Question

We use the form ŷ = a + bx for the least-squares line. In some computer printouts, the least-squares equation is not given directly. Instead, the value of the constant a is given, and the coefficient b of the explanatory or predictor variable is displayed. Sometimes a is referred to as the constant, and sometimes as the intercept. Data from a report showed the following relationship between elevation (in thousands of feet) and average number of frost-free days per year in a state.

A Minitab printout provides the following information.

Predictor	Coef	SE Coef	T	P
Constant	315.81	28.31	11.24	0.002
Elevation	-29.031	3.511	-8.79	0.003
S = 11.8603	R-Sq = 96.2%

Notice that "Elevation" is listed under "Predictor." This means that elevation is the explanatory variable x. Its coefficient is the slope b. "Constant" refers to a in the equation

ŷ = a + bx.

(a) Use the printout to write the least-squares equation.

ŷ =

+   x

(b) For each 1000-foot increase in elevation, how many fewer frost-free days are predicted? (Round your answer to three decimal places.)


(c) The printout gives the value of the coefficient of determination r². What is the value of r? Be sure to give the correct sign for r based on the sign of b. (Round your answer to four decimal places.)


(d) What percentage of the variation in y can be explained by the corresponding variation in x and the least-squares line?
%

What percentage is unexplained?
%

Answer 1

a)

The equation of the regression line is given as

y = 315.81 - 29.031x

b) For every 1000 feet increase in elevation, the frost-free days = b

= 29.031 ( since the elevation is in 1000's feet so x=1)

c) r = - sqrt(coefficient of dtermination) { negative because b is negative}

=sqrt(0.962)

= 0.9808

d) R-sq is the % of variation explained in Y

so % varaince explained = 96.2%

and % of variance not explained = (1-96.2) = 3.8%