Correlation Coefficient from Regression Lines
The correlation coefficient (\(r\)) between two variables \(x\) and \(y\) can be determined from the slopes of the two regression lines. If the regression line of \(y\) on \(x\) is \(y = b_{yx}x + c_1\) and the regression line of \(x\) on \(y\) is \(x = b_{xy}y + c_2\), then the square of the correlation coefficient is given by \(r^2 = b_{yx} \cdot b_{xy}\). The sign of \(r\) is the same as the sign of the slopes \(b_{yx}\) and \(b_{xy}\).
Identify Regression Line Slopes
We are given two lines:
- Line 1: \(x + y + 11 = 0\)
- Line 2: \(2x + 3y + 4 = 0\)
We need to determine which line corresponds to the regression of \(y\) on \(x\) (\(b_{yx}\)) and which corresponds to the regression of \(x\) on \(y\) (\(b_{xy}\)).
- Rewrite Line 1 as \(x\) in terms of \(y\): \(x = -y - 11\). The slope \(b_{xy}\) (coefficient of \(y\)) is -1.
- Rewrite Line 2 as \(y\) in terms of \(x\): \(3y = -2x - 4 \implies y = -\frac{2}{3}x - \frac{4}{3}\). The slope \(b_{yx}\) (coefficient of \(x\)) is \(-\frac{2}{3}\).
Let's check the alternative assignment:
- Line 1 as \(y\) on \(x\): \(y = -x - 11\), so \(b_{yx} = -1\).
- Line 2 as \(x\) on \(y\): \(2x = -3y - 4 \implies x = -\frac{3}{2}y - 2\), so \(b_{xy} = -\frac{3}{2}\).
Calculate \(r^2\) for this alternative assignment: \(r^2 = b_{yx} \cdot b_{xy} = (-1) \cdot (-\frac{3}{2}) = \frac{3}{2}\). Since \(r^2\) cannot be greater than 1, this assignment is incorrect.
Therefore, the correct slopes are \(b_{yx} = -\frac{2}{3}\) and \(b_{xy} = -1\).
Calculate Correlation Coefficient Squared (\(r^2\))
Using the correct slopes:
\(r^2 = b_{yx} \cdot b_{xy} = \left(-\frac{2}{3}\right) \times (-1) = \frac{2}{3}\)
Determine the Sign of Correlation Coefficient (\(r\))
The correlation coefficient \(r\) must have the same sign as the slopes \(b_{yx}\) and \(b_{xy}\). In this case, both \(b_{yx} = -\frac{2}{3}\) and \(b_{xy} = -1\) are negative.
Therefore, the correlation coefficient \(r\) must be negative.
\(r = -\sqrt{\frac{2}{3}}\)
The value of the correlation coefficient between \(x\) and \(y\) is \(-\sqrt{2/3}\).
Similar Questions
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 ?
The coefficient of correlation between ages of husband and wife at the time of marriage for a given set of 100 couples was noted to be 0.7. Assume that all these couples survive to celebrate the silver jubilee of their marriage. The coefficient of correlation at that point of time will be
Consider the following statements:
1. The coefficient of correlations r is \(\rm \frac{3}{4}\) .
2. The means of x and y are 3 and 4 respectively.
Which of the above statements is/are correct?
Consider the following statements:
1. The regression line of y on x is \(\rm y = \frac{3}{4}x+2\)
2. The regression line of x on y is \(\rm x = \frac{3}{4}y+\frac{1}{4}\)
Which of the above statements is/are correct?
If two variables X and Y are independent, then what is the correlation coefficient between them?
In which one of the following cases would you expect to get a negative correlation?
If the regression coefficient of Y on X is -6, and the correlation coefficient between X and Y is \( - \frac{1}{2},\) then the regression coefficient of X on Y would be
The set of bivariate observation (x 1, y 1) (x 2y 2) …., (x n, y n) are such that all the values are distinct and all the observations fall on a straight line with non-zero slope. Then the possible value of the correlation coefficient between x and y are
The correlation coefficient computed form a set of 30 observations is 0.8 then the percentage of variation not explained by linear regression is
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
If two regression coefficients are -0.8 and -0.2, then the value of coefficient of correlation is
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?
