If two regression coefficients are -0.1 and -0.9, then correlation coefficient is:
-0.3
Understanding Regression and Correlation Coefficients
The question asks us to find the correlation coefficient given two regression coefficients. There is a specific relationship that connects the two regression coefficients ($b_{yx}$ and $b_{xy}$) with the correlation coefficient ($r$).
Relationship Between Correlation and Regression Coefficients
The correlation coefficient ($r$) and the two regression coefficients ($b_{yx}$ and $b_{xy}$) are related by the formula:
\( r = \pm \sqrt{b_{yx} \times b_{xy}} \)
An important property is that the correlation coefficient ($r$) must have the same sign as both regression coefficients ($b_{yx}$ and $b_{xy}$). If both regression coefficients are positive, $r$ is positive. If both regression coefficients are negative, $r$ is negative.
Calculating the Correlation Coefficient
Given regression coefficients are:
- \( b_{yx} = -0.1 \)
- \( b_{xy} = -0.9 \)
Since both regression coefficients are negative, the correlation coefficient ($r$) must also be negative.
Using the formula \( r = \pm \sqrt{b_{yx} \times b_{xy}} \), and knowing $r$ is negative:
\( r = - \sqrt{(-0.1) \times (-0.9)} \)
\( r = - \sqrt{0.09} \)
Now, we calculate the square root of 0.09:
\( \sqrt{0.09} = 0.3 \)
Substituting this back into the equation for $r$:
\( r = -0.3 \)
Conclusion
The calculated correlation coefficient is -0.3.
| Given Values | Result |
|---|---|
| \( b_{yx} = -0.1 \) | \( r = -0.3 \) |
| \( b_{xy} = -0.9 \) |
Revision Table: Key Properties
Here are some important properties relating regression and correlation coefficients:
- The sign of \( r \), \( b_{yx} \), and \( b_{xy} \) is always the same.
- The geometric mean of the two regression coefficients is equal to the square of the correlation coefficient: \( \sqrt{b_{yx} \times b_{xy}} = |r| \).
- The arithmetic mean of the two regression coefficients is greater than or equal to the correlation coefficient: \( \frac{b_{yx} + b_{xy}}{2} \ge r \). This property holds true regardless of the sign of the coefficients, provided the coefficients and $r$ have the same sign.
- If one regression coefficient is greater than 1, the other must be less than 1. This is because their product \( b_{yx} \times b_{xy} = r^2 \), and \( r^2 \) is always between 0 and 1 (inclusive).
Additional Information on Correlation and Regression
Correlation Coefficient (\( r \)): Measures the strength and direction of a linear relationship between two variables. It ranges from -1 to +1.
- \( r = 1 \): Perfect positive linear correlation.
- \( r = -1 \): Perfect negative linear correlation.
- \( r = 0 \): No linear correlation.
Regression Coefficients (\( b_{yx} \) and \( b_{xy} \)): These represent the change in the dependent variable associated with a one-unit change in the independent variable.
- \( b_{yx} \): Regression coefficient of Y on X (change in Y for a unit change in X).
- \( b_{xy} \): Regression coefficient of X on Y (change in X for a unit change in Y).
Understanding the relationship \( r = \pm \sqrt{b_{yx} \times b_{xy}} \) is crucial for solving problems involving both concepts.
Important Questions from Correlation and Regression
If the regression coefficient of x on y and y on x are \(- \frac{1}{2}\) and \(- \frac{1}{8}\) respectively, then what is the correlation coefficient between x and y?
Variable Y regresses with variable X with the conditions that \(\overline X = 5.50\) , \(\overline Y = 3.50\) and b = 1.50 in the linear regression model (Y = a + bX), where \(\overline Y\) and \(\overline X\) are means of the respective variables and b refers to gradient of line of Y w. r. t X. Which one of the following values of parameter 'a' of the model is correct?
It is given that X̅ = 10, Y̅ = 90, σ X= 3, σ Y= 12 and r XY = 0.8. The regression equation of X on Y is
For 10 observations on price (x) and supply (y), the following data was obtained:
∑x = 130, ∑y = 220, ∑ x 2= 2288, ∑y 2= 5506 and ∑ xy = 3467.
What is the line of regression of y on x?If, in a series. there are m items whose ranks are common, then which factor is added for correction for each repeating value in both the series?
