Find the explained variation for the paired data.
-The paired data below consists of test scores and hours of preparation for 5 randomly selected students. The equation of the regression line is $\hat { y } = 44.8447 + 3.52427 x$. Find the explained variation.
$\begin{array}{c|rrrrr}
x \text { Hours of preparation } & 5 & 2 & 9 & 6 & 10 \
\hline y \text { Test of score } & 64 & 48 & 72 & 73 & 80
\end{array}$

A) $498.103$
B) $511.724$
C) $599.2$
D) $87.4757$

Question

Find the explained variation for the paired data.
-The paired data below consists of test scores and hours of preparation for 5 randomly selected students. The equation of the regression line is $\hat { y } = 44.8447 + 3.52427 x$. Find the explained variation.
$\begin{array}{c|rrrrr}
x \text { Hours of preparation } & 5 & 2 & 9 & 6 & 10 \
\hline y \text { Test of score } & 64 & 48 & 72 & 73 & 80
\end{array}$

A) $498.103$
B) $511.724$
C) $599.2$
D) $87.4757$

Quizplus · Accepted Answer

First, we need to find the total variation of the data set:
$$\sum (y-y_{avg})^2 = \sum y^2 - n(y_{avg})^2$$
where
$$\sum y = 337, \quad \sum y^2 = 25057, \quad n = 5$$
$$y_{avg} = \frac{\sum y}{n} = \frac{337}{5} = 67.4$$
So,
$$\sum (y-y_{avg})^2 = 25057 - 5(67.4)^2 = 1062.4$$
Next, we need to find the residual variation, which is the unexplained variation of the data around the regression line:
$$\sum (y-\hat y)^2 = \sum (y - (44.8447 + 3.52427x))^2 = 563.676$$
Finally, we can find the explained variation by subtracting the residual variation from the total variation:
$$\sum (\hat y - y_{avg})^2 = \sum y^2 - n(y_{avg})^2 - \sum (y-\hat y)^2 = 1062.4 - 563.676 = 498.724$$
Therefore, the answer is B.

Find the Explained Variation for the Paired Data y^=44.8447+3.52427x\hat { y } = 44.8447 + 3.52427 xy^​=44.8447+3.52427x

Find the Explained Variation for the Paired Data $\hat { y } = 44.8447 + 3.52427 x$