Question

Shown below are ages (𝑥) and the systolic blood pressures (𝑦) of 9 patients in a certain hospital. Find the regression equation.

Age (𝑥) 26 40 35 50 45 55 28 30 52

Blood pressure (𝑦) 110 140 120 145 130 150 150 125 142

Answer 1

Linear regression: $y=\theta_0+\theta_1x$, where $\theta_1 = \frac{\overline{xy}-\overline{x}\cdot\overline{y}}{\overline{x^2}-(\overline{x})^2}$, $\theta_0=\overline{y}-\theta_1\overline{x}$.

$x=[26, 40, 35, 50, 45, 55, 28, 30, 52]$

$y=[110, 140, 120, 145, 130, 150, 150, 125, 142]$

$x^2 = [676, 1600, 1225, 2500, 2025, 3025, 784, 900, 2704]$

$xy=[2860, 5600, 4200, 7250, 5850, 8250, 4200, 3750, 7384]$

$\overline{x}= \frac{1}{9}(26+40+ 35+ 50+ 45+ 55+ 28+ 30+ 52)= \frac{361}{9}$

$\overline{y}= \frac{1}{9}(110+ 140+ 120+ 145+ 130+ 150+ 150+ 125+ 142) = \frac{403}{3}$

$\overline{x^2}= \frac{1}{9}(676+ 1600+ 1225+ 2500+ 2025+ 3025+ 784+ 900+ 2704) = \frac{15439}{9}$

$\overline{xy}= \frac{1}{9}(2860+ 5600+ 4200+ 7250+ 5850+ 8250+ 4200+ 3750+ 7384) = \frac{16448}{3}$

$(\overline{x})^2= \frac{130321}{81}$

$\overline{x^2}-(\overline{x})^2 = \frac{15439}{9}-\frac{130321}{81}=\frac{8630}{81}$

$\overline{xy}-\overline{x}\cdot\overline{y}=\frac{16448}{3}-\frac{361}{3}\cdot\frac{403}{3}=-\frac{96139}{9}$

$\theta_1=\frac{8630}{81}:(-\frac{96139}{9})=-\frac{69040}{7787259}$

$\theta_0=\frac{403}{3}+\frac{69040}{7787259}\frac{361}{9}=\frac{9439719571}{70085331}$

$y=\frac{9439719571}{70085331} -\frac{69040}{7787259}x$

$y=40.111-0.009x$