The data obtained on X, the length of time in weeks that a promotional project has been in progress at a small business, and Y, the percentage increase in weekly sales over the period just prior to the beginning of the campaign.

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

Y |
10 | 10 | 18 | 20 | 11 | 15 | 12 | 15 | 17 | 19 | 13 | 16 |

Find the equation of the regression line to predict the percentage increase in sales if the campaign has been in progress for 1.5 weeks.

#### Solution

Here, X = Length of time in weeks,

Y = Percentage increase in weekly sales

X = x_{i} |
Y = y_{i} |
`x_i^2` |
x_{i} y_{i} |

1 | 10 | 1 | 10 |

2 | 10 | 4 | 20 |

3 | 18 | 9 | 54 |

4 | 20 | 16 | 80 |

1 | 11 | 1 | 11 |

3 | 15 | 9 | 45 |

1 | 12 | 1 | 12 |

2 | 15 | 4 | 30 |

3 | 17 | 9 | 51 |

4 | 19 | 16 | 76 |

2 | 13 | 4 | 26 |

4 | 16 | 16 | 64 |

30 | 176 | 90 | 479 |

From the table, we have

n = 12, ∑ xi = 30, ∑ yi = 176, `sum x_i^2 = 90`, ∑ x_{i} y_{i} = 479

∴ `bar x = (sum x_i)/"n" = 30/12 = 2.5`

`bar y = (sum y_i)/"n" = 176/12 = 14.67`

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

`= (479 - 12 xx 2.5 xx 14.67)/(90 - 12 xx (2.5)^2)`

`= (479 - 440.1)/(90 - 75) = 38.9/15 = 2.59`

Also,

`"a" = bar y - "b"_"YX" bar x`

= 14.67 - 2.59 × 2.5 = 14.67 - 6.475 = 8.195

∴ a ≈ 8.2

∴ The regression equation of percentage increase in weekly sales (Y) on length of weeks (X) is

Y = a + b_{YX} + X

i.e., Y = 8.2 + 2.59 X

For X = 1.5, we get

Y = 8.2 + 2.59(1.5) = 8.2 + 3.885 = 12.085

∴ Increase in sales is 12.085% if the campaign has been in progress for 1.5 weeks.