--- title: "Linear trend regression and forecast error" subject: "Industrial Engineering and Management" chapter: 6 key: "semester_6/industrial_engineering_and_management:linear-regression-trend-forecast" source: https://www.bctengineer.com/q/semester_6/industrial_engineering_and_management/linear-regression-trend-forecast site: BCT Engineering --- # Linear trend regression and forecast error Industrial Engineering and Management · Chapter 6: Forecasting - Asked in: year not given, 8 marks - Video: [03_03_P6 Demand Forecasting Using Linear Trend Line or Regression Equation (Using Excel)](https://www.youtube.com/watch?v=uhG9R_BeszY) — Industrial Engineering and Manufacturing - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_6/industrial_engineering_and_management/linear-regression-trend-forecast - Whole chapter: https://www.bctengineer.com/md/semester_6/industrial_engineering_and_management/chapter-6 ## Question Annual demand (in thousand units) for a product over six years is: Year 1: 42, Year 2: 45, Year 3: 51, Year 4: 53, Year 5: 59, Year 6: 62. (a) Fit a straight-line trend $y = a + bx$ by the least squares method. (b) Forecast the demand for years 7 and 8. (c) Compute MAD, MSE and MAPE of the fitted line, and comment on the fit. Also name the causal quantitative techniques and the difference from time-series methods. ## Short answer Least squares trend: y = 37.6 + 4.114x (thousand units). Year7 forecast 66.4, year8 forecast 70.5. MAD = 0.72, MSE = 0.63, MAPE = 1.4%. Causal methods use regression on related variables; time-series uses only historic values of the variable itself. ## Answer #### (a) Least squares line Take $x$ = year number, $n = 6$. | $x$ | $y$ | $xy$ | $x^2$ | |---|---|---|---| | 1 | 42 | 42 | 1 | | 2 | 45 | 90 | 4 | | 3 | 51 | 153 | 9 | | 4 | 53 | 212 | 16 | | 5 | 59 | 295 | 25 | | 6 | 62 | 372 | 36 | | **21** | **312** | **1164** | **91** | $$ b = \frac{n\sum xy - \sum x \sum y}{n\sum x^2 - (\sum x)^2} = \frac{6(1164) - 21(312)}{6(91) - 441} = \frac{6984 - 6552}{105} = 4.114 $$ $$ a = \bar{y} - b\bar{x} = 52 - 4.114(3.5) = 37.6 $$ Trend line: $y = 37.6 + 4.114x$ (demand rises by about 4.1 thousand units per year). #### (b) Forecasts - Year 7: $37.6 + 4.114(7) = 66.4$ thousand units - Year 8: $37.6 + 4.114(8) = 70.5$ thousand units #### (c) Forecast error of the fit | $x$ | Actual | Fitted | Error | |---|---|---|---| | 1 | 42 | 41.71 | 0.29 | | 2 | 45 | 45.83 | -0.83 | | 3 | 51 | 49.94 | 1.06 | | 4 | 53 | 54.06 | -1.06 | | 5 | 59 | 58.17 | 0.83 | | 6 | 62 | 62.29 | -0.29 | - MAD $= 4.34/6 = 0.72$ thousand units - MSE $= 3.77/6 = 0.63$ - MAPE $= 1.4\%$ The errors are small and change sign, so the straight line fits the data well (correlation coefficient $r \approx 0.994$). Forecasting two years ahead is reasonable, but the line should be revised as new data arrive. #### Causal methods **Causal (associative) techniques** forecast a variable from other variables that influence it, for example demand of cement from construction activity, using simple or multiple regression and correlation, or econometric models. **Time-series methods** (moving average, exponential smoothing, trend) use only the past values of the variable itself. **Answer: $y = 37.6 + 4.114x$; forecasts 66.4 (year 7) and 70.5 (year 8) thousand units; MAD = 0.72, MSE = 0.63, MAPE = 1.4%.** ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Which expression gives the slope b of the least squares line? A. (n sum xy - sum x sum y)/(n sum x^2 - (sum x)^2) B. (sum x sum y)/(n sum xy) C. sum y / sum x D. (sum x^2 - sum xy)/n Answer: A. With the sums given: (6984 - 6552)/105 = 4.114 thousand units per year. (Key point: Slope formula) 2. What is the fitted trend line? A. y = 52 + 3.5x B. y = 4.114 + 37.6x C. y = 37.6 + 4.114x D. y = 21 + 312x Answer: C. a = ybar - b xbar = 52 - 4.114(3.5) = 37.6, so y = 37.6 + 4.114x. (Key point: Trend line) 3. What is the forecast for year 8? A. 66.4 B. 74.6 C. 70.5 D. 62.3 Answer: C. 37.6 + 4.114 x 8 = 70.5 thousand units. (Key point: Forecast) 4. How do causal methods differ from time-series methods? A. Causal use only past values of the variable; time series use other variables B. Causal forecast from other influencing variables (regression); time series use only past values of the variable itself C. They are identical D. Causal methods are qualitative opinions Answer: B. E.g. cement demand from construction activity is causal; moving average and exponential smoothing are time-series. 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