Question

ASSERTION: The correlation between two variables “Intensity of cold” and “Sale of woollen clothes” is positive.
REASON: The angle between two regression lines is $\pi $.
${\text{(A)}}$ Both Assertion and Reason are correct and Reason is the correct explanation for Assertion.
${\text{(B)}}$ Both Assertion and Reason are correct but reason is not the correct explanation for assertion.
${\text{(C)}}$ Assertion is correct but Reason is incorrect.
${\text{(D)}}$ Both Assertion and Reason are incorrect.

Answer 1

Hint: The statistical association of linear relationship between two variables is known as Correlation. If the relationship between two variables moves in the same direction, it is said to be a positive correlation. (i.e) when a variable increases, the other variable increases too and vice versa.

Complete step-by-step solution:
A positive correlation graph can be plotted as follows.

Here, when the “Intensity of cold” increases, the “Sale of woollen clothes” too increases.
Let the variable “Intensity of cold” be X & the other variable “Sale of woollen clothes” be Y. So, we will get a linear graph of which the correlation is positive.
Therefore, the Assertion is true.
REASON:
It is given, “The angle between two regression lines is $\pi $”.
 For a positive correlation, the angle between two regression lines is always an acute angle. Since the angle lies between ${\text{0 to 9}}{{\text{0}}^ \circ }$ (i.e) ${\text{0 to }}\dfrac{{{\pi }}}{{\text{2}}}$, there is no way to be an obtuse angle. So, the angle $\pi $ is impossible.
Therefore, the Reason is incorrect.

Now, the Final answer is ${\text{(C)}}$ Assertion is true but Reason is incorrect.

Note: The solution can be found through logical reasoning itself, because it is a universal truth. When winter arrives, people prefer woollen clothes. So, there will be an increase in sales of woollen clothes. Obviously, it is a positive correlation.
And the angle of line regression is found by plotting the graph and using the formula $\tan \theta = \dfrac{{1 - {r^2}}}{r} \times \dfrac{{{\sigma _x}{\sigma _y}}}{{\sigma _x^2 + \sigma _y^2}}$. But, below is a simple strategy.
If the correlation between two variables is positive, then the angle between the two regression line is an acute angle (i.e.) ${\text{0 to }}\dfrac{{{\pi }}}{{\text{2}}}$
If the correlation between two variables is negative, then the angle between the two regression line is an obtuse angle (i.e.) \[\dfrac{{{\pi }}}{{\text{2}}}{{ {\text{to}} \pi }}\].