Question

Find the least squares regression line for the points. Use the regression capabilities of a graphing utility to verify your results. (0,4), (1, 5), (3, 6), (6,9), (8, 10) ya Use the graphing utility to plot the points and graph the regression line. у y 12 12 10 10 8 6A 2 2 2 + 6 8 10 6 8 10

Answer 1

Given data

x	y	Ꮖ - Ꮖ	y Y	72- 2)	(n-1) (2-0)
0	4	-3.6	-2.8	12.96	10.08
1	5	-2.6	-1.8	6.76	4.68
3	6	-0.6	-0.8	0.36	0.48
6	9	2.4	2.2	5.76	5.28
8	10	4.4	3.2	19.36	14.08

Mean of x

* = Σ/Ν

$\bar{x}=18/5$

$\bar{x}=3.6$

Mean of y

$\bar{y}=\sum y/N$

$\bar{y}=34/5$

$\bar{y}=6.8$

$\sum (x-\bar{x})^{2}= 12.96+6.76+0.36+5.76+19.36=45.2$

$\sum (x-\bar{x})(y-\bar{y})= 10.08+4.68+0.48+5.28+14.08=34.6$

Using linear regression considering that line passes through $\bar{x}$ and  $\bar{y}$ , we can write

$\bar{y}= a+b\bar{x}$

Here a is y intercept of the line.

b is the slope of the line.

Slope of the line is given by

$b=\sum (x-\bar{x})(y-\bar{y})/(\sum (x-\bar{x})^{2})$

$b=34.6/45.2=0.7655$

$\therefore$ Slope of the given line calculated using regression method is  $b=0.7655$.

We can calculate a using equation

  $\bar{y}= a+b\bar{x}$

$6.8=a+(0.7655\times 3.6)$

$6.8=a+2.755$

$a=4.0442$

$\therefore$ y-intercept of the given line calculated using regression method is  $a=4.0442$.

$\therefore$ The equation for the given best fit line is   $y=a+bx$ $y=4.0442+0.7655x$

We can plot this best fit line using excel.

12 10 y = 0.7655x + 4.0442 00 4 2 0 0 1 2 3 4 5 6 7 8 X

This looks similar to the graph 3 in the given question.