the American League and the National League. MLB collects statistics often used to evaluate pitching performance are as follows: wide variety of team and player statistics. Some of the 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 scored as a resu lt 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 season. Player ERA So/IP R/IP Team W L HR/IP 24 5 2.41 1.00 0.11 0.28 Verlander, J DET 13 0.10 0.34 7 2.89 0.91 BOS Beckett, J 16 7 2.93 0.92 0.07 0.41 Wilson, C TEX 19 3.01 0.98 0.06 0.37 Sabathia, C NYY 16 10 3.18 0.80 0.07 0.39 Haren, D LAA 0.42 3.33 0.72 0.05 McCarthy, B OAK 11 12 3.37 0.78 0.10 0.41 Santana, E LAA 0.94 15 3.46 0.11 0.39 Lester, J BOS 0.09 14 14 3.46 0.95 0.41 Hernandez, F SEA 13 3.60 0.53 0.10 0.46 CWS Buehrle, M 0.12 0.43 10 3.75 1.00 Pineda, M SEA 8 10 4.01 0.82 0.13 0.53 Colon, B NYY 12 4.25 0.53 0.16 0.49 CLE Tomlin, 13 4.29 0.46 0.10 0.55 Pavano, C MIN 8 0.78 0.12 0.52 12 4.32 Danks, J CWS 9 17 4.34 0.63 0.13 0.53 Guthrie, J BAL 14 10 4.41 0.85 0.17 0.52 Lewis, C TEX 0.88 0.14 0.52 15 9 4.43 DET Scherzer, M 0.12 11 10 4.44 0.56 0.51 Davis, W TE 4.75 0.10 14 9 0.58 0.57 Porcello, R DET a. What are the values of R and R (to 3 decimals), if the average number of runs is the dependent variable and the average number strikeouts per inning pitched and the average number of home runs per inning pitched are the independent variables. Enter negative value as negative number. R/IP =0.552 -0.265 sO/IP HR/IP 1.017 R2 = 0.602 R2 0.555 b. Does the estimated regression equation provide good fit to the data? Explain. (to 1 decimal) of the variability in the number of runs given up per inning pitched. because the nature of the data able to explain The fit is not bad c. Suppose the earned run average (ERA) is used as the dependent variable in part (a) instead of the average number of runs given up per inning pitch ed. What are the values of R2 and R? (to 3 decimals). Enter negative value as negative number. % So/IP HR/IP ERA R2 0.563 R2=0.512 Does the estimated regression equation provide a good fit to the data? Explain. (to 1 decimal) % of the variability in the ERA The fit is not bad because the nature of the data is able to explain

Answer 1

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.758632808
R Square	0.575523737
Adjusted R Square	0.525585353
Standard Error	0.054228651
Observations	20
ANOVA
	df	SS	MS	F	Significance F
Regression	2	0.067782308	0.033891154	11.52467687	0.000686679
Residual	17	0.049992692	0.002940747
Total	19	0.117775
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
Intercept	0.552756845	0.077706048	7.113433974	1.738E-06	0.388811415	0.716702275	0.388811415	0.716702275
SO/IP	-0.264414757	0.071814506	-3.681912917	0.001849044	-0.415930119	-0.112899395	-0.415930119	-0.112899395
HR/IP	0.988382143	0.407979736	2.422625577	0.026866574	0.127620149	1.849144137	0.127620149	1.849144137

a)

R/IP =  0.552 - 0.264 * SO/IP + 0.988 * HR/IP

R Square = 0.575

Adjusted R Square = 0.525

b)

The fit is not bad, because the nature of the data is able to explain 52.5% of the variability in the number of runs given up per inning pitched

c)

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.793797666
R Square	0.630114734
Adjusted R Square	0.586598821
Standard Error	0.423448366
Observations	20
ANOVA
	df	SS	MS	F	Significance F
Regression	2	5.192810183	2.596405092	14.48009895	0.000213093
Residual	17	3.048244817	0.179308519
Total	19	8.241055
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
Intercept	4.045311972	0.606773332	6.666924462	3.96996E-06	2.765132155	5.32549179	2.765132155	5.32549179
SO/IP	-1.934653148	0.560768794	-3.450001443	0.003058094	-3.117771873	-0.751534422	-3.117771873	-0.751534422
HR/IP	11.13629894	3.185739449	3.495671607	0.002769917	4.414976283	17.85762159	4.414976283	17.85762159

ERA =  4.045 - 1.935 * SO/IP + 11.136 * HR/IP

R Square = 0.630

Adjusted R Square = 0.586

The fit is not bad, because the nature of the data is able to explain 58.6% of the variability in the ERA