The equation of the regression line for the paired data below is $\hat { y } = 3 x$. Find the explained variation.
$\begin{array}{l|rrrr}
\mathrm{x} & 2 & 4 & 5 & 6 \
\hline \mathrm{y} & 7 & 11 & 13 & 20
\end{array}$
A) $88.75$
B) $10.00$
C) $72.45$
D) $78.75$

Question

Quizplus · Accepted Answer

The explained variation can be found by calculating the sum of the squares of the differences between the predicted y values (using the regression equation) and the mean of the actual y values. First, calculate the mean of y: $mean(y) = (7 + 11 + 13 + 20) / 4 = 51 / 4 = 12.75$. Then, calculate the predicted y values using the regression equation $\hat{y} = 3x$: for $x = 2, \hat{y} = 6$; for $x = 4, \hat{y} = 12$; for $x = 5, \hat{y} = 15$; for $x = 6, \hat{y} = 18$. Now, calculate the differences between these predicted values and the mean of y, square these differences, and sum them up: $(6 - 12.75)^2 + (12 - 12.75)^2 + (15 - 12.75)^2 + (18 - 12.75)^2 = 45.5625 + 0.5625 + 5.0625 + 27.5625 = 78.75$.

The Equation of the Regression Line for the Paired Data y^=3x\hat { y } = 3 xy^​=3x

The Equation of the Regression Line for the Paired Data $\hat { y } = 3 x$