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 $56 million in sales, an estimate of the earnings is  ($ millions).

Answer 1

a.

X Values
∑ = 499.4
Mean = 41.617
∑(X - M_x)² = SS_x = 7592.997

Y Values
∑ = 64.1
Mean = 5.342
∑(Y - M_y)² = SS_y = 116.009

X and Y Combined
N = 12
∑(X - M_x)(Y - M_y) = 647.842

R Calculation
r = ∑((X - M_y)(Y - M_x)) / √((SS_x)(SS_y))

r = 647.842 / √((7592.997)(116.009)) = 0.690

b. Hence coefficient of determination is

c.

Sum of X = 499.4
Sum of Y = 64.1
Mean X = 41.6167
Mean Y = 5.3417
Sum of squares (SS_X) = 7592.9967
Sum of products (SP) = 647.8417

Regression Equation = ŷ = bX + a

b = SP/SS_X = 647.84/7593 = 0.085

a = M_Y - bM_X = 5.34 - (0.09*41.62) = 1.791

ŷ = 0.085X + 1.791

d. For x=56, ŷ = (0.085*56) + 1.791=6.551