**For the following bivariate data obtain the equations of two regression lines:**

X |
1 | 2 | 3 | 4 | 5 |

Y |
5 | 7 | 9 | 11 | 13 |

#### Solution

X = x_{i} |
Y = y_{i} |
`"x"_"i"^2` |
`"y"_"i"^2` |
x_{i} y_{i} |

1 | 5 | 1 | 25 | 5 |

2 | 7 | 4 | 49 | 14 |

3 | 9 | 9 | 81 | 27 |

4 | 11 | 16 | 121 | 44 |

5 | 13 | 25 | 169 | 65 |

15 |
45 |
55 |
445 |
155 |

From the table, we have

n = 6, ∑ x_{i} = 15, ∑ y_{i} = 45, `sum "x"_"i"^2 = 55`, `sum "y"_"i"^2 = 445`, ∑ x_{i} y_{i} = 155

`bar x = (sum x_i)/"n" = 15/5 = 3`

`bar y = (sum y_i)/"n" = 45/5 = 9`

Now, for regression equation of Y on X,

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

`= (155 - 5 xx 3 xx 9)/(55 - 5(3)^2) = (155 - 135)/(55 - 45) = 20/10 = 2`

Also, `"a" = bar y - "b"_"XY" bar x` = 9 - 2(3) = 9 - 6 = 3

The regression analysis of Y on X is

Y = a + b_{YX} X

∴ Y = 3 + 2X

Now, for regression equation of X on Y,

`"b"_"XY" = (sum"x"_"i" "y"_"i" - "n" bar "x" bar "y")/(sum "y"_"i"^2 - "n" bar"y"^2)`

`= (155 - 5xx3xx9)/(445 - 5(9)^2) = (155 - 135)/(445 - 405) = 20/40 = 0.5`

Also, `"a"' = bar x - "b"_"XY" bar y`

= 3 - (0.5)(9) = 3 - 4.5 = - 1.5

The regression equation of X on Y is

X = a' + b_{XY} Y

∴ X = - 1.5 + 0.5Y