The Pearson correlation coefficient $r$ between two variables $x$ and $y$ is defined as:

$r = \frac{\text{Cov}(x, y)}{\sigma_x \cdot \sigma_y}$

Where:

• $r$ is the correlation coefficient.
• $\text{Cov}(x, y)$ is the covariance between $x$ and $y$.
• $\sigma_x$ and $\sigma_y$ are the standard deviations of $x$ and $y$, respectively.

Given:

• $r = 0.2$
• $\text{Cov}(x, y) = 2$
• $\sigma_x = 2$

We need to determine the interval for $\sigma_y$, the standard deviation of $y$.

Step 1: Substitute the known values into the formula

$0.2 = \frac{2}{2 \cdot \sigma_y}$

Step 2: Simplify the equation

$0.2 = \frac{1}{\sigma_y}$

Step 3: Solve for $\sigma_y$

$\sigma_y = \frac{1}{0.2} = 5$

Step 4: Determine the interval

The problem indicates that $\sigma_y$ belongs to an interval. Since the standard deviation is positive and there is no additional variation given, the standard deviation $\sigma_y$ must be exactly $5$, which implies the interval is:

$\sigma_y \in (5, 5)$

If any further uncertainty is implied, clarify the range of variability. Would you like a deeper exploration of this? Here are 5 related questions:

1. How is the Pearson correlation coefficient affected by the covariance and standard deviations of variables?
2. Can the correlation coefficient ever exceed the range of $[-1, 1]$?
3. What happens to the correlation coefficient if the standard deviation of one variable is 0?
4. How can we interpret the meaning of a correlation coefficient of $0.2$?
5. Is it possible to have high covariance but low correlation? Why or why not?

Tip: Always double-check calculations involving fractions and decimals in statistical formulas to avoid small errors.