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 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?
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 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 ?
If r is the coefficient of correlation between x and y, then what is the correlation coefficient between (3x + 4) and (-3y + 3)?
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?
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?
Let X and Y represent prices (in Rs) of a commodity in Kolkata and Mumbai respectively. It is given X̅ = 65, Y̅ = 67, σ X = 2.5, σ Y = 3.5 and r(X, Y) = 0.8. What is the equation of regression of Y on X ?
For the variables x and y, the two regression lines are 6x + y = 30 and 3x + 2y = 25. What are the values of x̅, y̅ and r respectively?
If two variables X and Y are independent, then what is the correlation coefficient between them?
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?
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?
If X 1, X 2, … X nis a simple random sample without replacement of size n from a finite population of N units with mean μ and variance σ 2, the covariance of (X i, Xj ) will be:
If x = X- \(\bar{X}\) and y = Y - \(\bar{Y}\) and the number of pairs (X, Y) is n, then the Karl Pearson's coefficient of correlation is:
