Question

Use the value of the linear correlation coefficient to calculate the coefficient of determination. What does this tell you about the explained variation of the data about the regression​ line? About the unexplained​ variation?

r=0.864
a) Calculate the coefficient of determination.
​(Round to three decimal places as​ needed.)
b) What does this tell you about the explained variation of the data about the regression​ line?
c) What % of the variation can be explained by the regression line.
​(Round to one decimal place as​ needed.)
About the unexplained​ variation?
d) What % of the variation is unexplained and is due to other factors or to sampling error.
(Round to one decimal place as​ needed.)

Answer 1

The coefficient of determination is 0.746, indicating that 74.6% of the variation can be explained by the regression line. The remaining 25.4% of the variation is unexplained and due to other factors or sampling error.

Step-by-step explanation:

To calculate the coefficient of determination based on the linear correlation coefficient r, you square the value of r. In this case, with r=0.864, the coefficient of determination is r² = 0.864² = 0.746496. When rounded to three decimal places, this becomes 0.746.

The coefficient of determination tells us the proportion of variance in the dependent variable that can be explained by the independent variable using the regression model. Therefore, the explained variation of the data about the regression line is 74.6% of the total variation in the dependent variable. This implies that the regression line explains a substantial amount of the variation in the data.

The percentage of the variation explained by the regression line is 74.6% when you multiply the coefficient of determination by 100. To find the unexplained variation, you subtract the coefficient of determination from 1. Hence, 1 - r² = 1 - 0.746 = 0.254, or 25.4% when expressed as a percentage. This percentage represents the variation in the dependent variable that cannot be explained by the independent variable and is presumably due to other factors or to sampling error.

Answer 2

Kindly check explanation

Step-by-step explanation:

Given the Coefficient of correlation (r) = 0.864

The Coefficient of determination (R^2) :

0.864^2 = 0.747 ( 3 decimal places)

B) the Coefficient of determinationR^2 tells about the proportion of variation in the dependent variable that is predicted by the independent variable, Hence, a proportion of about 0.75 should fall within the regression line.

C) percentage variation explained :

Coefficient of determination * 100%

0.747 * 100% = 74.7%

D) unexplained variation due to other factors or sampling error :

(total variation - explained variation) = (100 - 74.7)% = 25.3%