For the following data, find the regression line of Y on X Hence find the most likely value of y when x = 4.

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 | $"x"_"i"^2$ | 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, æ xi = 6, æ yi = 9, $sum "x"_"i"^2 = 14$, æ xi yi = 22

$bar x = (sum x_i)/"n" = 6/3 = 2$

$bar y = (sum y_i)/"n" = 9/3 = 3$

Now, $"b"_"YX" = (sum"x"_"i" "y"_"i" - "n" bar "x" bar "y")/(sum "x"_"i"^2 - "n" bar"x"^2)$

$= (22 - 3xx2xx3)/(14 - 3(2)^2) = (22 - 18)/(14 - 12) = 4/2 = 2$

Also,$"a" = bar y - "b"_"YX" bar x$ = 3 - 2(2) = - 1

The regression equation of Y on X is,

Y = a + bYX X

Y = - 1 + 2X

For X = 4,

Y = - 1 + 2(4) = - 1 + 8 = 7

The most likely value of Y for X = 4 is 7.