Question

We wish to predict the salary for baseball players (y) using the variables RBI (x1) and HR (x2), then we use a regression equation of the form ˆy=b0+b1x1+b2x2.

• HR - Home runs - hits on which the batter successfully touched all four bases, without the contribution of a fielding error.
• RBI - Run batted in - number of runners who scored due to a batters' action, except when batter grounded into double play or reached on an error
• Salary is in millions of dollars.

The following is a chart of baseball players' salaries and statistics from 2016.

Player Name	RBI's	HR's	Salary (in millions)
Miquel Cabrera	108	38	28.050
Yoenis Cespedes	86	31	27.500
Ryan Howard	59	25	25.000
Albert Pujols	119	31	25.000
Robinson Cano	103	39	24.050
Mark Teixeira	44	15	23.125
Joe Mauer	49	11	23.000
Hanley Ramirez	111	30	22.750
Justin Upton	87	31	22.125
Adrian Gonzalez	90	18	21.857
Jason Heyward	49	7	21.667
Jayson Werth	70	21	21.571
Matt Kemp	108	35	21.500
Jacoby Ellsbury	56	9	21.143
Chris Davis	84	38	21.119
Buster Posey	80	14	20.802
Shin-Soo Choo	17	7	20.000
Troy Tulowitzki	79	24	20.000
Ryan Braun	91	31	20.000
Joey Votto	97	29	20.000
Hunter Pence	57	13	18.500
Prince Fielder	44	8	18.000
Adrian Beltre	104	32	18.000
Victor Martinez	86	27	18.000
Carlos Gonzalez	100	25	17.454
Matt Holliday	62	20	17.000
Brian McCann	58	20	17.000
Mike Trout	100	29	16.083
David Ortiz	127	38	16.000
Adam Jones	83	29	16.000
Curtis Granderson	59	30	16.000
Colby Rasmus	54	15	15.800
Matt Wieters	66	17	15.800
J.D. Martinez	68	22	6.750
Brandon Crawford	84	12	6.000
Rajai Davis	48	12	5.950
Aaron Hill	38	10	12.000
Coco Crisp	55	13	11.000
Ben Zobrist	76	18	10.500
Justin Turner	90	27	5.100
Denard Span	53	11	5.000
Chris Iannetta	24	7	4.550
Leonys Martin	47	15	4.150
Justin Smoak	34	14	3.900
Jorge Soler	31	12	3.667
Evan Gattis	72	32	3.300
Logan Forsythe	52	20	2.750
Jean Segura	64	20	2.600

a) Use software to find the multiple linear regression equation. Enter the coefficients rounded to 4 decimal places.
ˆy= ______ + _____ x1 + ______ x2

b) Use the multiple linear regression equation to predict the salary for a baseball player with an RBI of 31 and HR of 20. Round your answer to 1 decimal place, do not convert numbers to dollars.
      ________ millions of dollars

c) Holding all other variables constant, what is the correct interpretation of the coefficient b1=0.111 in the multiple linear regression equation?

• For each HR, a baseball player's predicted salary increases by 0.111 million dollars.
• For each RBI, a baseball player's predicted salary increases by 0.111 million dollars.
• If the baseball player's salary increases by 0.111 million dollars, then the predicted RBI will increase by one.
• If the baseball player's salary increases by 0.111 million dollars, then the predicted RBI will increase by 0.0371.

d) Holding all other variables constant, what is the correct interpretation of the coefficient b2=0.0371 in the multiple linear regression equation?

• If the baseball player's salary increases by 0.0371 million dollars, then the predicted HR will increase by one.
• If the baseball player's salary increases by 0.0371 million dollars, then the predicted HR will increase by 0.111.
• For each RBI, a baseball player's predicted salary increases by 0.0371 million dollars.
• For each HR, a baseball player's predicted salary increases by 0.0371 million dollars.

Answer 1

Output using excel:

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.423071
R Square	0.178989
Adjusted R Square	0.1425
Standard Error	6.964542
Observations	48
ANOVA
	df	SS	MS	F	Significance F
Regression	2	475.8568	237.9284	4.90525	0.011826
Residual	45	2182.718	48.50485
Total	47	2658.575
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
Intercept	7.061791	2.964789	2.381886	0.021513	1.090399	13.03318	1.090399	13.03318
RBI's	0.110976	0.068915	1.610321	0.114321	-0.02783	0.249778	-0.02783	0.249778
HR's	0.03709	0.186675	0.198688	0.843401	-0.33889	0.413074	-0.33889	0.413074

a) Regression equation:

ŷ = 7.0618 + 0.1110 x1 + 0.0371 x2

b) Predicted salary for RBI = 31 and HR = 20

ŷ = 7.0618 + 0.1110 * 31 + 0.0371 *20 = 11.2 millions of dollars

c) Answer: For each RBI, a baseball player's predicted salary increases by 0.111 million dollars.

d) Answer: For each HR, a baseball player's predicted salary increases by 0.0371 million dollars.