The coefficient of correlations is r between X and Y having standard deviation σx and σy. The tangent of the between two lines of regression is:
The regression slopes are \(b_{yx} = r\dfrac{\sigma_y}{\sigma_x}\) and (rewriting X-on-Y in Y = ... form) \(m_2 = \dfrac{\sigma_y}{r\sigma_x}\). Using \(\tan\theta = \left|\dfrac{m_1 - m_2}{1 + m_1 m_2}\right|\):
Numerator: \(\dfrac{\sigma_y}{\sigma_x}\!\left(r - \dfrac{1}{r}\right) = \dfrac{\sigma_y(r^{2}-1)}{r\sigma_x}\)
Denominator: \(1 + \dfrac{\sigma_y^{2}}{\sigma_x^{2}} = \dfrac{\sigma_x^{2}+\sigma_y^{2}}{\sigma_x^{2}}\)
Dividing and simplifying:
\[\tan\theta = \frac{1-r^{2}}{r}\cdot\frac{\sigma_x \sigma_y}{\sigma_x^{2}+\sigma_y^{2}}\]
This matches option (4).
Similar Questions
The product of partial regression coefficient \(b_1{_2}{_.3}\) \(b_2{_3}{_.1}\) \(b_3{_1}{_.2}\) equals to:
If six hand-writings were ranked by two judges in a competition and the rankings are as follows:
Hand Writing 1 2 3 4 5 6 Judge1 6 5 4 3 2 1 Judge2 1 2 3 4 5 6
Then the value of Spearman's rank correlation coefficient isIf the difference between the rank of the 4 observations are 2.5, 0.5, -1.5, -1.5, then Spearman's rank correlation coefficient equals to:
For the data set
x: 1 2 3 4 y: 4 5 3 2 the regression coefficient b yx (y on x) equals to:
For the two variables X and Y, the following observations are tabulated.
X: 3 4 4 Y: 10 10 9 The Spearman's correlation coefficient is:
For the variables X and Y, we collect 4 observations with ∑x = 10, ∑y = 14, ∑x 2= 30, ∑y 2= 54, ∑xy = 31. The regression line y on x is
In Spearman rank correlation coefficient r s= 1 - \(\dfrac{6∑ d^2}{n(n^2-1)}\) , the maximum value of ∑d 2in case of untied rank is:
If the correlation coefficient between x and y is 0.5 and the regression coefficient of y on x is 0.5, then the regression coefficient of x on y is:
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:
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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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