Use trend projection with regression to forecast sales for weeks 10-13. What are the error measures (CFE, MSE, sigma MAD, and MAPE) for this forecasting procedure? How about r^2? Obtain the trend projection with regression forecast for weeks 10-13. (Enter your responses rounded to two decimal places.) Obtain the error measures. (Enter your responses rounded to two decimal places.)

## Expert Answer

**Trend Projection Method**

Least square method is applied to determine to a linear trend line which is given by following equation:

Y = a + bx

Where,

y = Sales

x = quarter number

Y = demand computed using regression equation

a = y-axis intercept

b = slope of the line

x | y | x^{2} |
x*y | y^{2} |

1 | 47 | 1 | 47 | 2209 |

2 | 50 | 4 | 100 | 2500 |

3 | 46 | 9 | 138 | 2116 |

4 | 51 | 16 | 204 | 2601 |

5 | 55 | 25 | 275 | 3025 |

6 | 57 | 36 | 342 | 3249 |

7 | 63 | 49 | 441 | 3969 |

8 | 60 | 64 | 480 | 3600 |

9 | 61 | 81 | 549 | 3721 |

45 |
490 |
285 |
2576 |
26990 |

The regression line is given by formula:

Y = a + bx = 43.94 + 2.1x

Forecast value for the next months are:

x |
Y = 43.9 + 2.1x |

10 | Y = 43.9 + 2.1*10 = 64.94 |

11 | Y = 43.9 + 2.1*11 = 67.04 |

12 | Y = 43.9 + 2.1*12 = 69.14 |

13 | Y = 43.9 + 2.1*13 = 71.24 |

Calculating Errors:

Forecast |
Error |
Absolute |
Sum of % E |
||||

x |
y |
y_{c}=43.9+2.1x |
E = y-y_{c} |
|E| |
CSE |
E^{2} |
100|E|/y |

1 | 47 | 46.0444 | 0.9556 | 0.9556 | 0.9556 | 0.9131 | 2.0331 |

2 | 50 | 48.1444 | 1.8556 | 1.8556 | 2.8111 | 3.4431 | 3.7111 |

3 | 46 | 50.2444 | -4.2444 | 4.2444 | -1.4333 | 18.0153 | 9.2271 |

4 | 51 | 52.3444 | -1.3444 | 1.3444 | -2.7778 | 1.8075 | 2.6362 |

5 | 55 | 54.4444 | 0.5556 | 0.5556 | -2.2222 | 0.3086 | 1.0101 |

6 | 57 | 56.5444 | 0.4556 | 0.4556 | -1.7667 | 0.2075 | 0.7992 |

7 | 63 | 58.6444 | 4.3556 | 4.3556 | 2.5889 | 18.9709 | 6.9136 |

8 | 60 | 60.7444 | -0.7444 | 0.7444 | 1.8444 | 0.5542 | 1.2407 |

9 | 61 | 62.8444 | -1.8444 | 1.8444 | 0.0000 | 3.4020 | 3.0237 |

Total |
0.0000 |
16.3556 |
0.0000 |
47.6222 |
30.5947 |

Forecast sales for each month are calculated by formula: y_{c} = 43.94 + 2.1x

Forecast Error (E) for each month = E = y – y_{c}

Cumulative forecast error= CFE = ∑( y – y_{c} ) = 0

Mean Squared Error (MSE) = [∑( y – y_{c} )^{2}]/n = [∑E^{2}]/9 = 47.622/9 =5.2913

MSE = 5.2913

Mean Absolute Deviation = [∑|y – y_{c}|]/n = [∑|E|]/9 = 16.3556/9 = 1.8172

Mean Absolute Percent Error = MAPE = (1/n) x (∑100|E|/y) = (1/9)x(30.5947)=3.34

Standard Error of Estimate (S_{y,x} ) = Standard deviation of regression (σ)