Question

The following data are expected to follow a linear relation of the form y = ax+b. Obtain the best linear relation in accordance with a least-squares analysis. Calculate the standard deviation of the data from the predicted straight-line relation :

x| 0.9| 2.3| 3.3| 4.5| 5.7 |6.7

y| 1.1 |1.6| 2.6| 3.2| 4.0| 5.0

Answer 1

Following table shows the calcualtion for regression line:

	X	Y	X^2	Y^2	XY
	0.9	1.1	0.81	1.21	0.99
	2.3	1.6	5.29	2.56	3.68
	3.3	2.6	10.89	6.76	8.58
	4.5	3.2	20.25	10.24	14.4
	5.7	4	32.49	16	22.8
	6.7	5	44.89	25	33.5
Total	23.4	17.5	114.62	61.77	83.95

Number of data points : n= 6

Now

$S_{yy}=\sum y^{2}-\frac{\left (\sum y \right )^{2}}{n}=10.7283$

$S_{xx}=\sum x^{2}-\frac{\left (\sum x \right )^{2}}{n}=23.36$

$S_{xy}=\sum xy-\frac{\left (\sum x \right )\left (\sum y \right )}{n}=15.7$

Slope of the regression equation is

$a=b_{1}=\frac{S_{xy}}{S_{xx}}=0.6721$

and intercept of the equation will be

$b=b_{0}=\frac{1}{n}(\sum y - b_{1} \sum x)=0.2955$

So least sqaure line between the data is

y = 0.6721 x + 0.2955

--------------

Let us find SSE first :

$SSE=\sum y^{2}-b_{0}\sum y-b_{1}\sum xy=0.1765$

So standard error of estimate will be

$SE=\sqrt{\frac{SSE}{n-2}}=0.2101$