**Example of Linear Regression **

**The historical data on the cost (in hundreds of $) for a project activity is given below. Develop a trend equation using Linear Regression Analysis and forecast the cost of this activity for period 10 and 15.**** **

t |
y |

1 |
58 |

2 |
57 |

3 |
61 |

4 |
64 |

5 |
67 |

6 |
71 |

7 |
71 |

8 |
72 |

9 |
71 |

The calculations for determining the slope and intercept of the regression line are shown below

t |
y |
t*y |
t^{2} |
y^{2 } |

1 |
58 |
58 |
1 |
3364 |

2 |
57 |
114 |
4 |
3249 |

3 |
61 |
183 |
9 |
3721 |

4 |
64 |
256 |
16 |
4096 |

5 |
67 |
335 |
25 |
4489 |

6 |
71 |
426 |
36 |
5041 |

7 |
71 |
497 |
49 |
5041 |

8 |
72 |
576 |
64 |
5184 |

9 |
71 |
639 |
81 |
5041 |

t = 45 |
y = 592 |
t*y = 3084 |
t^{2} = 285 |
y^{2} = 39226 |

The slope ** b **of the line is given by:

The intercept ** a **of the line is given by:

Hence the linear trend equation is given by:

**y**_{t} = a + bt = 55.44 + 2.067t, and** **

The forecast for period 10 is given by

y_{10} = 55.44 + 2.067*10 = 55.44 + 20.67 = 76.11

For t = 15, y_{15} = 55.44 + 2.0667 *15 = 86.445

In the discussion and example above on linear regression analysis, the independent variable was *t*--the time period. However, the linear regression technique can also be used determine association or causation between two variables. In such cases, we use the notation *x* for the independent variable and *y* for the dependent variable. The linear regression equation in such cases would be of the form

y, where_{c}= a + bx_{i}

The slope *b* of the regression line is given by:

.

The intercept *a* of the regression line is given by:

.