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	319.59	28.31	11.24	0.002
Elevation	-29.679	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

Let  

x=elevation (in thousands of feet)

y=average number of frost-free days per year in a state

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

using this

we get a=319.59 and b=-29.679

ans: the least-squares equation is

(b) For each 1000-foot increase in elevation, how many fewer frost-free days are predicted? (Round your answer to three decimal places.)
The x values: elevation is measured in  thousands of feet. This means 1 unit of x is equal to 1000 feet.

The slope coefficient is -29.679. The negative sign indicates that the elevation and number of frost-free days move in opposite direction. Thatis, for 1 unit increase in x (which is 1000 foot increase in elevation), the number of frost free days decreases by 29.679 days

ans: For each 1000-foot increase in elevation, 29.679 fewer frost-free days are predicted

(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.)
From this

we get the value of r-square as

The value of r is

However, the sign of the slope is -ve. Hence the sign of r should be negative.

ans: the value of r is -0.9808

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

ans: the percentage of variation y can be explained by the corresponding variation in x and the least-squares line is 96.2%

What percentage is unexplained?

The percentage unexplained is 100-96.2=3.8%

ans: The percentage is unexplained is 3.8%