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 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 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
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 a random variable (\(x\)) follows binomial distribution with mean 5 and variance 4, and \(5^{23}P(X = 3) = \lambda 4^{\lambda}\), then what is the value of \(\lambda\) ?
Consider the following statements:
1) If the correlation coefficient r xy = 0, then the two lines of regression are parallel to each other.
2) If the correlation coefficient r xy = + 1, then the two lines of regression are perpendicular to each other.
Which of the above statements is/are correct?
If 4x – 5y + 33 = 0 and 20x – 9y = 107 are two lines of regression, then what are the values of x̅ and y̅ respectively?
The regression coefficients of a bivariate distribution are -0.64 and -0.36. Then the correlation coefficient of the distribution is
- Let \(x - 3y+4 = 0\) and \(2x-7y+8 = 0\) be two lines of regression computed from some bivariate data. If \(b_{yx}\) and \(b_{xy}\) are regression coefficients of lines of regression of \(y\) on \(x\) and \(x\) on \(y\) respectively, then what is the value of \(b_{xy} + 7b_{yx}\)?
Which one of the following can be considered as appropriate pair of values of regression coefficient of y on x and regression coefficient of x on y?
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 ?
Important Questions from Correlation and Regression
- For testing the significance of overall regression the test to be used is
- Including relevant lagged values of the dependent variable on the right hand side of a regression equation could lead to which one of the following ?
- If OLS is used in the presence of auto-correlation, which of the following will be likely consequences ?
i. Coefficients estimates may be misleading.
ii. Hypothesis tests could reach the wrong conclusions.
iii. Forecasts made from the model could be biased.
iv. Standard errors may be inappropriate. - Regression analysis is of particular interest
- Which of the following can be linked with those providing method for estimating black money in India ?
I. Kaldor’s approach
II. NIPFP method
III. O.P. Chopra’s method
IV. Arjun Sengupta
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