Question

Assertion – The angle between two regression lines is \[{{90}^{\circ }}\] then \[r=0\], where r is correlation coefficient.
Reason – The angle between the regression line is given by \[\tan \theta =\dfrac{1-{{r}^{2}}}{r}\dfrac{{{\sigma }_{x}}{{\sigma }_{y}}}{{{\left( {{\sigma }_{x}} \right)}^{2}}+{{\left( {{\sigma }_{x}} \right)}^{2}}}\]
(A) Both Assertion and Reason are correct and Reason is the correct explanation for Assertion.
(B) Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion.
(C) Assertion is correct but Reason is incorrect.
(D) Both Assertion and Reason are incorrect.

Answer 1

Hint: We are given an assertion in the above question and following the assertion we are given the reason for the assertion. We are asked to find whether the assertion and the following reason are correct or not or whether they comply with each other. We know that a regression line is a straight line that is used to predict the value of y as the value of x changes. So regression lines have a right angle between them when there is no coefficient of correlation. The angle can be found using the expression of tangent function. Hence, we will have the correct option.

Complete step by step answer:
According to the given question, we are given an assertion in the above question and following the assertion we are given the reason for the assertion. We are asked in the question to check whether the assertion and the reason are correct or not or whether they comply with each other.
A regression line is generally a straight line which describes how the y variable will respond when the variable x is changed. Since it is straight, it can be represented using the equation \[y=mx+c\], where m is the slope. A regression line is used to predict how y will respond with respect to x.
Two regression lines refer to the two variables, say, x and y. If one line represents regression of x upon y then the other shows the regression of y upon x.
So, the two regression lines will have an angle of \[{{90}^{\circ }}\], when there is zero coefficient of correlation.
And if the angle in general can be found using the expression of tangent function, which is, \[\tan \theta =\dfrac{1-{{r}^{2}}}{r}\dfrac{{{\sigma }_{x}}{{\sigma }_{y}}}{{{\left( {{\sigma }_{x}} \right)}^{2}}+{{\left( {{\sigma }_{x}} \right)}^{2}}}\]

So, the correct answer is “Option A”.

Note: Regression lines should be known well. Its predictability is what increases the scope to be used in various fields. Also, the correct formula of the angle between the regression lines must be known. While answering a question as above, the question must be read clearly and correctly in order to avoid any terms or words getting overlooked which in turn may increase the chances of wrong conclusion.