Question

assume that you have paired values consisting of heights (in inches) and weights (in lb) from 40 randomly selected men. the linear correlation coefficient r is 0.479. find the value of the coefficient of determination. what practical information does the coefficient of determination provide? choose the correct answer below. a. the coefficient of determination is 0.229. 77.1% of the variation is explained by the linear correlation, and 22.9% is explained by other factors. b. the coefficient of determination is 0.771. 22.9% of the variation is explained by the linear correlation, and 77.1% is explained by other factors. c. the coefficient of determination is 0.771. 77.1% of the variation is explained by the linear correlation, and 22.9% is explained by other factors. d. the coefficient of determination is 0.229. 22.9% of the variation is explained by the linear correlation, and 77.1% is explained by other factors.

Explanation:

Step1: Recall formula for coefficient of determination

The coefficient of determination \( r^2 \) is the square of the linear correlation coefficient \( r \).
Given \( r = 0.479 \), we calculate \( r^2=(0.479)^2\).
\( 0.479\times0.479 = 0.229441\approx0.229 \).

Step2: Interpret coefficient of determination

The coefficient of determination \( r^2 \) represents the proportion of the variation in the dependent variable (weight, in this case) that is explained by the linear relationship with the independent variable (height).
So, the percentage of variation explained is \( r^2\times100\%= 0.229\times 100\% = 22.9\% \).
The remaining variation (explained by other factors) is \( 100\% - 22.9\%=77.1\% \).

Answer:

D. The coefficient of determination is 0.229. 22.9% of the variation is explained by the linear correlation, and 77.1% is explained by other factors.