The problem asks to calculate the fraction of variation in $y$ that can be explained by the variation in $x$, given the linear regression model:

$y = 5.8 + 1.95x$

You are also provided with the standard deviations for $x$ and $y$:

• Standard deviation of $x$: $\sigma_x = 2.44$
• Standard deviation of $y$: $\sigma_y = 5.86$

The fraction of the variation in $y$ explained by $x$ is given by the coefficient of determination, $R^2$, which can be found by calculating the correlation coefficient $r$ first. The correlation coefficient $r$ can be estimated as:

$r = \frac{b \cdot \sigma_x}{\sigma_y}$

where $b = 1.95$ is the slope of the regression equation. Substituting the values:

$r = \frac{1.95 \cdot 2.44}{5.86} = \frac{4.758}{5.86} \approx 0.812$

Now, to find $R^2$:

$R^2 = r^2 = (0.812)^2 \approx 0.659$

So, the fraction of the variation in $y$ explained by the variation in $x$ is approximately 0.659.

Would you like a detailed explanation or have any additional questions?

Here are five related questions for deeper understanding:

1. How do you interpret the $R^2$ value in regression analysis?
2. What does a higher $R^2$ value indicate about the relationship between $x$ and $y$?
3. How is the slope of the regression line related to the strength of the correlation?
4. What is the difference between the standard deviation and the coefficient of variation?
5. How would the $R^2$ value change if the standard deviation of $x$ were higher?

Tip: Always check whether the $R^2$ value is close to 1, as it indicates a stronger linear relationship between the variables.