Question

When $r=-1$, the lines of regression will be _______.
(A) Negative slope
(B) Positive slope
(C) Running downwards from left to right
(D) Both (A) and (C)

Answer 1

Hint: We solve this question by first considering the formula for regression lines of y on x and x on y, $\left( y-\overline{y} \right)=r\dfrac{{{S}_{y}}}{{{S}_{x}}}\left( x-\overline{x} \right)$ and $\left( x-\overline{x} \right)=r\dfrac{{{S}_{x}}}{{{S}_{y}}}\left( y-\overline{y} \right)$ respectively. Then we substitute the value $r=-1$ in them and find their slopes by comparing them with the general form of line, $y=mx+c$. Then we consider the property that standard deviation is always greater than or equal to zero and find whether the slope is positive or negative.

Complete step by step answer:
First let us consider the regression lines.
There are two types of regression lines
1) Regression line of y on x
2) Regression line of x on y
The regression line of y on x is given as
$\Rightarrow \left( y-\overline{y} \right)=r\dfrac{{{S}_{y}}}{{{S}_{x}}}\left( x-\overline{x} \right)$
Similarly, we can give the regression line of x on y as,
$\Rightarrow \left( x-\overline{x} \right)=r\dfrac{{{S}_{x}}}{{{S}_{y}}}\left( y-\overline{y} \right)$
Where ${{S}_{y}}=$ Standard deviation of y
${{S}_{x}}=$ Standard deviation of x
$r=$ Correlation co-efficient
We are given that $r=-1$. Then the lines will become,
The regression line of y on x becomes
$\begin{align}
  & \Rightarrow \left( y-\overline{y} \right)=\left( -1 \right)\dfrac{{{S}_{y}}}{{{S}_{x}}}\left( x-\overline{x} \right) \\
 & \Rightarrow \left( y-\overline{y} \right)=-\dfrac{{{S}_{y}}}{{{S}_{x}}}\left( x-\overline{x} \right) \\
\end{align}$
Now let us consider the general form of equation of a line, $y=mx+c$. Comparing the regression line with this formula we get the slope of the regression line as,
$\Rightarrow Slope=-\dfrac{{{S}_{y}}}{{{S}_{x}}}..........\left( 1 \right)$
Similarly, the regression line of x on y becomes,
$\begin{align}
  & \Rightarrow \left( x-\overline{x} \right)=\left( -1 \right)\dfrac{{{S}_{x}}}{{{S}_{y}}}\left( y-\overline{y} \right) \\
 & \Rightarrow \left( x-\overline{x} \right)=-\dfrac{{{S}_{x}}}{{{S}_{y}}}\left( y-\overline{y} \right) \\
 & \Rightarrow \left( y-\overline{y} \right)=-\dfrac{{{S}_{y}}}{{{S}_{x}}}\left( x-\overline{x} \right) \\
\end{align}$
Now let us consider the general form of equation of a line, $y=mx+c$. Comparing the regression line with this formula we get the slope of the regression line as,
$\Rightarrow Slope=-\dfrac{{{S}_{y}}}{{{S}_{x}}}...........\left( 2 \right)$
From equation (1) and (2) we get the slopes of regression lines are same when $r=-1$, that is,
$\Rightarrow Slope=-\dfrac{{{S}_{y}}}{{{S}_{x}}}$
Now let us consider the property of standard deviation. Standard deviation is always greater than or equal to zero.
So, we can say that,
\[\begin{align}
  & \Rightarrow \dfrac{{{S}_{y}}}{{{S}_{x}}}>0 \\
 & \Rightarrow -\dfrac{{{S}_{y}}}{{{S}_{x}}}<0 \\
\end{align}\]
So, we get that,
$\therefore Slope=-\dfrac{{{S}_{y}}}{{{S}_{x}}}<0$
So, we get that when $r=-1$, the regression lines will have negative slope.
As the slope is negative the line runs downward from left to right.

Hence the answer is Option D.

Note:
The common mistake one makes while solving this question is one might solve the question till finding the slope and mark the answer as Option A without checking the possibility of another correct option here, that is the regression line runs downward from left to right.