Coefficient of correlation - class-XI

Given:-

Perfect correlation occurs when the correlation coefficient r is exactly 1 (perfect positive) or -1 (perfect negative).

$\Rightarrow$  Correlation coefficients are expressed as values between $-1$ and $+1$. 

In statistics, a correlation coefficient r between 0.5 and 0.699 is generally categorized as a moderate degree of correlation.

A correlation coefficient r between 0.7 and 0.99 is typically categorized as a high degree of correlation.

The formula provided in option C is a standard representation of the correlation coefficient using deviations from the mean or assumed mean (d_x and d_y).

The value of correlation coefficient lies between $-1$ to $+1$.

Correlation coefficient is the measure of the degree of linear relationship between two variables, usually labelled X and Y. While in regression the emphasis is on predicting one variable from the other, in correlation the emphasis is on the degree to which a linear model may describe the relationship between two variables. In regression the interest is directional, one variable is predicted and the other is the predictor; in correlation the interest is non-directional, the relationship is the critical aspect.

$-1\le r\le 1$

Hence statement is false

Two independent variables have coefficient of correlation $0$, So they are always uncorrelated.

Solution:

We know that, for two variables $x$ and $y$

Let $7x-3y-18=0$ represents the regression line of $y$ on $x$
$\Rightarrow y=-6+\dfrac { 7 }{ 3 } x\ \Rightarrow { b } _{ yx }=\dfrac { 7 }{ 3 } $
Then $4x-y-11=0$ is the regression line of $x$ on $y$.
$\Rightarrow x=\frac { 11 }{ 4 } +\dfrac { y }{ 4 } \ \Rightarrow { b } _{ xy }=\dfrac { 1 }{ 4 } $
Both ${ b } _{ yx }$ and ${ b } _{ yx }$ are positive. 
$\Rightarrow r=\sqrt { { b } _{ yx }\times { b } _{ xy } } \ \Rightarrow r=\sqrt { \dfrac { 7 }{ 3 } \times \dfrac { 1 }{ 4 }  } =0.7638$
So, option C is correct.