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?
Understanding Regression and Correlation Coefficients
In statistics, regression coefficients measure the change in the dependent variable for a one-unit change in the independent variable. We are given two regression coefficients:
- Regression coefficient of x on y, denoted as \(b_{xy}\)
- Regression coefficient of y on x, denoted as \(b_{yx}\)
The correlation coefficient, denoted as \(r\), measures the strength and direction of the linear relationship between two variables, x and y. It ranges from -1 to +1.
Relationship Between Regression and Correlation Coefficients
There is a direct relationship between the correlation coefficient and the two regression coefficients. The correlation coefficient is the geometric mean of the two regression coefficients. The formula connecting them is:
\[r = \pm \sqrt{b_{xy} \times b_{yx}}\]The sign of the correlation coefficient (\(r\)) is always the same as the sign of the two regression coefficients (\(b_{xy}\) and \(b_{yx}\)). This means if both regression coefficients are positive, the correlation coefficient is positive. If both are negative, the correlation coefficient is negative. It is important to note that \(b_{xy}\) and \(b_{yx}\) must always have the same sign.
Calculating the Correlation Coefficient
We are given the following values:
- Regression coefficient of x on y, \(b_{xy} = -\frac{1}{2}\)
- Regression coefficient of y on x, \(b_{yx} = -\frac{1}{8}\)
Both regression coefficients are negative, which means the correlation coefficient \(r\) must also be negative.
Now, we can substitute these values into the formula:
\[r = - \sqrt{\left(-\frac{1}{2}\right) \times \left(-\frac{1}{8}\right)}\]First, multiply the two regression coefficients:
\[\left(-\frac{1}{2}\right) \times \left(-\frac{1}{8}\right) = \frac{1 \times 1}{2 \times 8} = \frac{1}{16}\]Now, take the square root of the result and apply the negative sign:
\[r = - \sqrt{\frac{1}{16}}\] \[r = - \frac{\sqrt{1}}{\sqrt{16}}\] \[r = - \frac{1}{4}\]Thus, the correlation coefficient between x and y is \(-\frac{1}{4}\).
Conclusion on Correlation Coefficient Value
The calculated correlation coefficient is \(-\frac{1}{4}\). This value falls between -1 and +1, which is consistent with the range of a correlation coefficient. Since the value is negative, it indicates a negative linear relationship between x and y; as one variable increases, the other tends to decrease.
Let's compare this result with the given options:
| Option | Value |
|---|---|
| 1 | \(-\frac{1}{4}\) |
| 2 | \(-\frac{1}{{16}}\) |
| 3 | \(\frac{1}{{16}}\) |
| 4 | \(\frac{1}{4}\) |
Our calculated value \(-\frac{1}{4}\) matches Option 1.
Revision Table: Key Concepts in Regression and Correlation
| Concept | Description | Notation | Relationship to Others |
|---|---|---|---|
| Regression Coeff. of x on y | Measures change in x for unit change in y | \(b_{xy}\) | \(r = \pm \sqrt{b_{xy} b_{yx}}\) \(b_{xy}\) and \(b_{yx}\) have same sign as \(r\) |
| Regression Coeff. of y on x | Measures change in y for unit change in x | \(b_{yx}\) | |
| Correlation Coeff. | Measures strength and direction of linear relationship | \(r\) | \(r^2 = b_{xy} \times b_{yx}\) -1 \(\le\) r \(\le\) 1 |
Additional Information: Properties of Correlation and Regression
- The correlation coefficient \(r\) is unitless and symmetric, meaning the correlation between x and y is the same as the correlation between y and x.
- The regression coefficients \(b_{xy}\) and \(b_{yx}\) are generally not equal and depend on the units of measurement of x and y.
- The product of the two regression coefficients is equal to the square of the correlation coefficient: \(b_{xy} \times b_{yx} = r^2\).
- If the correlation coefficient is 0, it indicates no linear relationship between the variables.
- If the absolute value of the correlation coefficient is 1 (\(r=1\) or \(r=-1\)), it indicates a perfect linear relationship, and the regression lines are identical.
- Regression analysis helps predict the value of one variable based on the value of another, while correlation analysis quantifies the association between them.
Important Questions from Correlation and Regression
If two lines of regression are x + 4y + 1 = 0 and 4x + 9y + 7 = 0, then what is the value of x when y = -3 ?
A bivariate data set contains only two points (-1, 1) and (3, 2). What will be the line of regression of y on x ?
Two variates, x and y, are uncorrelated and have standard deviations σ xand σ yrespectively. What is the correlation coefficient between x + y and x – y?
If the covariance between x and y is 30, variance of x is 25 and variance of y is 144, then what is the correlation coefficient?
The coefficient of correlation when coefficients of regression are 0.2 and 1.8 is
