Linear Regression Line

A regression line is the straight line which gives the best fit by using the least square cocept to the given sets of data.

To get the line of regression we need to chose the sum of the squares of derivations parallel to the axis of y and minimize the same. It is called that the line of regression of y on x and it gives the best estimate of y for any given value of x.

The equation of y on x is given byy = a + bx.

To get this equation, we use the following normal equation.

sum y = na + bsumx

sum xy = asum x + bsum x 2

By solving them, we get ‘a’ and ‘b’, hence we get the regression line y on x as y = a + bx. Same principle will be applied to get the line x on y.

Now let us see few problems on linear regression line.

Example Problem on Linear Regression Line.

Ex 1: Find the regression line y on x for the following data

Table_1

Soln:

Regression Table1

Here n = 7.

The normal equations are:

sum y = an + bsum x = 56 => 7a + 56b ……….. (1)

sum xy = asum x + bsum x 2 => 469 = 56 a + 476 b ………. (2)

Therefore 8 xx (1) => 448 = 56a + 448 b

(2)=> 469 = 56a + 476 b

Simplifying them, we get: 21 = 28 b => b = 21 / 28  = 3 / 4 .

Therefore (1) => 56 = 7a + 56 (3 / 4)

=> 7a = 56 – 42 = 14 => a = 2

Therefore the equation y on x is y = 2 + 3 / 4 x.

I am planning to write more post on Finding Limits at Infinity . Keep checking my blog.

Example Problem on Linear Regression Line.

Ex 2: Find the regression line x on y for the following data

Table_2

soln:

Regression Table2

Here n = 9.

The normal equations are:

Σx = an + bΣy = 45 = 9a + 96b ……….. (1)

Σxy = aΣy + bΣy 2 = 536 = 96 a + 1224 b ………. (2)

Therefore 96 xx (1) = 4320 = 864a + 9216 b

9 xx (2) = 4824 = 864a + 11016b

Therefore on simplification, we get: 504 = 1800 b => b = 7 / 25 .

Therefore (1) => 45 = 9a + 96 (7 / 25)

=> 9a = 45 – 26.88 = 18.12 => a = 2.01

Therefore the equation x on y is x = 2.01 + 7 / 25 y.