Problem Statement
For 10 pairs: $\sum x = 130, \sum y = 220, \sum x^2 = 2288, \sum y^2 = 5506, \sum xy = 3468$.
(i) Find regression equations of $y$ on $x$ and $x$ on $y$.
(ii) Estimate $y$ when $x = 16$.
(iii) Calculate correlation coefficient $r$.
Verified Solution & Marking Scheme
Means and Coefficients
$\bar{x} = 13, \bar{y} = 22$. $b_{yx} = \frac{10(3468) - (130)(220)}{10(2288) - 130^2} = \frac{6080}{5980} = 1.0167$.
$b_{xy} = \frac{6080}{10(5506) - 220^2} = \frac{6080}{6660} = 0.9129$.
Regression Lines
$y$ on $x$: $y - 22 = 1.0167(x - 13) \implies y = 1.0167x + 8.783$.
$x$ on $y$: $x - 13 = 0.9129(y - 22) \implies x = 0.9129y - 7.084$.
Estimate and r
When $x = 16$: $y = 1.0167(16) + 8.783 \approx 25.05$.
$r = \sqrt{b_{yx} \times b_{xy}} = \sqrt{1.0167 \times 0.9129} = 0.963$.