If two regression coefficients are -0.8 and -0.2, then the value of coefficient of correlation is
-0.40
Understanding Regression Coefficients and Correlation
The question asks us to find the coefficient of correlation (r) given two regression coefficients. Regression coefficients measure the change in one variable associated with a unit change in the other variable. The coefficient of correlation measures the strength and direction of the linear relationship between two variables.
Relationship Between Regression and Correlation Coefficients
There is a specific relationship connecting the coefficient of correlation (r) and the two regression coefficients, say \(b_{yx}\) (regression of Y on X) and \(b_{xy}\) (regression of X on Y). The relationship is given by the formula:
\(r = \pm\sqrt{b_{yx} \times b_{xy}}\)
An important property is that the sign of the correlation coefficient (r) must be the same as the sign of both regression coefficients (\(b_{yx}\) and \(b_{xy}\)). This is because the direction of the relationship indicated by correlation and regression is the same.
Solving the Problem
We are given the two regression coefficients as -0.8 and -0.2.
- Let \(b_{yx} = -0.8\)
- Let \(b_{xy} = -0.2\)
Both coefficients are negative. Therefore, the coefficient of correlation (r) must also be negative.
Using the formula \(r = \pm\sqrt{b_{yx} \times b_{xy}}\), and knowing r must be negative:
\(r = -\sqrt{(-0.8) \times (-0.2)}\)
First, multiply the two regression coefficients:
\((-0.8) \times (-0.2) = 0.16\)
Now, take the square root of the product:
\(\sqrt{0.16} = 0.4\)
Since r must have the same sign as the regression coefficients (which are negative), we take the negative square root:
\(r = -0.4\)
Thus, the value of the coefficient of correlation is -0.4.
Summary of Calculation
| Given regression coefficients | \(b_{yx} = -0.8\), \(b_{xy} = -0.2\) |
| Formula for correlation coefficient | \(r = \pm\sqrt{b_{yx} \times b_{xy}}\) |
| Sign of r | Negative (same as \(b_{yx}\) and \(b_{xy}\)) |
| Calculation | \(r = -\sqrt{(-0.8) \times (-0.2)} = -\sqrt{0.16} = -0.4\) |
| Result | \(r = -0.4\) |
The calculated coefficient of correlation is -0.4.
Revision Table: Key Statistical Concepts
| Concept | Description | Range | Relationship with other coefficients |
|---|---|---|---|
| Regression Coefficient (\(b_{yx}\) or \(b_{xy}\)) | Measures change in dependent variable per unit change in independent variable. | \(-\infty\) to \(+\infty\) | Product is \(r^2\) |
| Coefficient of Correlation (r) | Measures strength and direction of linear relationship. | -1 to +1 | \(r = \pm\sqrt{b_{yx} \times b_{xy}}\). Has same sign as regression coefficients. |
| Coefficient of Determination (\(r^2\)) | Proportion of variance in dependent variable predictable from independent variable. | 0 to 1 | \(r^2 = b_{yx} \times b_{xy}\) |
Additional Information on Correlation and Regression
The coefficient of correlation, denoted by r, ranges from -1 to +1. A value of +1 indicates a perfect positive linear relationship, -1 indicates a perfect negative linear relationship, and 0 indicates no linear relationship.
- If r is positive, as one variable increases, the other tends to increase.
- If r is negative, as one variable increases, the other tends to decrease.
- The closer |r| is to 1, the stronger the linear relationship.
Regression analysis helps in understanding the nature of the relationship and predicting the value of a dependent variable based on an independent variable.
The two regression lines (Y on X and X on Y) intersect at the point \((\bar{X}, \bar{Y})\), where \(\bar{X}\) and \(\bar{Y}\) are the means of the variables X and Y respectively.
The geometric mean of the two regression coefficients is the coefficient of correlation (ignoring the sign for a moment, or considering \(r^2\)): \(|r| = \sqrt{|b_{yx} \times b_{xy}|}\). The sign of r is then applied based on the sign of the regression coefficients.
Important Questions from Correlation and Regression
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?
Given below are two statements:
Statement l: One of the assumptions under OLS method states that the regression model is linear in the parameters, though it may or may not be linear in the variables.
Statement ll: The variance of the error, or disturbance, term is not the same regardless of the value of the explanatory variable under OLS.
In light of the above statements, choose the most appropriate answer from the options given below
Which of the following statements relating to Correlation and Regression are true?
(a) The Coefficient of Correlation is independent of change of origin and scale.
(b) The Coefficient of Correlation between the two variables is the arithmetic average of the two Regression Coefficients.
(c) The probable error of the Coefficient Correlation is 0.6745 times its standard error.
(d) Coefficient of Correlation multiplied by the ratio between the standard deviations of the two variables denotes the slope of the regression line.
Code:
If two regression coefficients are 0.8 and 1.2, which one of the following is the value of coefficient of correlation?
For 4 data points of two correlated variables x and y, it is given that
∑ x = 24, ∑ y = 11, ∑ x 2= 202, ∑ xy = 84, ∑ y 2= 39
Fit a least squares line to this data using x as independent variable.
