Question

The following table lists a portion of Major League Baseball’s (MLB’s) leading pitchers, each pitcher’s salary (In $ millions), and earned run average (ERA) for 2008.

a-1. Estimate the model: Salary = β₀ + β₁ERA + ε. (Negative values should be indicated by a minus sign. Enter your answers, in millions, rounded to 2 decimal places.)Click here for the Excel Data File

	Salary	ERA
J. Santana	17.0	2.25
C. Lee	2.0	2.37
T. Lincecum	0.1	2.57
C. Sabathia	10.0	2.11
R. Halladay	7.0	2.52
J. Peavy	5.7	2.77
D. Matsuzaka	6.7	2.44
R. Dempster	6.6	2.89
B. Sheets	11.5	2.90
C. Hamels	0.3	2.91

Salaryˆ=Salary^=    +  ERA

a-2. Interpret the coefficient of ERA.

• A one-unit increase in ERA, predicted salary decreases by $6.37 million.

• A one-unit increase in ERA, predicted salary increases by $6.37 million.

• A one-unit increase in ERA, predicted salary decreases by $16.70 million.

• A one-unit increase in ERA, predicted salary increases by $16.70 million.

b. Use the estimated model to predict salary for each player, given his ERA. For example, use the sample regression equation to predict the salary for J. Santana with ERA = 2.25. (Round coefficient estimates to at least 4 decimal places and final answers, in millions, to 2 decimal places.)

c. Derive the corresponding residuals. (Negative values should be indicated by a minus sign. Round coefficient estimates to at least 4 decimal places and final answers, in millions, to 2 decimal places.)

Answer 1

a-1. Salary = 23.0715-6.3667*ERA

we need to use regression function in excel data analysis to get the results as below:

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.3480
R Square	0.1211
Adjusted R Square	0.0113
Standard Error	5.2169
Observations	10
ANOVA
	df	SS	MS	F	Significance F
Regression	1	30.0044	30.0044	1.1025	0.3244
Residual	8	217.7246	27.2156
Total	9	247.7290
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	23.0715	15.6886	1.4706	0.1796	-13.1066	59.2496
ERA	-6.3667	6.0636	-1.0500	0.3244	-20.3494	7.6160

a-2. Option 1
From the above table we can observe that the coefficient is negative indicates about the negative relationship, hence one unit increase in ERA decreases salary by $ 6.37 million

b.

Salary	ERA	Forecast
17	2.25	23.0715-6.3667*2.25 = 8.75
2	2.37	23.0715-6.3667*2.37 = 7.98
0.1	2.57	23.0715-6.3667*2.57 = 6.71
10	2.11	23.0715-6.3667*2.11 = 9.64
7	2.52	23.0715-6.3667*2.52=7.03
5.7	2.77	23.0715-6.3667*2.77 =5.44
6.7	2.44	23.0715-6.3667*2.44=7.54
6.6	2.89	23.0715-6.3667*2.89=4.67
11.5	2.9	23.0715-6.3667*2.90=4.61
0.3	2.91	23.0715-6.3667*2.91=4.54

c.

Salary	ERA	Forecast	Residuals = Actual-Forecast
17	2.25	8.75	=17-8.75 = 8.25
2	2.37	7.98	=2-7.98 = -5.98
0.1	2.57	6.71	=0.1-6.71 =-6.61
10	2.11	9.64	=10-9.64 = 0.36
7	2.52	7.03	=7-7.03 = -0.03
5.7	2.77	5.44	=5.7-5.44 = 0.26
6.7	2.44	7.54	= 6.7-7.54 = -0.84
6.6	2.89	4.67	= 6.6-4.67 = 1.93
11.5	2.9	4.61	=11.5-4.61 = 6.89
0.3	2.91	4.54	=0.3-4.54 = -4.24