Calculating the Coefficient of Correlation from Regression Coefficients

The question asks us to find the value of the coefficient of correlation ($r$) given the two regression coefficients. We are given that the two regression coefficients are 0.8 and 1.2.

In the context of linear regression between two variables, say X and Y, there are two regression coefficients:

• $b_{yx}$: The regression coefficient of Y on X. It measures the change in Y for a unit change in X.
• $b_{xy}$: The regression coefficient of X on Y. It measures the change in X for a unit change in Y.

The relationship between the coefficient of correlation ($r$) and the two regression coefficients ($b_{yx}$ and $b_{xy}$) is given by the formula:

\( r = \pm \sqrt{b_{yx} \times b_{xy}} \)

The sign of the coefficient of correlation ($r$) is the same as the sign of the regression coefficients ($b_{yx}$ and $b_{xy}$). Since both given regression coefficients (0.8 and 1.2) are positive, the coefficient of correlation ($r$) will also be positive.

Let's assume the given regression coefficients are $b_{yx} = 0.8$ and $b_{xy} = 1.2$ (the order doesn't matter for the product).

Now, we can substitute these values into the formula:

\( r = + \sqrt{0.8 \times 1.2} \)

First, calculate the product of the two regression coefficients:

\( 0.8 \times 1.2 = 0.96 \)

So, the formula becomes:

\( r = \sqrt{0.96} \)

Now, we need to calculate the square root of 0.96:

\( \sqrt{0.96} \approx 0.97979 \)

Rounding this value to two decimal places, we get approximately 0.98.

Let's look at the given options:

1. 0.97
2. 0.96
3. 1.00
4. 0.80

Comparing our calculated value ($\approx 0.9798$) with the options, the closest value is 0.97.

Therefore, the value of the coefficient of correlation is approximately 0.97.

It is important to note that the geometric mean of the two regression coefficients ($b_{yx}$ and $b_{xy}$) is equal to the coefficient of correlation ($r$) when both coefficients are positive (or negative, but the sign matches). Also, the correlation coefficient must lie between -1 and +1, inclusive. $r^2 = b_{yx} \times b_{xy}$ is a key relationship.

Revision Table: Key Concepts

Additional Information: Properties of Regression and Correlation

• Both regression coefficients ($b_{yx}, b_{xy}$) and the correlation coefficient ($r$) must have the same sign. If one regression coefficient is positive, the other must also be positive, and the correlation coefficient must be positive. If one is negative, the other must be negative, and the correlation coefficient must be negative. This is consistent with the given values (0.8 and 1.2 are both positive).
• The magnitude of the correlation coefficient ($|r|$) is the geometric mean of the magnitudes of the regression coefficients: \(|r| = \sqrt{|b_{yx}| \times |b_{xy}|}\).
• The coefficient of correlation ($r$) is a symmetric measure, meaning the correlation between X and Y is the same as the correlation between Y and X.
• Regression coefficients are not symmetric ($b_{yx}$ is generally not equal to $b_{xy}$) unless the standard deviations of X and Y are equal.
• If the regression lines are perpendicular, the product $b_{yx} \times b_{xy} = 0$, which implies $r=0$ (no linear correlation).
• If the regression lines are identical, the product $b_{yx} \times b_{xy} = 1$, which implies $r=\pm 1$ (perfect linear correlation).
• In this problem, $b_{yx} \times b_{xy} = 0.8 \times 1.2 = 0.96$. Since $0 < 0.96 < 1$, the correlation must be strong but not perfect ($0 < |r| < 1$). Our calculated value $\sqrt{0.96} \approx 0.9798$ falls within this range and confirms a strong positive correlation.