Question

2. “In Salary Arbitration, Old Math Matters Most," published in the New York Times, explained that for baseball players, "Salary arbitration hearings allow the club and the player to argue for one salary or another based largely on the player's statistics and how the statistics compare to those of similar players." Below are batting averages and following year's salaries (in millions) for some players the preceding year, along with number of homeruns hit the previous season: Player Batting Average Salary HomeRuns Lance Berkman 0.316 10.500 30 Carlos Lee 0.305 8.000 Mike Lowell 0.276 6.500 Aubrey Huff 0.297 4.917 Craig Wilson 0.264 3.000 Erubiel Durazo 0.259 2.100 21 a) Use Excel or your calculator to produce a scatterplot of salary (y) versus batting average (x), and verify that its appearance is linear. Regress salary on batting average: report the equation of the regression line, the value of the correlation, and the p-value. b) Based on the small p-value, choose two appropriate conclusions from the following: i) There is evidence of a relationship between salary and batting average in the larger population of players. ii) There is no evidence of a relationship between salary and batting average in the larger population of players. 111) There is evidence that the slope of the regression line relating salary to batting average in the larger population of players is not zero. iv) The slope of the regression line relating salary to batting average in the larger population of players could be zero. c) Regress salary (y) on homeruns (x): report the equation of the regression line, the value of the correlation, and the p-value.

Answer 1

a)

Scatterplot у 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

Regression Statistics
Multiple R	0.8948
R Square	0.8007
Adjusted R Square	0.7509
Standard Error	1.5739
Observations	6
ANOVA
	df	SS	MS	F	Significance F
Regression	1	39.8	39.8	16.07	0.0160
Residual	4	9.9	2.5
Total	5	49.7
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	-28.9	8.700	-3.327	0.0292	-53.1019	-4.7916
X	121.5	30.319	4.009	0.0160	37.3688	205.7266

so, regression line is   Ŷ =   -28.947   +   121.548   *x
correlation coefficient ,    r =   0.8948
      

p-value =    0.0160

b)

i) There is evidence of a relationship between salary and batting average in the larger population of players.

iii) There is evidence that the slope of the regression line relating salary to batting average in the larger population of players is not zero.

c)

Regression Statistics
Multiple R	0.6591
R Square	0.4345
Adjusted R Square	0.2931
Standard Error	2.6514
Observations	6
ANOVA
	df	SS	MS	F	Significance F
Regression	1	21.6	21.6	3.07	0.1545
Residual	4	28.1	7.0
Total	5	49.7
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	-9.3	8.711	-1.069	0.3451	-33.4994	14.8698
X	0.5	0.302	1.753	0.1545	-0.3086	1.3656

Ŷ =   -9.315   +   0.529   *x
correlation , r=   0.6591
p-value =    0.1545