Question

Compute the least square regression line for a sample of 6 pairs of observations, given that

x_i = 5, 4, 6, 3, 8, 7,    Ʃy = 150,  Ʃxy = 404,  (y-ybar)2=37, (y-yhat)2=15

Also calculate the Co-efficient of determination and interpret the result.

Answer 1

We obtain the parameters of the least square equation by solving the simultaneous equations

$\sum y_i= m\sum x_i + c$

$\sum x_i y_i= m\sum x_i ^2 + c \sum x_i$

$\sum x_i = 5+4+6+3+8+7=33$

$\sum x_i ^2= 5^2+4^2+6^2+3^2+8^2+7^2=199$

Therefore, we have

$150=33m + c$ ------- equation i

$404=199m + 33c$ -------- equation ii

From equation i

$c = 150-33m$

Substitute c in equation ii

$404=199m-33(150-33m)$

$404= 199m + 1089m - 4950$

$1288m = 5354$

$m=\frac{5353}{1288}=4.1568$

Therefore

$c=\frac{404-199(4.1568)}{33}=-12.8245$

The least square equation is given as

$y=4.1568x-12.8245$

Coefficient of determination

$R^2 = 1- \frac{RSS}{TSS}=1-\frac{15}{37}= 0.5944$

This means that the random variable x explains 59.44% of the variation in variable y. Other factors not considered in the least square regression are responsible for the remaining 40.56% of the variations in y