Question

Which of the following two sets of regression lines are the true representatives for the information that is from the given bivariate population?
I. \[x + 4y = 15\] and \[3x + y = 12\] at \[\bar x = 3\], \[\bar y = 3\].
II. \[3x + 4y = 9\] and \[4x + y = 1\] at \[\bar x = - \dfrac{5}{{10}}\], \[\bar y = \dfrac{{30}}{{13}}\].
A. Only I
B. Both I and II
C. None of these
D. Only II

Answer 1

Hint: The given problem revolves around the concepts of the certain equation of satisfying the regression lines particularly. As a result, solving all the given equations by substituting the certain values, that is bivariate population (which means the two respective parameters). If the certain values equate/satisfies the actual equation and/or, it contradicts; then, the required conclusion is obtained.

Complete step-by-step solution:
Since,
Considering the certain equations in statement I that is,
The two regression lines in the first statement is,
\[ \Rightarrow x + 4y = 15\] … (i)
And,
\[ \Rightarrow 3x + y = 12\] … (ii)
Since,
We have given that the values for the bivariate population that is x and y respectively is \[\bar x = 3\] and \[\bar y = 3\] where the ‘bar’ sign above the certain variables represents the constraint boundary condition satisfying the regression lines.
As a result,
To check whether this regression lines truly satisfying the bivariate parameters,
Substituting the respective values of ‘\[x\]’ and ‘\[y\]’ that is \[\bar x = 3\] and \[\bar y = 3\] respectively, in both the equations (i) and (ii) simultaneously, we get
\[ \Rightarrow 3 + 4 \times 3 = 3 + 12\]
And,
\[ \Rightarrow 3 \times 3 + 3 = 9 + 3\]
Hence, solving the equations mathematically, we get
\[ \Rightarrow 3 + 4 \times 3 = 15\] … (iii)
And,
\[ \Rightarrow 3 \times 3 + 3 = 12\] … (iv)
Now,
From (i), (ii), (iii) and (iv), it seems that, after substituting the values equates the actual equations of the regression lines i.e.
\[x + 4y = 15\]
And,
\[3x + y = 12\].
\[ \Rightarrow \therefore \]The set of regression lines consisting of the equations \[x + 4y = 15\] and \[3x + y = 12\] are absolutely correct for the certain bivariate parameters that is \[\bar x = 3\] and \[\bar y = 3\] respectively.
Similarly,
Considering the certain equations in statement II that is,
The two regression lines in the second statement is,
\[ \Rightarrow 3x + 4y = 9\] … (v)
And,
\[ \Rightarrow 4x + y = 1\] … (vi)
Since,
We have given that the values for the bivariate population that is x and y respectively is \[\bar x = - \dfrac{5}{{10}}\] and \[\bar y = \dfrac{{30}}{{13}}\] where the ‘bar’ sign above the certain variables represents the constraint boundary condition satisfying the regression lines.
As a result,
To check whether this regression lines truly satisfying the bivariate parameters,
Substituting the respective values of ‘\[x\]’ and ‘\[y\]’ that is \[\bar x = - \dfrac{5}{{10}}\] and \[\bar y = \dfrac{{30}}{{13}}\] respectively, in both the equations (v) and (vi) simultaneously, we get
\[ \Rightarrow 3 \times \left( { - \dfrac{5}{{10}}} \right) + 4 \times \dfrac{{30}}{{13}} = - \dfrac{{15}}{{10}} + \dfrac{{120}}{{13}}\]
And,
\[ \Rightarrow 4 \times \left( { - \dfrac{5}{{10}}} \right) + \dfrac{{30}}{{13}} = - \dfrac{{20}}{{10}} + \dfrac{{30}}{{13}}\]
Hence, solving the equations mathematically, we get
\[ \Rightarrow 3 \times \left( { - \dfrac{5}{{10}}} \right) + 4 \times \dfrac{{30}}{{13}} = \dfrac{{ - 195 + 1200}}{{130}}\]
And,
\[ \Rightarrow 4 \times \left( { - \dfrac{5}{{10}}} \right) + \dfrac{{30}}{{13}} = \dfrac{{ - 260 + 300}}{{130}}\]
Hence, we get
\[ \Rightarrow 3 \times \left( { - \dfrac{5}{{10}}} \right) + 4 \times \dfrac{{30}}{{13}} = \dfrac{{1005}}{{130}} = \dfrac{{201}}{{26}}\] … (vii)
And,
\[ \Rightarrow 4 \times \left( { - \dfrac{5}{{10}}} \right) + \dfrac{{30}}{{13}} = \dfrac{{40}}{{130}} = \dfrac{4}{{13}}\] … (viii)
Now,
From (v), (vi), (vii) and (viii), it seems that
Substituting the values contradicts the given bivariate condition i.e.
\[3x + 4y \ne 9\]
And,
\[4x + y \ne 1\].
\[ \Rightarrow \therefore \]The set of regression lines consisting of the equations \[3x + 4y = 9\] and \[4x + y = 1\] are absolutely correct for the certain bivariate parameters that is \[\bar x = - \dfrac{5}{{10}}\] and \[\bar y = \dfrac{{30}}{{13}}\] respectively.
Hence, from the above solution, it seems that
Only statement I, satisfy the given information.
\[ \Rightarrow \therefore \]Option A is correct.

Note: In this case remember, \[\left| r \right| = \sqrt {{b_{xy}}{b_{yx}}} \] where, \[{b_{xy}}\], \[{b_{yx}}\] are the regression of the certain lines w.r.t. ‘\[y\]’ and ‘\[x\]’ respectively, which represents the formula to find the coefficient of correlation of two regression lines. One must be able to relate the certain equations which satisfy the geometric figure such as line, circle, etc. mathematically equates in terms of \[{x^2} + {y^2} = 1\], \[y = mx\], etc., so as to be sure of our final answer.