0
7.6kviews
Find the correlation coefficient between the height of father & height of son from the following data
Height of Father 65 66 67 68 69 71 73
Height of Son 67 68 64 68 72 69 70

Subject: Applied Mathematics 4

Topic: Correlation & Regression

Difficulty: Medium

1 Answer
0
347views

Assume mean for x is 68 and for y is 68

Change of scale is u = x - 68 and v = y - 68

x y u = x-68 v = y-68 uv u$^2$ v$^2$
65 67 -3 -1 3 9 1
66 68 -2 0 0 4 0
67 64 -1 -4 4 1 16
68 68 0 0 0 0 0
69 72 1 4 4 1 16
71 69 3 1 3 9 1
73 70 5 2 10 25 4
3 2 24 49 38

$ \bar{u} = \frac{1}{n}(\sum u) = \frac{1}{7}(3) = 0.42857 \\ \bar{v} = \frac{1}{n}(\sum v) = \frac{1}{7}(3) = 0.285714 \\ \sigma_u^2 = \frac{1}{n}(\sum u^2) - (\bar{u})^2 = \frac{1}{7}(49) - (0.42857)^2 = 6.81632775 \\ \sigma_v^2 = \frac{1}{n}(\sum u^2) - (\bar{v})^2 = \frac{1}{7}(38) - (0.285714)^2 = 5.346938939 \\ \sigma_u = 2.610809788 \hspace{0.5cm} \sigma_v = 2.3123449 $

$ Cov(u,v) = \frac{1}{n}(\sum uv) - (\bar{u})(\bar{v}) = \frac{1}{7}(24) - (0.42857)(0.285714) = 3.30612298 \\ r_{uv} = \frac{Cov(u,v)}{\sigma_u \sigma_v} = \frac{3.30612298}{(2.610809788)(2.3123449)} = 0.5476349536 \sim 0.55 \\ r_{uv} = r_{xy} = r = 0.55 $

Please log in to add an answer.