Question

the value of y & their corresponding values of y are shown in the table below
x 0 1 2 3 4

y 2 3 5 4 6


a) find the least square regression line y=ax+b
b) estimate the value of y when x=10​

Answer 1

Given : The values ​​of x and their corresponding values ​​of y

x  0 1 2 3 4

y 2 3 5 4 6  

To Find : least square regression line y = ax + b

Estimate the value of y when x = 10

Solution:

y = a + bx

$y-\overline{y}=b(x-\overline{x})$

$b=\dfrac{\sum(x-\overline{x}).(y-\overline{y})}{\sum(x-\overline{x})^2}$

$\overline{x}=\dfrac{0+1+2+3+4}{5} =2$

$\overline{y}=\dfrac{ 2+3+4+5+6}{5} =4$

x y $x-\overline{x}$     $y-\overline{y}$ ($x-\overline{x}$) ($y-\overline{y}$) ($x-\overline{x}$) ²  

0 2 -2    -2          4             4

1 3 -1    -1          1              1

2 4 0     0          0              0

3 5 1          1           1              1

4 6 2   2           4              4

           10             10

b = 10/10

=> b = 1

Either use

$\overline{y}=a+b\overline{x}$   to find a

=> 4 = a  + 1* 2  => a  = 2

y = 2 + x

or use  $y-\overline{y}=b(x-\overline{x})$

=> y - 4 = 1(x - 2)

=> y - 4 = x - 2

=> y = 2 + x

y = 2 + x is  the least square regression line

a = 2  , b = 1

value of y when x = 10

=> y = 2 + 10 = 12

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