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Answer the following on regression and correlation.
The correlation coefficient between x and y is 0.6. If the variance of x is 225, the variance of y…
The correlation coefficient between $x$ and $y$ is $0.6$. If the variance of $x$ is $225$, the variance of $y$ is $400$, mean of $x$ is $10$ and mean of $y$ is $20$, find:
(i)[2]
the equations of two regression lines.
(ii)[2]
the expected value of $y$ when $x = 2$.
Answer
Answer (i)
Mean of x: 10
Mean of y: 20
Regression coefficient of y on x: 0.8
Regression coefficient of x on y: 0.45
Correlation coefficient r: 0.6
Line of y on x: y = 0.8x + 12
Line of x on y: x = 0.45y + 1
$\sigma_x = \sqrt{225} = 15$, $\sigma_y = \sqrt{400} = 20$, $r = 0.6$.
$b_{yx} = r\frac{\sigma_y}{\sigma_x} = 0.6 \times \frac{20}{15} = 0.8$ and $b_{xy} = r\frac{\sigma_x}{\sigma_y} = 0.6 \times \frac{15}{20} = 0.45$.
Regression line of $y$ on $x$: $y - 20 = 0.8(x - 10)$, so $y = 0.8x + 12$.
Regression line of $x$ on $y$: $x - 10 = 0.45(y - 20)$, so $x = 0.45y + 1$.
Answer (ii)
Using the line of $y$ on $x$, $y = 0.8x + 12$, at $x = 2$: $y = 0.8(2) + 12 = 13.6$.
Final answer: $13.6$