Question

Which of the following two sets of regression lines are the true representative of the information from the bivariate population?

I. $x+4y=15$ and $y+3x=12$, $\overline{x}=3,\overline{y}=3$.

II. $3x+4y=9$ and $4x+y=1$, $\overline{x}=-\frac{5}{10}$ and $\overline{y}=\frac{30}{13}$

• A. Both I and II
• B. II only
• C. I only
• D. None of these

Answer 1

The correct option is C

I only

Explanation of correct option:

For statement I. $x+4y=15$ and $y+3x=12$, $\overline{x}=3,\overline{y}=3$.

Substitute $\overline{x}=3,\overline{y}=3$ in L.H.S. of $x+4y=15$

L.H.S.

$$\begin{gathered} x+4y=3+4\times 3 \\ =3+12 \\ =15 \\ =R.H.S. \end{gathered}$$

Substitute $\overline{x}=3,\overline{y}=3$ in L.H.S. of $y+3x=12$

$$\begin{gathered} y+3x=3+3\times 3 \\ =3+9 \\ =12 \\ =R.H.S. \end{gathered}$$

For statement II. $3x+4y=9$ and $4x+y=1$, $\overline{x}=-\frac{5}{10}$ and $\overline{y}=\frac{30}{13}$

Substitute $\overline{x}=-\frac{5}{10}$ and $\overline{y}=\frac{30}{13}$ in L.H.S. of $3x+4y=9$

$$\begin{gathered} 3x+4y=3\times \left(-\frac{5}{10}\right)+4\times \frac{30}{13} \\ =-\frac{3}{2}+\frac{120}{13} \\ =\frac{-39+240}{26} \\ =\frac{201}{26} \\ \ne R.H.S. \end{gathered}$$

Substitute $\overline{x}=-\frac{5}{10}$ and $\overline{y}=\frac{30}{13}$ in L.H.S. of $4x+y=1$

$$\begin{gathered} 4x+y=4\times \left(-\frac{5}{10}\right)+1\times \frac{30}{13} \\ =-2+\frac{30}{13} \\ =\frac{-26+30}{13} \\ =\frac{4}{13} \\ \ne R.H.S. \end{gathered}$$

Therefore, statement II is incorrect statement.

Hence, the correct option is option (C).