Question

For the following data, find the regression line of Y on X

X	1	2	3
Y	2	1	6

Hence find the most likely value of y when x = 4.

Answer 1

X = xi	Y = yi	${\displaystyle {\text{x}}_{\text{i}}^2}$	xi yi
1	2	1	2
2	1	4	2
3	6	9	18
6	9	14	22

From the table, we have

n = 3, ∑ x_i = 6, ∑ y_i = 9, ${\displaystyle \sum {\text{x}}_{\text{i}}^2=14}$, ∑ x_i y_i = 22

${\displaystyle \overline{x}=\frac{\sum x_i}{\text{n}}=\frac{6}{3}=2}$

${\displaystyle \overline{y}=\frac{\sum y_i}{\text{n}}=\frac{9}{3}=3}$

Now, ${\displaystyle {\text{b}}_{\text{YX}}=\frac{\sum {\text{x}}_{\text{i}}{\text{y}}_{\text{i}}-\text{n}\overline{\text{x}}\overline{\text{y}}}{\sum {\text{x}}_{\text{i}}^2-\text{n}{\overline{\text{x}}}^2}}$

${\displaystyle =\frac{22-3\times 2\times 3}{14-3{\left(2\right)}^2}}$

${\displaystyle =\frac{22-18}{14-12}}$

${\displaystyle =\frac{4}{2}}$

= 2

The regression equation of Y on X is,

${\displaystyle (y-\overline{y})={\text{b}}_{\text{YX}}(x-\overline{x})}$

y − 3 = 2(x − 2)

y − 3 = 2x − 4

y = 2x − 4 + 3

y = 2x − 1 is the regression equation of Y on X.   ...(1)

When x = 4, y = ?

Substituting x = 4 in equation (1)

y = 2(4) − 1

= 8 − 1

y = 7