Answer to Question #160382 in Statistics and Probability for Abu Ubayda

Question

Find the regression line of y on x for the following data:

X 40 70 50 60 80 50 90 40 60 60

Y 2.5 6.0 4.5 5.0 4.5 2.0 5.5 3.0 4.5 3.0

Draw the line. Estimate the value of y when x = 10.

Answer 1

$$\bar{x}=\dfrac{600}{10}=60, \bar{y}=\dfrac{40.5}{10}=4.05$$

$$SS_{xx}=\sum x_i^2-n\bar{x}^2=2400$$

$$SS_{yy}=\sum y_i^2-n\bar{y}^2=16.225$$

$$SS_{xy}=\sum x_iy_i-n\bar{x}\bar{y}=140$$

$$b=\dfrac{SS_{xy}}{SS_{xx}}=\dfrac{140}{2400}=\dfrac{7}{120}=0.0583$$

$$a=\bar{y}-b\bar{x}=4.05-\dfrac{7}{120}(60)=0.55$$

$$Y=a+bX$$

$$Y=0.55+0.0583X$$

$$r=\dfrac{SS_{xy}}{\sqrt{SS_{xx}}\sqrt{S_{yy}}}=\dfrac{140}{\sqrt{2400}\sqrt{16.225}}=0.7095$$

$$r^2=0.5033$$

Moderate positive correlation.

$X=10$

$$Y=0.55+0.0583(10)=1.133$$