To determine the correlation coefficient ($r_{xy}$) given the tangent of the angle between the regression lines and the ratio of standard deviations ($\sigma_x = \frac{1}{2} \sigma_y$), let’s solve step by step:

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Step 1: Understand the formula

The tangent of the angle between the regression lines is related to the correlation coefficient $r_{xy}$ by the formula: $\tan \theta = \frac{2(1 - r_{xy}^2)^{1/2}}{r_{xy}\left(\frac{\sigma_y}{\sigma_x} + \frac{\sigma_x}{\sigma_y}\right)}$ where:

• $\theta$ is the angle between the regression lines,
• $\sigma_x$ and $\sigma_y$ are the standard deviations of $x$ and $y$,
• $r_{xy}$ is the correlation coefficient.

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Step 2: Substitute known values

Given:

• $\tan \theta = 0.6$,
• $\sigma_x = \frac{1}{2} \sigma_y \implies \frac{\sigma_y}{\sigma_x} = 2$.

Substituting $\frac{\sigma_y}{\sigma_x} = 2$ into the denominator of the formula: $\frac{\sigma_y}{\sigma_x} + \frac{\sigma_x}{\sigma_y} = 2 + \frac{1}{2} = 2.5$

Thus, the formula becomes: $0.6 = \frac{2(1 - r_{xy}^2)^{1/2}}{r_{xy} \cdot 2.5}$

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Step 3: Rearrange the equation

Rearranging for $r_{xy}$, we get: $0.6 \cdot 2.5 \cdot r_{xy} = 2(1 - r_{xy}^2)^{1/2}$ $1.5r_{xy} = 2(1 - r_{xy}^2)^{1/2}$ Square both sides to eliminate the square root: $(1.5r_{xy})^2 = 4(1 - r_{xy}^2)$ $2.25r_{xy}^2 = 4 - 4r_{xy}^2$

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Step 4: Solve for $r_{xy}^2$

Rearrange terms: $2.25r_{xy}^2 + 4r_{xy}^2 = 4$ $6.25r_{xy}^2 = 4$ $r_{xy}^2 = \frac{4}{6.25} = 0.64$

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Step 5: Solve for $r_{xy}$

Taking the square root: $r_{xy} = \pm \sqrt{0.64} = \pm 0.8$

Since the sign of $r_{xy}$ is not specified in the problem, the correlation coefficient could be either $+0.8$ or $-0.8$.

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Final Answer:

$r_{xy} = \pm 0.8$

Would you like a more detailed explanation or clarification on any part of the derivation? 😊

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Related Questions:

1. How does the angle between regression lines affect the correlation coefficient?
2. What happens to $r_{xy}$ if $\tan \theta$ becomes zero?
3. Can $\sigma_x > \sigma_y$ in a similar scenario? How would that affect $r_{xy}$?
4. Why does $|r_{xy}| \leq 1$ always hold in statistical calculations?
5. How are regression coefficients related to $r_{xy}$ in general?

Tip:

Always double-check the ratio of $\sigma_x$ and $\sigma_y$ to correctly calculate terms like $\frac{\sigma_y}{\sigma_x}$.