Question

If \[y={{m}_{1}}x+{{c}_{1}}\] is the regression line of \[y\] on \[x\] and \[y={{m}_{2}}x+{{c}_{2}}\] is the regression line of \[x\] on \[y\], then which is true?
1. \[{{m}_{1}}{{m}_{2}}<1\]
2. \[0\le \sqrt{{{m}_{1}}{{m}_{2}}}\le 1\]
3. \[-1\le \dfrac{\sqrt{{{m}_{1}}}}{\sqrt{{{m}_{2}}}}\]
4. \[-1\le \sqrt{\dfrac{{{m}_{2}}}{{{m}_{1}}}}\]

Answer 1

Hint: To solve this question you should know all the concepts related to the regression lines and the correlation coefficient. To solve this, firstly get the equation of both the regression lines i.e. \[y\] on \[x\] and \[x\] on \[y\]. After that, find out the slopes of both the lines, then apply the condition of the correlation coefficient.

Complete step-by-step solution:
The equation of the regression line of \[y\] on \[x\] is
\[y=a+bx\]
Here \[y\]is the dependent variable and \[x\] is the independent variable whereas \[b\] is the slope of the line and \[a\] is the \[y-\] intercept that is the point at which the pint cuts the \[y-\] axis.
Similarly, The equation of the regression line of \[x\] on \[y\] is
\[x=c+dy\]
Here \[x\]is the dependent variable and \[y\] is the independent variable whereas \[d\] is the slope of the line and \[c\] is the \[x-\] intercept that is the point at which the pint cuts the \[x-\] axis.
And it is given that the equation of regression line of \[y\] on \[x\] is
\[y={{m}_{1}}x+{{c}_{1}}\]
From this we can say that the slope of the line is
\[{{b}_{yx}}={{m}_{1}}\]\[........(1)\]
Similarly, it is given that the equation of regression line of \[x\] on \[y\] is
\[y={{m}_{2}}x+{{c}_{2}}\]
But we have to find the equation in terms of \[x\] . So we can write it as
\[{{m}_{2}}x=y-{{c}_{2}}\]
Now divide by \[{{m}_{2}}\] on both sides of the equation, we get
\[x=\dfrac{1}{{{m}_{2}}}y-\dfrac{{{c}_{2}}}{{{m}_{2}}}\]
From this equation we can clearly say that the slope of this line is \[\dfrac{1}{{{m}_{2}}}\]. So it is represented as
\[{{b}_{xy}}=\dfrac{1}{{{m}_{2}}}\]
The correlation coefficient that is \[r\] in terms of slope of the two lines is
\[r=\sqrt{{{b}_{yx}}\times {{b}_{xy}}}\]
Substituting the values of slope of the two lines, we will get
\[r=\sqrt{\dfrac{{{m}_{1}}}{{{m}_{2}}}}\]
And as per the logic of the correlation coefficient we know that the range of the coefficient is from \[-1 \] to \[+1\].
So by this concept we can say that,
\[-1\le \sqrt{\dfrac{{{m}_{1}}}{{{m}_{2}}}}\le +1\]
As per the given options we can observe that option \[(3)\] is the only one which satisfies the condition.
Hence we can conclude that option \[(3)\] is correct.

Note: If the value of correlation coefficient is \[0\] then it shows that there is no linear relationship, and if the value of correlation coefficient is \[+1\] then it shows that there is a perfect relationship that means if one variable increases the other variable will also increase whereas if the value of correlation coefficient is \[-1\] then it shows the negative linear relationship i.e. one variable increases other will decrease.