Question

A recent article in Bloomberg Businessweek listed the “Best Small Companies.” We are interested in the current results of the companies’ sales and earnings. A random sample of 12 companies was selected and the sales and earnings, in millions of dollars, are reported below.

Company	Sales ($ millions)			Earnings ($ millions)			Company	Sales ($ millions)			Earnings ($ millions)
Papa John's International	$	87.4		$	4.9		Checkmate Electronics	$	17.5		$	2.6
Applied Innovation		18.6			4.4		Royal Grip		9.8			1.7
Integracare		17.4			1.3		M-Wave		19.6			3.5
Wall Data		71.7			8.0		Serving-N-Slide		53.7			8.2
Davidson & Associates		58.6			6.6		Daig		28.6			6.0
Chico's FAS		47.3			4.1		Cobra Golf		69.2			12.8

Let sales be the independent variable and earnings be the dependent variable. (Round your answers to 3 decimal places.)

a.	The coefficient of correlation is .
b.	The coefficient of determination is .
c.	The regression equation, with the coefficients, is Y' =  +  X
d.	For a small company with $44 million in sales, an estimate of the earnings is  ($ millions).

Answer 1

a.	The coefficient of correlation is 0.690

b.	The coefficient of determination is 0.4761. Here we assumed to be a linear relationship between X and Y. However the cubic regression model seems more effective with coefficient of determination = 0.665

c.	The regression equation, with the coefficients, is Y = 1.791 + 0.085X. However, it is simple regression model. If we use the cubic regression model the regression equation becomes Y=3.248-0.151X^3+0.008X^2-0.001X^3

d.	For a small company with $44 million in sales, an estimate of the earnings is 5.545 ($ millions). Here we assume a simple linear regression model. For cubic regression model, for a small company with $44 million in sales, an estimate of the earnings is 6.456 ($ millions).