In a simple linear regression problem,the following sum of squares are produced: $$\sum \left( y _ { i } - \bar { y } \right) ^ { 2 } = 200$$ , $$\sum \left( \gamma _ { i } - \hat { y } _ { i } \right) ^ { 2 } = 50$$ ,and $$\sum \left( \tilde { y } _ { i } - \bar { y } \right) ^ { 2 } = 150$$ .The percentage of the variation in y that is explained by the variation in x is:
A) 25%
B) 75%
C) 33%
D) 50%

Question

Quizplus · Accepted Answer

In simple linear regression, the sum of squares total can be decomposed into two parts: the sum of squares explained by the regression (also known as sum of squares regression or model sum of squares) and the sum of squares unexplained by the regression (also known as sum of squares residual or error sum of squares).

In this problem, we are given the following information:

$SS_{total} = \sum \left( y_i - \bar{y} ight)^2 = 200$

$SS_{residual} = \sum \left( \gamma_i - \hat{y}_i ight)^2 = 50$

$SS_{regression} = SS_{total} - SS_{residual} = \sum \left( 	ilde{y}_i - \bar{y} ight)^2 = 150$

The percentage of the variation in y that is explained by the variation in x, also known as the coefficient of determination or R-squared, is defined as:

$R^2 = \frac{SS_{regression}}{SS_{total}}$

Plugging in the values given in the problem, we get:

$R^2 = \frac{150}{200} = 0.75$

Multiplying by 100% to get a percentage, we get:

$R^2 = 75\%$

Therefore, the answer is B) 75%. 75% of the variation in y is explained by the variation in x, as captured by the simple linear regression model.

In a Simple Linear Regression Problem,the Following Sum of Squares ∑(yi−yˉ)2=200\sum \left( y _ { i } - \bar { y } \right) ^ { 2 } = 200∑(yi​−yˉ​)2=200

In a Simple Linear Regression Problem,the Following Sum of Squares $\sum \left( y _ { i } - \bar { y } \right) ^ { 2 } = 200$