written 6.1 years ago by | • modified 2.2 years ago |
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
written 6.1 years ago by | • modified 2.2 years ago |
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
written 5.8 years ago by |
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 $