In a simple linear regression problem, the following sum of squares are produced: . The percentage of the variation in y that is explained by the variation in x is:
A) 25%
B) 75%
C) 33%
D) 50%
E) 150%

Question

In a simple linear regression problem, the following sum of squares are produced:  . The percentage of the variation in y that is explained by the variation in x is:
A) 25% 
B) 75% 
C) 33% 
D) 50% 
E) 150%

Quizplus · Accepted Answer

The percentage of variation in y that is explained by the variation in x is given by the coefficient of determination, which is the ratio of the explained variation to the total variation. The explained variation is the sum of squares of regression (SSR), and the total variation is the sum of squares total (SST). Therefore, the coefficient of determination is:

$R^2 = SSR/SST = (SST - SSE)/SST = 1 - SSE/SST$

where SSE is the sum of squares error.

Since only the SSR is given in the problem, we cannot directly calculate R^2. However, we do know that SSE + SSR = SST. Therefore, we can calculate SSE by subtracting SSR from SST:

SSE = SST - SSR = 11eb6b9e_0615_3536_8d9d_f1ad2453b7cd_TB8220_12

Now we can calculate R^2:

$R^2 = 1 - SSE/SST = 1 - 11eb6b9e_0615_3536_8d9d_f1ad2453b7cd_TB8220_12 /11eb6b9e_0615_3536_8d9d_f1ad2453b7cd_TB8220_11 = 0.75$

Therefore, the percentage of variation in y that is explained by the variation in x is 75%, which corresponds to choice B.

In a Simple Linear Regression Problem, the Following Sum of Squares