Using your favorite statistics sof tware package, you generate a scatter plot that display a linear form Youfing and the standard deviation for both variables. The standard deviation for \( x \) is 2.45 , and the standard deviation regression equation is reported as \( \hat{y}=-5.8+1.7 x \) What frection of the variation in y can be explained by the variation in the values of \( x \) ? Express your answer as a decimal rounded to three decimal places. Type your answer.

To determine the fraction of the variation in \( y \) that can be explained by the variation in the values of \( x \), we need to calculate the coefficient of determination, denoted as \( R^2 \). You can find it using the formula: \[ R^2 = \frac{(b \cdot \text{SD}_x)^2}{\text{SD}_y^2} \] In this case, you have the slope \( b = 1.7 \) and the standard deviation for \( x \) is \( 2.45 \). However, we need the standard deviation for \( y \) to compute \( R^2 \). Still, we can assume you either have that information or can calculate it based on residuals from the regression. Assuming the calculation provides a standard deviation for \( y \), you would substitute that value in. As a completely made-up example, let’s say the standard deviation for \( y \) is \( 3.5 \). Plugging in the values gives: \[ R^2 = \frac{(1.7 \cdot 2.45)^2}{3.5^2} = \frac{(4.165)^2}{12.25} \approx 1.1334 / 12.25 \approx 0.093 \] This would mean that approximately \( 0.093 \) of the variation in \( y \) can be explained by \( x \). To answer your question, if you have the correct \( SD_y \), compute it similarly with provided and actual values to find a final decimal rounded to three decimal places. Given that you haven't specified what the standard deviation of \( y \) is, I'm going to focus on that, and say: The fraction of the variation in \( y \) that can be explained by the variation in the values of \( x \) is approximately \( 0.093 \) (or 9.3%) based on this hypothetical example. Please enter the actual SD_y to find the precise answer tailored to your situation.