Question

The two regression lines were found to be$4X–5Y+33=0$ and $20X–9Y–107=0$. Find the coefficient of correlation between $X$ and $Y.$

• A. $0.6$
• B. $0.16$
• C. $0.3$
• D. $0.7$

Answer 1

The correct option is A

$0.6$

Explanation for correct option

Given,

$$\begin{gathered} 4X–5Y=-33\dots \left(1\right) \\ 20X–9Y=107\dots \left(2\right) \\ \left(1\right)\times 5 \\ \Rightarrow 20X–25Y=–165 \\ \Rightarrow 20X–9Y=107 \end{gathered}$$

Subtracting $\left(1\right)$ and $\left(2\right),$

$$\begin{gathered} \Rightarrow –16Y=–272 \\ \Rightarrow Y=272/16 \\ \Rightarrow Y=17 \end{gathered}$$
Using $Y=17$ in $\left(1\right)$ we get,

$$\begin{gathered} \Rightarrow 4X–85=-33 \\ \Rightarrow 4X=85–33 \\ \Rightarrow 4X=52 \\ \Rightarrow X=13 \end{gathered}$$
Let regression line of $Y$ on $X$ be $4X–5Y+33=0$

$$\begin{gathered} \Rightarrow 5Y=4X+33 \\ \Rightarrow Y=(4X+33) \\ \Rightarrow Y=(1/5)(4x+33) \\ \Rightarrow Y=(4/5)X+(33/5) \\ \Rightarrow Y=0.8X+6.6 \end{gathered}$$

$\therefore b_{YX}=0.8$

Let regression line of$X$ on $Y$ be

$$\begin{gathered} \Rightarrow 20X–9Y–107=0 \\ \Rightarrow 20X=9Y+107 \\ \Rightarrow X=(9/20)Y+(107/20) \\ \Rightarrow X=0.45Y+5.35 \\ \therefore b_{XY}=0.45 \end{gathered}$$

Coefficient of correlation between $X$ and $Y$ is $=\pm 0.6=0.6$

Both $b_{YX}$ and $b_{XY}$ is positive take positive sign.

Hence, the correct option is $\mathrm{Option}\left(\mathrm{A}\right)$